It can be an interesting challenge to be presented with a formula and asked how it was derived. This becomes a bigger challenge when the formula is only approximate, so we have to figure out how to arrive at this particular approximation. But it is impressive when several different approaches all naturally lead to the same approximation! We’ll bring in geometry, algebra, trigonometry, and some results from calculus to attack this puzzle.

This question came in last month, from Cvetača:

Hello,

I have an equation as shown in the picture. That equation is an approximation of the distance from a chord to an arc. My question is, how is that equation derived? I have no idea how to explain it.

Here we have a chord of a circle, divided into two parts, *m* and *n*; the only other thing we know is the radius *R* of the circle. We want to find a formula for the perpendicular distance to the arc.

An exact formula would be considerably more complicated than the formula shown, \(\displaystyle p \approx \frac{mn}{2R}\). Where did this simple formula come from? And can we derive an exact formula for comparison? Finally, what is the significance of the markings for the two arcs in the picture? Do they suggest a particular derivation?

After pondering the form of the formula, I saw one way to obtain it, by a favorite geometry theorem:

I’d be interested to know where it came from, and what was said about it. But I have a guess.

The formula implies that

mn=2Rp. This is reminiscent of theintersecting chord theorem, which says thatmn=pq, whereqwould be the distance on segmentpextended downward, until it intersects the circle.

Therefore, the approximation assumes that

qis nearly the diameter of the circle, i.e. the arc is small compared to the circle. In the image, I have shown the locus of the point X located by your formula, showing that it is close to the actual arc, but always smaller; if I move chord MN lower, it becomes considerably farther from the arc, but if I move it upward, the approximation improves considerably.Now, the approximation may well have been derived differently, but this seems to be a particularly simple way to explain it. Others of us may well have additional ideas.

So one way to derive the formula is to assume that the arc is a sufficiently small part of the circle that we can ignore the difference between my *q* and the diameter. This might have been motivated by the theorem I referred to. But most approximations arise by other means.

Though Cvetača had no further comment, and no other Math Doctors made suggestions, I remained curious, and have subsequently played with the problem more, discovering several interesting approaches.

First, let’s try applying the Pythagorean Theorem to the problem, solving for *p* algebraically and making an approximation by ignoring small terms at some point.

Since \(BN=\frac{m+n}{2}\) (half of the chord), triangle ABN yields the equation $$ \left(\frac{m+n}{2}\right)^2 + d^2 = R^2$$ where *d* is the distance of the chord from the center of the circle.

Since \(AD=\frac{n-m}{2}\) (the distance from the midpoint of the chord to O), triangle ADP yields the equation $$ \left(\frac{n-m}{2}\right)^2 + \left(d+p\right)^2 = R^2.$$

Expanding and subtracting the second from the first yields $$ mn – 2dp – p^2 = 0.$$

Assuming *p* is small (so that its square is negligible) and *d* is nearly *R*, we get, approximately, $$ mn – 2Rp = 0,$$ from which we obtain our goal, $$p=\frac{mn}{2R}$$.

But can we solve to get an exact formula, and then simplify that? Yes, since we can express *d* in terms of *m*, *n*, and *R*. First, let’s solve that quadratic equation for *p*. We have $$ p^2 + 2dp – mn = 0,$$ so $$p = \frac{-2d\pm\sqrt{4d^2 + 4mn}}{2} = -d + \sqrt{d^2 + mn}.$$ But we have $$d^2 = R^2 – \left(\frac{m+n}{2}\right)^2,$$ and when we put that into our expression for *p*, with a little manipulation we get $$p = \sqrt{R^2 – \left(\frac{m-n}{2}\right)^2} – \sqrt{R^2 – \left(\frac{m+n}{2}\right)^2}.$$ That is our exact formula.

Now, there is a well-known approximation to the square root, which can be obtained from the extended binomial theorem, or from the power series, or from differential approximation: $$\sqrt{1 – x} \approx 1 – \frac{1}{2}x.$$ We can generalize this to $$\sqrt{a^2 – b^2} \approx a – \frac{b^2}{2a}.$$ Applying this, we get $$p \approx \left(R – \frac{(m-n)^2}{8R}\right) – \left(R – \frac{(m+n)^2}{8R}\right) = \frac{mn}{2R}.$$ There we are: a third derivation of the formula.

As successful as we have been so far, we haven’t used the notations on the original picture showing the arc lengths as approximating the parts of the chord (which is not true in my picture, but would be for a sufficiently shallow arc). Surely they must be part of the original derivation. So let’s try focusing on those arcs and their corresponding angles.

If we take arc MP as approximately *m* and arc PN as approximately *n*, then arc CN is \(\frac{m+n}{2}\), and arc PC is \(\frac{n-m}{2}\). The corresponding angles (in radians) are the arc lengths divided by *R*, namely \(\alpha = \frac{m+n}{2R}\) and \(\beta = \frac{n-m}{2R}\).

Now \(p = EB = AE – AB = R\cos\beta – R\cos\alpha\).

Again, we can use an approximation, \(\cos\theta \approx 1 – \frac{1}{2}\theta^2\). This gives us $$p = R\left(\cos\beta – \cos\alpha\right) = R\left(\left(1 – \frac{1}{2}\beta^2\right) – \left(1 – \frac{1}{2}\alpha^2\right)\right) = \frac{R}{2}\left(\alpha^2 – \beta^2\right).$$

Replacing the angles with their approximations, we get $$p = \frac{R}{2}\left(\alpha^2 – \beta^2\right) = \frac{R}{2}\left(\left(\frac{m+n}{2R}\right)^2 – \left(\frac{n-m}{2R}\right)^2\right) = \frac{mn}{2R}.$$

So now we have four ways to derive the formula. I imagine the last one, or something like it, is most likely the original.

We have received dozens of questions like the following, looking for the same result with different “givens”:

Formula for Laying Out an Arc I work as a layout carpenter and do not have the advantage of the newer transits. I am working on a building having a 317' radius with steel centers along it, and will not be able to swing this radius with a measuring tape. If I could figure the layout from a chord it would be a great help. There must be a formula for this. The radius is known, the distance from the center of the chord at 90 deg. to the arc is known, the distance from the center of the chord along the chord to a point is known - call it A. What would be the distance from A to the point where it intersects the arc at 90 deg. off the chord. Your help would be greatly appreciated.

The difference from the current question lies in what is known. Here, we know the radius *r*, the height *h* of the entire circular segment (from the chord to the middle of the arc), and the distance *x* from the center of the chord. The latter is closely related to our *m* and *n*, but above we didn’t know the height, which is what made it harder.

Here is a picture:

Doctor Rick answered this one:

I think I understand what you're saying, and it's not hard to work out using coordinate geometry. We can lay out our coordinate system so that the chord is along the x-axis with the center of the chord at the origin (0,0). If the radius is r and the perpendicular distance from the center of the chord to the arc is h, then the center of the circle is at (0, h-r), so the equation for the circle is x^2 + (y-(h-r))^2 = r^2 We can solve for y in terms of x: y = sqrt(r^2 - x^2) - r + h This is what you're looking for: x is the distance along the chord from the center of the chord to your point A, and y is the perpendicular distance from point A to the arc.

Carl asked another question, and later told us he was regularly using this formula in his work.

Many others who have asked about the same calculation don’t know the radius, but only the height and the length of the chord. We have referred them to this page (or others like it):

Calculating the Radius from a Chord If I know the chord length and chord height, is it possible to determine the radius of the circle? Is there a formula?

Here is my answer:

Hi, Katie. I assume that by "chord height" you mean the distance from the chord to the middle of the arc it cuts off. If so, there's a simple formula, and I'll even show you how to find it, if you can do a little algebra. Here's my picture: ******* * |h * * d | d * *----------+----------* * | / * * r-h| /r * * | / * *------------+------------* I've called the length of the chord 2d, to simplify the calculations, so half the chord is d. The distance from the chord to the arc (your "chord height") is h, and the unknown radius of the circle is r. Then the distance from the center to the chord is (r-h). Incidentally, the technical term for h is the "sagitta" of the chord, and r-h is the "apothem." You don't hear those words used much. "Sagitta" is Latin for arrow. If you think of the arc as the bow and the chord as the string, you can see why. Look at the right triangle with sides d, r-h, and r. Using the Pythagorean Theorem, we can say that: d^2 + (r-h)^2 = r^2 ("^2" means squared) which expands to: d^2 + r^2 - 2rh + h^2 = r^2 and by subtracting r^2 from both sides we get: d^2 - 2rh + h^2 = 0 Now we can add 2rh to both sides: d^2 + h^2 = 2rh and divide both sides by 2h to get: d^2 + h^2 r = --------- 2h So the radius is just the sum of the squares of the height and half the length, divided by twice the height. If you look back at the picture, d^2 + h^2 is just the square of the straight-line distance from one end of the chord to the middle. There's a simple geometric way to get the formula in that form, if you know enough about similar triangles.

In our case, though, we know the radius and the chord length, but not the height; we can solve this formula for *h*, which requires the quadratic formula:

$$r = \frac{d^2 + h^2}{2h}$$

$$2rh = d^2 + h^2$$

$$h^2 – 2rh + d^2 = 0$$

$$h = \frac{2r \pm \sqrt{4r^2 – 4d^2}}{2} = r \pm \sqrt{r^2 – d^2}$$

The two values this gives are for the major and minor arcs; we can choose the negative sign since we will only be interested in the minor arc.

If we put this expression for *h* into Doctor Rick’s formula for *y*, taking \(d = \frac{m+n}{2}\) and \(x = \frac{n-m}{2}\), we get $$p = \sqrt{R^2 – x^2} – R + h = \sqrt{R^2 – \left(\frac{n-m}{2}\right)^2} + \sqrt{R^2 – \left(\frac{m+n}{2}\right)^2},$$ which looks familiar from my work above. We’ve come full circle, so to speak.

The question, from 2006, started with the idea of a percent increase:

Percent Increase and "Increase by a Factor of ..." A math doctor here recently explained percent increase this way: If we start with 1 apple today and tomorrow have 2 apples, then because 2 - 1 = 1 and 1/1 = 1, we have a100 percent increase. But can't I also say there was anincrease by a factor of 2? Two divided by 1 equals 2, an increase by a factor of 2 -- and also anincrease by 200 percent?This is what is confusing me! I'd never been confused about saying "increased by a factor of" and "increased by percent of" until I saw the Dr. Math conversation about finding percentages ... which is a good thing, I guess, because now I know what I didn't know! Thank you for any help.

Unfortunately, Joseph didn’t directly quote from the page he had in mind, and we have never said exactly what he said; so we couldn’t be sure which page it was. Everything he said, however, was correct.

Doctor Rick was the first to answer:

Hi, Joseph. Yes, this can be very confusing, becausesome statements about increases are ambiguous. When we say "increased by a factor of 2,"the word "factor" makes it clear that we mean "multiplied by 2."When we say "increased by 10%," there is only one reasonable interpretation: the amount of the increase is 10% of the original amount. If we meant multiplication by 10%, that would be a decrease -- not an increase! Even when we say "increased by 100%," there is only one reasonable interpretation, since multiplication by 100% is the same as multiplication by 1, and that's still not an increase.When we want to speak of an increase that is greater than the original amount, then ambiguity can arise. In that situation, I much prefer "increase by a factor of 3" or "by a factor of 2.5," etc. I don't know what page you saw -- but have you seen this one? Percent Greater Than vs. Increased http://mathforum.org/library/drmath/view/61774.html See also the page linked there, about the even more confusing phrase "___ times more than" and the like. I am on the side of avoiding the confusing phrases, as a basic principle of communication. If you saw another page and you are still confused by it, please tell me the URL of that page so I can review it with you.

Before we get back to this conversation, we should take a look at the page he referred to, which is a good starting point:

Percent Greater Than vs. Increased What is the difference between the following statements: My profits are200% biggerthan they were last year. and My profits from last year haveincreased 200%. This is one of the questions we have to answer in my Middle school methods course and I have looked everywhere for the answer. I hope you can help.

I answered this one:

As far as I can see,they mean the same thing; in fact, both aresimilarly ambiguous. Taken literally, "200% bigger" (or, more formally, larger or greater) and "increased 200%" (or, more completely, increased _by_ 200%) bothmean that the increase from one year to the next is 200% of the first year's value, so that the second year's profit is 3 times the first. But both statements aremore likely to have been made with the intention of saying that this year's profit is twice last years. English is not very clear in cases like this.

So there is a **literal** meaning (which mathematicians tend to see as best), and an **idiomatic** meaning (which ordinary people are more likely to have intended to say). I referred to a page we’ll be looking at below, and then quoted a favorite book of mine that gives a lexicographer’s perspective:

Since writing that, I found a good reference on "two times greater," although it doesn't mention your "200% greater." It is in Merriam Webster's _Dictionary of English Usage_, which under "times" writes The argument in this case is that _times more_ (or _times larger_, _times stronger_, _times brighter_, etc.) is ambiguous, so that "He has five times more money than you" can be misunderstood as meaning "He has six times as much money as you."It is, in fact, possible to misunderstand _times more_ in this way, but it takes a good deal of effort.If you have $100, five times that is $500, which means that "five times more than $100" can mean (the commentators claim) "$500 more than $100," which equals "$600," which equals "six times as much as $100." The commentators regard this as a serious ambiguity, and they advise you to avoid it by always saying "times as much" instead of "times more." Here again,it seems that they are paying homage to mathematics at the expense of language. The fact is that "five times more" and "five times as much" areidiomatic phraseswhich have - and are understood to have - exactly the same meaning. The "ambiguity" of _times more_ is imaginary:in the world of actual speech and writing, the meaning of _times more_ is clear and unequivocal. It is an idiom that has existed in our language for more than four centuries, and there is no real reason to avoid its use. I think the same applies to "X percent bigger" and "increased [by] X%." There is just enough ambiguityin a technical contextthat I would want toask what was intendedbefore assuming anything, but there is no reason to say that they definitely mean different things, or mean something different than "X percent of" or "increased to X percent."I myself would avoid saying these things, just because there are enough people who have heard that they are ambiguous, and would therefore take them the wrong way (whichever that is!).

As a result of my side interest in linguistics, I recognize that human speech is not as logical as we might wish; what a word means is a matter of actual usage in a culture, rather than pure logic. So rather than state that either understanding of “times bigger” is “correct”, I just recognize that people take it in two ways, so you have to ask, or use contextual cues, in order to decide on what is meant.

Back to the original discussion: Joseph responded with specific references, the first of which was that link of mine that Doctor Rick said to “see also”:

I'm sorry, I should have specified the site. In fact, there were two -- and I still don't see the difference between them. Here is the first example, from Larger Than and As Large As http://mathforum.org/library/drmath/view/52338.html 1)"Three times as large as N" means "3 * N."2)"Three times larger than N" means "4 * N"-- but only if you stop to think about it, as many people do not. Here, I don't understand how something can be 3 times larger and be 4 times N. That sounds really weird to me. If you asked "What is something that is three times as large as N?" then I would say 3N ... but apparently I'd be wrong! I just don't see where my thinking is wrong.

His thinking **isn’t** wrong on this point: 3N **is** three times as large as N! He seems just to be letting what he’s read sow doubt about everything.

Here is the second example, from Percentage of Increase http://mathforum.org/library/drmath/view/58131.html You can choose two ways to express your answer now. One is to say: there will be a550% increaseby the year 2000. Or you can say: in the year 2000 the (new value) -- you didn't say what the numbers represented, so I'm a little confused right here -- will be aboutfive and a half times greaterthan what it was in 1995. Many people don't quite grasp those phrases, especially the latter one. Instead you might wish to say it this way: in 2000 the (new value) will be6 and a half times what it wasin 1995. The difference in the wording is subtle, of course, but important. The number 6 1/2 comes from 325,000 --------- = 6.5 or 6 1/2 50,000 which is NOT a percent increase situation. In this problem, I don't understand the difference between the way the doctor explains the two different ways you can talk about the increase, and the implications of each. The doctor says that 6.5 times is not a percent increase; but can you still say it's 650 percent OF the original? I'm sorry -- this is all very confusing at this point!

So his specific question is this:

- Why wouldn’t “three times as large” and “three times larger” mean the same thing?
- How can “5 1/2 times greater” and “6 1/2 times what it was” mean the same thing?

The two pages quoted are by me (1999) and Doctor Terrel (1997). The first is particularly worth reading in its entirety, as there is a lot more there.

Doctor Greenie responded, arguing against laxness on the matter, and making the case for the literalistic interpretation:

I'm going to jump in here, because this is one of my pet peeves. Mathematics is commonly called the exact science. Mathematics must be exact; if it is not, it all falls apart.We can't use ambiguous language in mathematics.I agree that the use of the phrase"x times larger than" is best avoided. However, as a mathematician who believes in using unambiguous language, I cannot accept the proposition that we should be able to interpret "5 times larger than 10" as either 50 or 60. It HAS TO BE ONE OR THE OTHER. And grammatically, "5 times larger than" means the "new" number is5 times largerthan the "old" number; this in turn means thedifferencebetween the new and old numbers is 5 times the old number, making the new number 6 times the old number. So the number which is 5 times larger than 10 is 10 + 5(10) = 10 + 50 = 60 (The phrase "...larger than..."implies comparison by subtraction; the phrase "...as large as..."implies comparison by division. Sixty is 6 times as large as 10, because 60/10 = 6. But 60 is 5 times larger than 10, because [60 - 10]/10 = 50/10 = 5.) Yes, we hear it all the time in everyday life. Sometimes, we even hear it in the supposedly rigorous world of science -- "an earthquake of magnitude 5 is 10 times greater than one of magnitude 4," and such. But the common idiom of using "10 times greater than" -- when the actual meaning is "10 times as great as" -- has no place in mathematics.

He concluded with an accidental overstatement of what the “other side” says:

I disagree with many of the concessions that other math doctors here have made in interpreting the phrase "x times larger than" as being the same as "x times as large as." On one of the pages I saw, a fellow doctor said that "50% larger than" and "50% as large" mean the same thing. But if my weekly salary last year was $1000 and it is 50% LARGER this year, then it is now $1000 + 50%($1000) = $1000 + $500 = $1500 While if it was $1000 last year and it is 50% AS LARGE this year, then it is now 50%($1000) = $500 If something is 50% larger, then it is larger; if it is 50% as large, then it is smaller. They can't be the same; that is nonsense.

In fact, as he admitted in a subsequent private discussion, he had misremembered what others had said; none of us have claimed that “50% larger than” and “50% as large” mean the same thing. What we say is that, when the percentage or multiplier is **greater than 100%**, we recognize **ambiguity** in the likely intent. I think we agree that the phrase **should not be used in a mathematical context**, and that we both grudgingly interpret it as intended elsewhere.

Then it was my turn to respond, as the author of the first page Joseph had asked about, wanting to make sure he understood both why people take it literally as they do, and how we should think about it.

First, on “Larger Than and As Large As”, I said this:

Joseph, your thinking is RIGHT: if M is three times AS LARGE AS N, then M = 3N. That's what statement (1) above says. But if we break statement (2) apart carefully (some would say TOO carefully :-)), then it means something different from what people usually mean by it. If I said "M is 50 larger than N," I would mean that if you ADD 50 to N, you get M: M = N + 50. And if I said, "M is 50% larger than N," I would mean that if you add 50% OF N to N, you get M; that is, I mean that M is 50% of N added to N: M = N + (0.50)N. Now, though I'm not entirely sure I agree with this, technically minded people often apply the same thinking to (2), for the sake of consistency. The"larger than" means we addsomething to N. And what do we add? Three times N. So by this thinking, M = N + 3N = 4N. So "three times LARGER THAN N" means the same as "four times AS LARGE AS N."

I then referred to the usage book quoted above, adding:

English usage experts think it is nonsense. My feeling is thatthis thinking puts a little too much weight on consistency, and is just too weird for the general public to follow. English is not known for consistency! So we need to recognize thatin everyday usage, (1) and (2) really mean the same thing. When we accept that, though,we set ourselves up for the opposite confusion: Cases like "50% larger" and "3 times larger" no longer follow the same pattern, and our language becomes inconsistent, which really bothers mathematicians! As Dr. Rick pointed out, this means thatthere are gray areas where it's hard to be sure what someone means, so it may be best just to avoid using these phrases in mathematical contexts.

Finally, I commented on Joseph’s other quote, from Doctor Terrel:

Joseph, here the doctor was saying that a 550% INCREASE means adding 550% of the original value to the original value, which means 650% OF the original value. In the other terminology, "5 1/2 times greater" (there again, "greater" is taken to refer to the increase) is the same as "6 1/2 times as much."When he says that the 6 1/2 is not a percent increase, he doesn't mean that it hasn't increased, just that he is talking about multiplying by 650% rather than adding 650%. When you think in terms of increase (adding), it is a 550% increase. Now,the "percent increase" case is pretty standard, because it IS technical terminology(though ordinary people reading it can get confused, so it's still risky).The "times greater" case is more disputable, since that sounds less technical.Most people don't demand absolute consistency from language; they are happy tounderstand "times greater" idiomatically. I hope that clears up some of the confusion. It isn't all cleared up yet at our end. You will definitely get different opinions as to what it all REALLY means!

Joseph replied, returning to his initial question:

That really cleared things up for me and I appreciate your time in driving home the differences! The last question I would like to ask is, how do you deal with factors? If someone says something haschanged by a factor of ...or is less/greater than by a factor of ..., do we use the same rules that you've discussed above? Or when using the word "factor," are things a bit different?

I answered:

As Dr. Rick pointed out in the first response,"factor" is used to make it clear that multiplication, rather than addition, is the cause of an increase. Just as "increased by a factor of 2" means "twice as large," so does "greater by a factor of 2." And "decreased by a factor of 2" and "less by a factor of 2" both mean "half as much" (divided by 2). I can't think of a context in which that would not be true -- but English is flexible enough that I probably shouldn't guarantee anything!

Isn’t English fun?

In closing, here is a more recent question (2015) where I summarized this complicated issue:

As Big As vs. Bigger Than I'm having difficulty convincing my 5th graders that "as big as" and "bigger than" do not mean the same thing. For example, when asked, "How many times larger is 10,000 than than 100?" they answer "100." Their tests and homework are full of this misunderstanding! How would you suggest telling them they are wrong?

I referred to most of the pages we’ve seen above, and added:

To be honest, my feeling (basically unchanged since the first of those) is that although your understanding is common among thoughtful people, it is a case of]]>excessive consistency. Mathematical people want a certain word structure to always have the same meaning, so we relate "x times bigger" to "x percent bigger," and that to "x bigger," and want to take all in an incremental (additive) sense. But taken on its own,it is perfectly logical to interpret "x times bigger" as "bigger, as a result of multiplication by x."And human language is not completely consistent; we have idioms all over the place that we interpret with no trouble. Having said that,I think that math books should avoid that form, because there is just enough truth to your thinking, and enough extra expectation of careful use of words in a math book, that it can be confusing. What I would do is to make a brief mention of the fact that many people take it as you do, but then point out that your book is using the phrase in the way it is usually intended in the real world. If I were writing the textbook, I would reverse this:almost always use "x times as big," but mention somewhere the fact that many people use "x times bigger" to mean the same thing, and briefly discuss the controversy before moving on to other things.

Here is a question from 2003:

Clarifying Percentages vs. Percentage Points What is the difference between measuring usingpercentagesversus measuring usingpercentage points? What is meant by a percentage point?

I answered, starting with why we need the terms:

The term "percentage point" is used to get around an ambiguity in English when we are comparing two different percentages. The problem is that "percent" implicitly refers to arelative change(some fraction of an original amount, like a salary increase of 10%) rather than anabsolute change(some specified amount, like a salary increase of $1000). What do we say when we want to treat a percentage as an absolute amount? If, for example, the current tax rate were 10% and we increased it to 12%, we might say that weincreased it by 2 percent. But that would be taken to mean that we increased it by2% _of the original 10%_(that is, by 2/100 of 10%, or 0.2%), to 10.2%. The question is, are we using "percent" to mean one of the units called percent, or a percentage of that percentage? To avoid this problem, we say instead that we areincreasing the tax rate by "two percentage points". This unambiguously refers to the number 2% itself as a unit, rather than to 2% of something else.

So the *percent increase* from 10% to 12% is the difference, 2%, divided by the original amount, 10%, which is 0.20, or 20%. But the *percentage point* increase is just the 2%. If we replaced the percentages given with a *unit*, say dollars, we would say that the *percent increase* from $10 to $12 is the difference, $2, divided by the original amount, $10, which is 0.20, or 20%; the *dollar* increase is $2. This is what I mean by saying we treat percentage points as a unit.

On the other hand, if we actually wanted to say that the tax increased to 10.2%, it would be a good idea to clarify that as well, perhaps by saying explicitly that itincreased by 2% of its old rate, or by stating the old and new values. Technically, however, it is correct to say that it increased by 2%. In summary, I wouldn't say that we "measure" using one or the other; rather, we use the one term to clarify our meaning where the other would be ambiguous, becausewe are switching perspective from thinking of a percentage as a fraction of something else, to treating it as a number that stands on its own. A percentage change is a difference divided by some base number, while a percentage _point_ change is a simple addition or subtraction.

The issue with my student was in this area. She was working on the following problem:

In 1950, Americans spent 22% of their budget on food. This has decreased at an average rate of approximately

0.25% per yearsince then.Find a linear function in slope-intercept form that models this description. It should model the percentage of total spending, p(x), by Americans x years after 1950.

She knew that a decrease of 0.25% means subtracting 0.25% of one year’s amount from that amount, equivalent to multiplying by 99.75% (1 – 0.0025 = 0.9975) each year. But this didn’t fit with anything she had learned about. (In fact, it would correspond to an exponential decrease, not a linear function, but she hadn’t learned about that.)

In answer, I pointed out that the statement was ambiguous, and we need to interpret it in a way that is consistent with the mention of a linear function. What the problem should have said, for clarity, is

In 1950, Americans spent 22% of their budget on food. This has decreased at an average rate of approximately

0.25 percentage pointsper year since then.

That is, the number 22 is to be reduced by 0.25 each year; that will be the slope of the function. Both 22 and 0.25 are to be thought of as measured in the same unit, percentage points, rather than the latter being a percentage *of* the former.

A different sort of issue arises when we mix together a fraction and a percentage in one measurement. Here is a question from 2005:

How to Pronounce a Fraction of a Percentage Is it correct to say "one tenth of one percent" as opposed to saying "one tenth percent" for 0.1%? Why or why not? It seems wrong to refer to a percentage as a fraction of a percentage, but news people and the financial industry do it all the time. I can't seem to find out what the precedence is for this.

I answered:

Both mean the same thing; neither is wrong.The reason for the longer phrase is probably the usual reason for using a longer phrase: to avoid ambiguity or possible confusion.Many people are not quite clear on what percentages mean, and might well take "one-tenth percent" as if it were just "one-tenth" (which, of course, is really 10%). So people tend to expand it to make it clear that they are using BOTH a fraction AND a percent; that is, 0.1% = 1/10 * 1% = 1/10 * 1/100 = 1/1000 I suppose you could compare this to using "a quarter OF A dollar" or "twenty-five hundredths OF a dollar", rather than just reading "$0.25" as "a quarter dollar" or "twenty-five hundredths dollars".

The latter just feels subtly wrong (despite the fact that American coins do say “quarter dollar”). I imagine it is not much more than habit, just as many other aspects of language feel right or wrong though we can’t point to a rule for it, or explain why it should be that way.

I was reminded, in writing this just now, of a fascinating conversation with a French reader in 2002 about a different issue that, we decided, depended heavily on what language we were using:

Use of Plural with Decimal Numbers

In part of my answer to this question about whether we write, for example, 1.5 *degrees* or 1.5 *degree* (we say the former in English, while they say the latter in French, for a very interesting reason), I said this:

My understanding is that we consider ONLY the number 1 as singular; in particular, zero is a plural: we say "0 degrees" or "0 (no) apples," not "0 degree" or "0 apple."We do not use fractions as adjectives at all, but say "half (of) an apple" or "two thirds of a degree" with the fraction standing alone as a noun phrase, so it would not be quite accurate to say that a proper fraction is singular. With a mixed number, we tend to use a plural: "one and a half apples."

This ties in to what I said above about “a quarter (of a) dollar”; possibly the real reason we say “a quarter of a percent” is exactly the same: The fraction is treated as a noun, not as an adjective or adverb; and “percent” is treated as a unit.

A distantly related issue came up in the following question from 2008:

Difference between Percent and Percentage What is the difference between percent and percentage are there any difference? I think both are out of 100.

Again I put on my “Ask Doctor Grammar” hat and answered:

The only difference is in how they are used grammatically--and people differ even on that.I take "percentage" to refer to the concept, and "percent" to be a unit, much like "voltage" vs. "volts" in electricity, or "mileage" vs. "miles" in distance. Themileageyou put on a car during a trip might be 40miles; thepercentageof people who expect gas prices to rise might be 40percent(40%). Here is one dictionary's take on it (www.merriam-webster.com): Percentage: noun 1 a: a part of a whole expressed in hundredths <a high percentage of students attended> b: the result obtained by multiplying a number by a percent <the percentage equals the rate times the base> Percent: adverb : in the hundred : of each hundred The main difference is that they report "percentage" as anounand "percent" as anadverb. That fits my understanding. Note that "percentage" is not used with a number, while "percent" is (and not without).

Looking at the current definitions, I see that I must have missed two of three entries:

percent (adverb)

: in the hundred : of each hundred

percent (noun)

1

pluralpercenta : one part in a hundred

b : percentage

a largepercentof their income2 percents

plural, British: securities bearing a specified rate of interestpercent (adjective)

1 : reckoned on the basis of a whole divided into 100 parts

2 : paying interest at a specified percent

Most uses are probably nouns, as in “20 percent of people”, though the adjective use is also common (“a 20 percent solution”). But note that they say “percent” is also used as a synonym for “percentage”, which I would take as a concession to common but mistaken usage.

I then referred to a previous question from 2002:

Percent vs. Percentage

There, Doctor Sarah quoted a dictionary and two usage books.

Doctor Sarah had also answered a similar question earlier that year:

Percent or Percentage? Would you please explain the exact difference between the words percent and percentage? I have used textbooks that use them as if they are the same. I have have always explained it as percent is a % andpercentage is a number that is the same unit as the base number.

This questioner seems to take “percentage” in sense 1b from my dictionary reference above, as the amount itself rather than the number of hundredths, which I have seen used in some textbooks but don’t think I really use in practice. (They might say “percentage = percent times whole”; I’d rather say, “part = percentage times whole”.) I’m not sure either Doctor Sarah or I recognized this detail. Her response was:

You're on the right track, and even a regular dictionary can help with this. percent - one part in a hundred percentage - a part of a whole expressed in hundredths; the result obtained by multiplying a number by a percent From the Guide to Grammar and Writing on the Web ("Notorious Confusables"): http://ccc.commnet.edu/grammar/notorious2.htm "We use the word percent as part of a numerical expression (e.g., Onlytwo percentof the students failed.). We use the word percentage to suggest a portion (e.g.,The percentage of students who failhas decreased.)." Unfortunately, you will find percent and percentage incorrectly used everywhere on the Web and in textbooks.

Although the dictionary definition quoted can be taken as representing the part of the whole itself, not the fraction, the example from the grammar site doesn’t have that meaning, as it is not the actual *number* of students who fail, but the *fraction*, that has decreased.

The same page includes a 2003 question on the same topic:

Is there a difference between the meaning of these two words, or are they totally interchangeable? I always thought the word percent required the correct notation using the symbol % and that percentage was referred to as AN AMOUNT BASED ON A GIVEN TOTAL, NOT NECESSARILY BASED ON 100. For example: Given 4/16 The percentage is 4 out of a total of 16, the percent is 25%

I replied:

To answer your specific question, I would say "thefractionused is 4 out of 16;the percentage is 25%[read as 25percent]." That is, both "percent" and "percentage" refer to an amount "out of 100" (since that is what "per cent" means), and the only difference is how they are used in a sentence. We (should) use "percent" only in phrases like "25 percent" where it can be directly replaced by the phrase "out of 100"; we use "percentage" as a name for the concept.

Although I don’t always defer to dictionaries in areas related to math, I do like to refer to them, and especially to what they say about common usage.

My American Heritage dictionary has this usage note: _Percent_ and _percentage_ are both used to express quantity with relation to a whole. _Percent_ is employed only specifically andalways with a number or numeral. _Percentage_ is never preceded by such a figure, but should be qualified by a general term to indicate size (since _percentage_ does not necessarily imply smallness). The number of the noun that follows _percent_ or _percentage_, or is understood to follow them, governs the choice of the verb: _Forty percentof his estate is in securities.A large percentageof the patients are children._

Because they are probably looking largely at non-technical material, I think they have missed some usages of “percentage”, as all their examples of it tend to be, as they say, with “a general term”. It can also be specific: the percentage of patients who are children is 75%, or whatever.

]]>Here is the question, with an included picture:

Q) In my book it is written that – “a gcd of 12 and 18 is 6. We observe that -6 also is a gcd of 12 and 18.”

Gcd is greatest common divisor and I don’t think that -6 is greatest so is the above line wrong? I think it should say that -6 is just a common divisor. The picture of the line in my book is given below.

Here is a typical definition of Greatest Common Factor (at an elementary level), from Math Is Fun:

The greatest number that is a factor of two (or more) other numbers.

When we find all the factors of two or more numbers, and some factors are the same (“common”), then the largest of those common factors is the Greatest Common Factor.

How could -6 be “greatest” or “largest”?

How about the more careful definition of Greatest Common Divisor from Wikipedia?

In mathematics, the

greatest common divisor(gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For example, the gcd of 8 and 12 is 4.

This definition requires the GCD to be a positive integer, so this certainly doesn’t apply.

So what is the book talking about? I pointed out a hint from what was quoted:

I suspect that if you showed us the previous paragraph, that would contain the answer to your question.

They say, “… it is largest,

only in the sense of (ii)“. Thus, they have given you a meaning for “largest” that is not what you are expecting. If they are referring to the line I can half-see just above your clip, which begins with “(ii)”, that makes sense. You are expecting “largest” in the sense of “greatest integer”. They are defining it, I think, such that a divisor is “largest”if any other common divisor is a divisor of this one.

In preparing this post, I searched for the relevant phrase and found what appears to be a copy of the textbook (either a different edition, or a related book by the same authors):

The notation says that (i) *r* must evenly divide both *m* and *n*, and (ii) if any other integer *l* divides *m* and *n*, then *l* divides *r*. That is, any other common divisor is a divisor of *r*, just as I had guessed. In the example, -6 is a divisor of both 12 and 18, and any other common divisor (1, 2, 3, 6, -1, -2, -3) is a divisor of -6. So it is a “greatest” common divisor **in this sense**. And this is why the book called it **a** gcd (not **the** gcd).

I continued:

This definition is used, for example, in the first section of this page:

https://proofwiki.org/wiki/Definition:Greatest_Common_Divisor

There they are defining the GCD for something called an “integral domain” — that is, for any set of “numbers” that has certain properties that are a generalization of integers, but may not even have an ordering at all, so that you can’t define “greatest” literally. But this definition

canbe used in that context. They go on to state the usual definition for integers, and later discuss its equivalence.Then your book goes on to state that if you restrict the gcd to positive integers, it becomes unique, as you expect.

So what is happening here is that this textbook, *Challenge and Thrill of Pre-College Mathematics *by V Krishnamurthy and C R Pranesachar, like *proofwiki*, is taking the concept of the GCD and extending it to provide a definition that doesn’t depend on being able to say that one number is greater than another. This extended definition turns out, as they point out, to yield more than one GCD (a positive one and a negative one), but when applied to integers, we can adjust the definition to give the expected result.

In fact, did you notice the section title and last line in the larger selection I quoted above? What they are about to do is to apply this new definition of GCD to *polynomials* rather than to *integers* — and since polynomials don’t have the property that any polynomial is either less than, equal to, or greater than any other, this new definition is just what they need. So the context tells us all we need to know to make sense of what they have said.

(When I read to my children when they were young, I would often pause to ponder with them what might be happening in the story, and one of them would say, “Just keep reading — it will tell you!” That is true of children’s books, and it is true of math — unless in fact they already told you, and you forgot.)

That was the end of this exchange; but in preparing this post, I checked to see if we have ever discussed this, or any other, variant definition of the GCD, and found that we have. That usually happens.

First, the specific subject of the GCD of polynomials underlies the following page, which is at a much higher level than my target audience, so I’ll let you read it on your own if you choose:

What Makes Polynomials Relatively Prime?

But a very similar issue with regard to the LCM (Least Common Multiple) arose in the following long discussion:

Least Common Multiple with Zero I'm trying to find any reference about the least common multiple of two numbers when one (or both) is zero. Could you help me, please?

Doctor Rick answered this question using the standard definition:

No number except zero is a multiple of zero, because zero times anything is zero. The only multiple that, say, 0 and 5 have in common is 0. Thus,if the LCM of 0 and 5 exists at all, it must be 0.We do not count zero as a common multiple.If we did, then zero would be the least common multiple of any two numbers (unless we also counted negative multiples, in which case there would be no least common multiple of any two numbers). Either we make an exception in this case, so that the LCM of zero and any number is zero, or we make no exception, in which case the LCM of zero and any number does not exist. To meit makes more sense to say that the LCM is defined only for positive numbers. See the definition of LCM here: Least Common Multiple - Eric Weisstein, World of Mathematics http://mathworld.wolfram.com/LeastCommonMultiple.html It says, "The least common multiple of two numbers a and b is the smallest number m for which there exist POSITIVE integers n_a and n_b such that n_a*a = n_b*b = m." [Emphasis is mine.] If the LCM of 0 and 5 were 0, we'd have a = 0, b = 5, n_a = any number, and n_b = 0 - which is not a positive integer, so it fails this definition. Thus, while the definition does not explicitly say that the two numbers must be positive, this is implied by the definition. I have to ask: Why do you care?Is there a context in which you need the LCM of zeroand another number?

“Pops”, who had asked the question, then gave his context:

I'm a computer science professor and I'm proposing to the students a program to obtain the LCM of two numbers. My aim is to explain the correct answers in all possible "legal" situations.

So what he wanted was just a conventional definition of the LCM that would include all reasonable inputs.

But 12 years later, a math major names Danny wrote in to object to part of what Doctor Rick said, starting with a different definition of LCM — one that is parallel to the one we discussed above, taking divisibility as a stand-in for “less than”:

Doctor Rick states that if we allow 0 to be the LCM of any non-zero number, then 0 will be the LCM of all numbers. Where a|c reads "a divides c,"the definition of c = LCM{a,b} is: i) a|c and b|c ii) if a|d and b|d, then c|dNow suppose that c = LCM{3,2} = 0. Then let d = 6. We have 3|d and 2|d. But 0 = c does not divide d, since no number m exists such that m|0 = 6 = d. So in fact, letting 0 be the LCM of a pair of numbers will lead to a contradiction in the usual setting. I believe the author mixed upthe usual order relationin the definition of LCM, thinking that a least common multiple must beless (in numerical value)than other common multiples.

Danny has missed the fact that the definition Doctor Rick explicitly used *does* involve “the usual order relation”, and that this is the usual definition outside of abstract math. After Doctor Rick pointed this out, Danny gave good reasons why his more sophisticated definition helps (bear with him, if you are not up to his level):

I believe that this definition of LCM is useful when dealing with group theory, ordered sets in general, and lattices. In particular, it remains defined in dealing with the case of infinite sets, and sets where the natural order has been changed or altered. Alsothe definition works in the case where LCM is required for two numbers, one of which is zero. Zero will be a common multiple of all pairs of numbers, so using divisibility as an order relation 0 becomes the top element of an infinite ordered set like the naturals with zero. This set then forms a complete lattice with bottom element 1 and top element 0. It is a lattice because every pair of elements has a least upper bound, and greatest lower bound; and it is complete because the entire set has a least upper bound, 0, and a greatest lower bound, 1.The LCM of, for example, 0 and 3, will be 0, while the LCM of 2 and 3 will be 6.I suspect that the definition you give is a useful version that works well in most cases; however, perhaps the definition I gave is a way that the LCM can be more general. Could you give me an example of why your definition might be more useful?

At Doctor Rick’s request, Doctor Jacques answered, first agreeing with Danny’s conclusion:

In fact, I would say that there is no problem in considering that 0 is a common multiple of a pair of integers: after all, it is a multiple of each of them.... The point is that it is not the least such multiple, where"least" must be understood with respect to the partial ordering induced by divisibility, since this is the meaning used implicitly in "least common multiple."The problem is maybe in the "implicit" aspect. In any case, I think you know all this.

That is, to a mathematician, “least” here doesn’t mean what we usually think it means, and that is important.

He went on to talk about why we have two different definitions, which is something mathematicians in isolation can easily forget:

The last question is why we have the "simpler" definition. It is already taughtin elementary school, at a time wherethe general definition would be out of reach. Even at that stage, the definition can be very useful in practical applications (like adding fractions); the same is true for greatest common divisor (GCD).For many people, that is about all the mathematics they will need.I think that, if you want to learn math seriously, you have first to unlearn many of the (false or incomplete) things you were taught in school, because it was not possible at that time to give you strictly correct and complete definitions.

I then joined the conversation to sum up, since this matter of definitions has been of interest to me:

I'd like to tie things up with a comment on the big picture.It is quite common for a concept to start with a simple idea and a "naive" definition, and later be generalized.In this particular case, as has been mentioned, both definitions MUST continue in use in different contexts, becauseonly the naive initial definitionis understandable by most people who need the concept, whileonly the sophisticated general definitionapplies to cases beyond natural numbers.Most online sources, including Wikipedia and MathWorld, give the definition applicable to natural numbers.This is the appropriate definition for use as the Least Common Denominator of fractions, since denominators can't be zero. It also fits the name: it is exactly what it says, the LEAST (positive integer) multiple of the given numbers. This definition is undoubtedly the source of the entire concept. Some sources give that same definition, but thenadd that if one of the numbers is zero, the LCM is 0(with or without explaining why this extension makes sense). This is probably the answer that should have been given to the original question (from a computer science context, just looking for a reasonable value to give in this case).

We have two very different contexts: ordinary arithmetic with whole numbers, and abstract algebra (which is algebra, or in this case number theory, applied to all sorts of entities that may or may not be numbers). Each has its own needs; if either forced its definitions on the other, it would fail.

This elementary definition had to be extended in order to cover other contexts, as you noted. As Doctor Rick pointed out, simply applying the basic definition to such cases would not work. The definition you are using is the result of a search for an appropriate extension, and isbased on a theorem that is true in the natural number context, and turns out to be usable as a definition in the general case. It certainly would not be appropriate to start with this as a definition in elementary grades, but it would be possible to make the transition before getting to abstract algebra -- though it would never be particularly helpful in understanding the concept in its everyday applications.

This is a common way in which definitions are extended: Theorems in one mathematical realm become definitions or axioms in a larger realm, where the original foundations have been left behind.

I had trouble searching for a source for your definition, sincethe elementary definition is overwhelmingly common. As I mentioned, Wikipedia gives the elementary definition, but adds https://en.wikipedia.org/wiki/Least_common_multiple Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define LCM(a,0) as 0 for all a,which is the result of taking the LCM to be the least upper bound in the lattice of divisibility. This last thought leads to your definition, which is given at the bottom of the page in defining the lattice of divisibility: The least common multiple can be defined generally over commutative rings as follows: Let a and b be elements of a commutative ring R. A common multiple of a and b is an element m of R such that both a and b divide m (i.e., there exist elements x and y of R such that ax = m and by = m). A least common multiple of a and b is a common multiple that is minimal in the sense that for any other common multiple n of a and b, m divides n. In general, two elements in a commutative ring can have no least common multiple or more than one. However, any two least common multiples of the same pair of elements are associates. In a unique factorization domain, any two elements have a least common multiple. In a principal ideal domain, the least common multiple of a and b can be characterised as a generator of the intersection of the ideals generated by a and b (the intersection of a collection of ideals is always an ideal).

If you live in Danny’s world of higher math, you may understand this. If not, just know that up there where the air is thin, the ideas of GCD and LCM change their foundations and look very different, but, having undergone a reconstruction, continue to be useful in much the same ways. And the conclusion (that an LCM can be zero) filters back down to the mundane world.

So, summing things up,]]>the definition Doctor Rick gave is very useful in the everyday world;yours is useful in higher-level mathematics-- and both can peacefully coexist becausethey give the same result where both apply. Both are "the correct definition" within their own context.

Having looked at the matter of faces, edges, and vertices from several different perspectives, I want to look at one more question and answer, to tie it all together.

The question is from 2008:

Definitions of Edge and Face in 2D and 3D Different resources define "edge" in different ways.What is the "official" definition of "edge", specifically is an edge restricted to the intersection of two non-coplanar faces ordo two dimensional shapes have edges?I also have a similar question about "faces"? How many faces does a two-dimensional shape have? The use of edges and faces arose in a problem aboutEuler's Formularelating numbers of faces, edges and vertices in a three-dimensional shape.The same terminology was applied to the two-dimensional net for the three-dimensional shape. In exploring the definitions some Geometry books use the terms without defining them well and many internet sites haveconflicting definitions and uses of these terms. Is it that there is no consistent definition? What source would be definitive for future definition questions? I could really use a mathematical dictionary or encyclopedia at home. The internet sites brought up by a search are not guaranteed to be scholarly or correct. An entry on this site defines edges for two-dimensional shapes. Other sites restricted edges to intersections of two faces on three dimensional shapes. I realize that I need to refine my own conceptualization of edge and face so I can use them correctly. I teach high school and coach high school math teachers so the correct use and understanding of these terms is something important to me.I respect the scholarship of this site. I had no way of evaluating the information on other sites.

Definitions, and their variability, have been one of the things that fascinate me as a Math Doctor. Since we get questions from all over the map (both in the global sense, and in terms of the many fields of math), we get to see how each field, and each country, can have its own use of words. Context is everything. There are no single official definitions; and having them would in a sense actually be bad for math.

Sue here is troubled by conflicting definitions; part of this is due to poor sources, as she suspects, but some is also due to different contexts. I answered:

Just as in ordinary usage,mathematical definitions depend on context. One word can have severalslightly different definitions centered around a common idea, depending on how it is being used. One problem we find is that elementary texts often try to use words in ways that mathematicians don't, wrongly assuming they have a uniform meaning that can be applied in all cases. Another problem is trying to use a general definition without considering the special restrictions imposed by a specific context.

You can see the variation of definitions by looking in any dictionary. Choosing an arbitrary word, consider “paper”. On the surface (no pun intended), this seems straightforward; but consider these variations in its use:

- A sheet of paper
- The substance that sheet is made from
- A document
- A specific type of document: a scholarly paper
- A newspaper
- Financial documents
- Identification documents

These are all, as I said above, “centered around a common idea” (not unrelated words that happen to look alike, which also happens); but each usage is adapted from that for a particular use. Depending on context, you would know which I meant if I just said, “Here are my papers”, though the definitions are different.

There are two quite different contexts in which “face” and “edge” are used, namely geometry and topology; and within geometry, we have both plane and solid geometry. In each context, we use these words with different nuances. What makes this confusing is that these fields are all interrelated, and it can be hard to be sure which we are really in!

I first looked at geometry:

Insolid geometry, the basic definition of "edge" is "the intersection of two faces of a polyhedron". It also applies to the segments comprising a polygon inplane geometry. Similarly, a face is one of the (flat)polygons comprising the surface of a polyhedron. The term is not really relevant in plane geometry, though it could be applied without trouble toa polygon that is part of a plane figure. It is also irrelevant to discussions of curved surfaces, such as cylinders, cones, and spheres, within solid geometry.

In these contexts, an edge is a line segment and a face is a polygon. Why? Because those are the elements we are interested in there. (Faces as part of a plane figure typically arise in the context of topology, to be discussed below, though one often doesn’t realize that.)

Outside of the context of polygons and polyhedra, there areno standard definitionsfor "edge" and "face". If one wants to talk about the "edges" or "faces" of acylinder, it is necessary to eitherextend the definitionsin a way that fits this new context, or to keep the restricted definitions anduse some new termsfor "curved edges and faces". This would be done on a case by case basis--if someone has a reason to make such a modified definition (in order to be able to state certain theorems efficiently, say), he will state his definitions at the top of his paper. Unfortunately, many elementary texts evidently make up a variety of solutions to this issue, so that kids who learn one thing from their text but see something different on the web get very confused. The right thing would be not to use these terms at all except for polyhedra.

The reality is that mathematicians don’t usually have a reason to talk about “faces” of a cylinder, so there is no clearly defined term for either the circular or tubular portions of its surface. When we do, as I said, we can just state how we are using terms in our own local context. Therefore, as I have said previously, curriculum writers are doing their students a disservice when they teach some particular usage that those students will find conflicts with other sources when they do research (which includes just trying to cheat by looking on the Internet!).

Now I turned to topology, the study of geometrical entities with regard only to their connectedness, dropping all considerations of length or direction.

But there's more. The same words are used with related but different meanings intopology. Here, straightness and flatness are irrelevant, but connections matter:an edge must be a curve with two endpoints(which, for some purposes, must be distinct), anda face must be a simply connected region bounded by edges. It is here that Euler's formula arises, so straightness doesn't matter, but the theorem imposes other restrictions--which are too often ignored or oversimplified in elementary treatments--namely the connectedness issues I just mentioned. The formula can't be blindly applied to any solid (or plane) figure. For a discussion of the restrictions, and how to relate the geometrical and the topological aspects, see this page: Faces, Vertices, and Edges of Cylinders, Cones, and Spheres http://mathforum.org/library/drmath/view/64540.html In order to apply the formula, the vertices, edges, and faces have to meet certain connectedness criteria, and the entire surface must be equivalent to a sphere--its interior must be simply connected. For example, as the link at the bottom of that page discusses, a torus needs a more general formula called the Euler characteristic.

A major source of the confusion over these issues is that Euler’s formula is not really about geometry proper, but about topology. In its original form, applied to polyhedra, it was thought of geometrically; but it was eventually realized that it was far more general, and relied only on how parts of a figure are connected. As a result, the conditions for it to hold are not those we think of in geometry; and the way it is typically stated at an elementary level can be misleading.

For our purposes the important thing is that the definitions we apply in talking about the formula are not those we use in geometry. The apparent inconsistency Sue found in definitions comes from the easily-overlooked difference in context.

]]>Here is a question from 2001 to start us off:

Parts of a Cone I am a second grade teacher and we are currently teaching a unit on shapes. The question came up as to whether or not a solid cone has any edges. My contention is thatthe definition of an edge is where two planes intersect, and therefore a cone cannot have an edge. Another teacher says that the curved surface of a cone represents an infinite number of planes, and therefore represents an infinite number of edges. I would very much appreciate your response, and don't be afraid to get technical. This is as much to satisfy my own curiosity as to let the kids know the proper answer.

We had previously answered several questions that touched on this, but had not really addressed the issue. I took the question, which was the start of a “career” of specializing in this question:

We get this question from time to time, and can never really give a definite answer. The word "edge" is used in different ways; often people get in trouble byintroducing the concept of "edge" in the context of polyhedra(where it does mean the intersection of two flat faces), butthen talking about curved surfaces like cones without additional comment. Here's the definition in the Academic Press Dictionary of Science and Technology: 1.in graph theory, a member of one of two (usually finite) sets of elements that determine a graph; i.e., an element of the edge set. The other set is called the vertex set; each element of the edge set is determined by a pair of elements of the vertex set... 2. a straight line that is the intersection of two faces of asolid figure. 3. a boundary of aplane geometric figure. In the latter sense (which I think is appropriate in discussing a cone, even though the dictionary only mentioned plane figures and not curved surfaces), the cone has one edge. I definitely would not bring in the idea of "an infinite number of edges"; that kind of reasoning generally leads to trouble! I would simply say that we can extend the concept of edge either from the world of polyhedra (definition 2) or from the world of plane geometry (definition 3) to apply to possibly curved boundaries of possibly curved surfaces, as long as we say that we are doing so. This also agrees with definition 1, which likewise does not require straightness (indeed, there is no such concept in graph theory), and which relates to boundaries when we consider planar graphs (as in Euler's polyhedral formula).

In other words, I did not find a proper definition of “edge” that fits this context, but it is reasonable to extend the definition, when we extend the context, to the nearest analogue of a polyhedron edge, which is the intersection of surfaces. But we need to state our definition when we do so — it is not necessarily something that others will have defined the same way.

What definition you use depends on what you are going to do with it. If you are just describing objects, my loose definition is fine. If you are going to prove theorems involving planes and angles, you'll want to restrict yourself to the polygonal definition, but then you won't be asking any questions about cones. I think people often fail to realize thateven though we are very particular about definitions in math, those definitions may vary from field to field, as they are adapted to a certain context. That's what I'm trying to do here. The same questions arise concerning faces and vertices, and it's even harder to decide in those cases.

Mathematicians are accustomed to making up definitions for their particular context like this. But we’ll have more to say about that.

A year later, we got this question along similar lines:

Does a Cone have an Edge? A Vertex? Our 4th grade math textbook defines a cone as "A solid figure withone circular face and one vertex." This sounds reasonable until you read the textbook's definitions for face, edge, and vertex. The textbook defines a face as "Aflat surfaceof a solid." It defines an edge as "A line segmentwhere two faces of a solid meet." It defines a vertex as "A pointwhere two or more edges meet." Assuming that these definitions are accurate and that I'm not misinterpreting them, a cone must not have a vertex. If a cone has only one face, then it can't possibly have an edge. Therefore, if it doesn't have an edge, it can't have a vertex.

Again, the book is giving definitions suitable for a polyhedron, but applying them inconsistently to non-polyhedra. I responded again, after referring to that previous answer:

Elementary texts (and high school texts, for that matter) are not always very careful about definitions. The problem really is that the same word can be used with slightly different but related definitions, and we don't always bother to specify how to modify the definitions when we move to a different context. The definitions given are for a polyhedron. When you talk about a cone or cylinder, you have toeither use a different set of words, since "edge" and "vertex" as defined don't apply at all, and "face" applies only to one of the two surfaces of a cone; or you have tomodify the definitionsto allow curved edges and faces. Taking the latter approach, the cone will have two faces, one curved, and one curved edge. I'm not sure I've ever seen such modified definitions actually stated, butI have no trouble allowing them, as long as we state them clearly! What really bothers me is when a book is consistent enough not to call the curved "face" of a cone a face, but doesn't bother to define a word that we _can_ use for it. When they then go ahead and say it does have an edge or a vertex, children are bound to be confused. The really tricky part here is thatthe "vertex" of a cone has nothing to do with edges, so it needs a whole new definition; and I can't think of a really good elementary-level definition for what they obviously mean, which is simply a "point." I prefer to use the word "apex" and avoid the problem.

I later discussed the “vertex” of a cone here:

Conical Vertex?

A month later we got yet another question about this, which got a long answer:

Number of Cylinder Edges My 8-year-old son was asked "how many edges are there on a solid cylinder?" on a recent math examination. His answer was "2" and it was marked as incorrect. He truly believes in his answer and has asked for my assistance in researching.

I answered first, again having referred to the previous answer as background:

It depends on how "edge" was defined in his class, which may not agree with his intuitive definition.Often, an edge is required to be straight, in which case a cylinder has no edges. Unfortunately, elementary texts are not always very careful about definitions, and they can ask questions like this that are really worthless.The only definition of "edge" that would make sense in this context would be the one your son is naturally using(a boundary between smooth surfaces making up an object), which would allow a cylinder to have two edges. Asking this question with the other definition only invites confusion, so I wish they wouldn't ask it. I'd like to hear how they did define the word.

If you are going to use the word “edge” in talking about a cone, you must use a definition that is appropriate, just as I previously commented about needing some word that describes the curved surface of a cone, if you are going to ask about it at all.

But this discussion went further, outside of Ask Dr. Math. Doctor Sarah added this:

David asked the same question in the discussion group geometry-pre-college, where it received these responses: http://mathforum.org/kb/message.jspa?messageID=1077941

She copied in two responses to the question. The first, by Walter Whiteley, dug into issues related to Euler’s formula, which I discussed last week, and will talk about more next time. I will just quote pieces of it that relate most directly to our present issue of definition:

This is a common issue among elementary teachers, and some elementary text book writers. Basically different sources put down different answers.The underlying issue is: What is the context?What is the larger mathematics one wants to engage with? Without this, there are too many plausible responses. ...

He talks about convex polyhedra, then about topology, where straightness and flatness are irrelevant, concluding that the intuitive definitions make sense, with caution.

However, some elementary texts and test writers decide they know best and give distinct definitions of 'faces', 'edges', and 'vertices'. When doing so, there should be some good mathematical reason for doing that. Some set of situations one is trying to make sense of. Simple extrapolation on one basis or another, without investigating the good and bad patterns, is a source of trouble. That, unfortunately, routinely happens in elementary (and some high school) materials.If faces are 'flat regions' and 'edges' are straight lines, then a cylinder has two faces, no edges, and there is no real purpose in the answer. It does not even help you calculate the surface area!If faces are regions, and edges are where two faces meet, then a cylinder has three faces and two edges (no vertices). This still does not seem to be a mathematically interesting description.

Some authors even require a face to be a polygon, so that a cylinder has no faces and no edges.

Definitions with no purpose are contrary to the spirit of mathematics, as well as to pedagogy.

I suspect that whatever answer this particular test expected, it is based on a particular discussion in a particular text.I can show you different materials with different answers, but seldom is there a mathematical discussion.Some people have concluded that, as a result, it is simply a bad idea (distracting without learning) to use the words faces, vertices, edges for such objects. I do not quite agree - but the only really useful context I know is the larger topology, and you can see that this takes a larger understanding, something I only learned at graduate school, and only teach in some upper level undergraduate courses (courses most teachers have not taken). Odds are this discussion in the source text or materials did NOT give enough context to explain why one would bother with these words for this object. What is the MATHEMATICS one is trying to do! That is where one needs to start.

The only purpose I see for using these terms is to be able to talk about an object, not necessarily even to do math with it. But to do that, the terms should have universal definitions, as these do not.

For a similar question from 2004, see:

Do Cones and Cylinders Have Faces or Surfaces?

I’ll close with a question from a 12-year-old in 2008:

Is a Curved Surface a Face? Is a curved surface a face or not? Like in a cylinder is the curved surface considered a face?Some people tell me that a curved surface is a face and some say it's not.When I search in Google I also don't get a straight answer. I just want to find out. I think a curved surface is not a face.

I took this as a chance to put together a larger perspective on these questions.

As you've discovered, there is no straight answer to this. In mathematics, we define terms to meet a need.If something is worth talking about, we give it a name, and define exactly what that name means.Mathematicians talk about faces, edges, and vertices commonly in the context of polyhedra, where faces are all flat, and therefore are always polygons, and edges are always straight line segments.We have not found it very useful to extend this idea to other shapes, such as cylinders or cones, which have curves, so we have not made a standard definition for these terms in that context. If we happen to need to do so, we would give our definitions at the start of our paper, and would use whatever definitions make it easy to talk about what we want to talk about.

In our experience answering random questions with little or no stated context, it quickly becomes clear that context determines what definitions apply, so we can’t answer effectively without that background. We also see that without a universal definition, mathematicians happily just define them *ad hoc*.

There are several ways we COULD extend the definitions. We could leave them just as they are,requiring faces to be polygons, and edges to be straight; but then since cylinders and cones have surfaces that are not faces,we need extra termsfor those. "Curved surface" is a reasonable name; probably we would also talk about "curved edges". Another possibility is tochange the definitions to fit curved objects. We might require a face to be flat, but not necessarily a polygon, so that the circular bases of a cylinder would be faces, but the "curved surface" would not be. Or, we might call any surface a face. The question would be,why do we need to use the terms? Are there theorems that apply only to what we are calling "faces", and not to other surfaces?That would determine what is the best set of definitions to use.

I have varied on which approach I prefer. That’s largely because there is no particular reason to choose one over the other. That is also why books disagree.

Elementary textbook authors seem to feel a need to have a word for everything, and to be able to apply each word to all the shapes they want to talk about--to be able to answer the question "How many faces does this have" for any object. So they decide for themselves (possibly without a valid mathematical reason) how they want to define these terms. As a result, you get books that use different definitions. I wish they wouldn't do that, because it confuses a lot of children when they look up an answer and find it disagrees with their book or teacher. The best thing is just not to bother asking the question at all.

The basic rule ought to be “when in doubt, don’t”. I have taught courses in Mathematics for Elementary Teachers several times, and observed that their textbooks did not ask about faces of cylinders; there was no need to do so. This wisdom needs to be passed down to the elementary curricula themselves.

So the answer to your question is: SOME people consider a curved surface to be a face, and others do not.Those whose opinion matters most, don't have an opinion(or would ask you for the context of your question before attempting an answer). If you are asking just for yourself, your answer is fine: a cylinder has two flat surfaces and a curved surface, and two curved edges. If you are answering a question asked on a test, you'll have to find out what your own text says.

That, unfortunately, is the bottom line.

]]>Having discussed counting earlier this week, let’s take a look at a different kind of counting. The subject of combinatorics (the study of counting) arises in many guises: probability, sets, geometry. Here, we look at a relatively basic type of problem that involves the same sort of organized thinking that we used to count vertices.

Here is the problem, from a couple months ago:

Q) How many ways are there to place two kings on a chessboard so that they are safe from each other?

I tried this question starting that if the king is at corner blocks then there would be total 12 places where the other king cannot be. Then in corner rows I am not able to find as there are overlappings. Please help me how to calculate in the corner rows. Please send the solution till the end so that I don’t have problems latest in this question.

Our first issue in answering this question is to agree on terminology (what is a “corner block” and a “corner row”?); then we need to determine how the student is looking at the problem itself. Doctor Rick replied:

“I tried this question starting that if the king is at corner blocks then there would be total 12 places where the other king cannot be.”Let’s start by understanding this part. I think I see where you got 12: is it the total number of places where the other king cannot be, for

all fourcorner positions of the first king? If so, that’s not what matters here. Rather, take the number of places where the second king cannot be, if the first king is in anyoneof the corner positions. Use that to find the number of places where the second kingcanbe in each case, and multiply that by the number of corner positions (4). This will give you the total number of “safe” placements of two kings for which the first king is in a corner square.

In approaching a counting question, we first have to be sure what is being asked; often, questions are asked from a perspective unfamiliar to students new to the topic. Here, the question is about “ways to place two kings”, which might be restated as “pairs of squares on the board such that kings in them are not attacking one another”. The concept of counting *pairs*, rather than individual items, is not something we do every day! So it is understandable that this student at first thinks of counting *all* squares where *any* king might be, rather than taking them separately for each place where the first could be.

So far, we have four places for the first king to be (the corners), each of which excludes four squares for the other (where the first king is, and three squares attacking it):

Doctor Rick continued:

“Then in corner rows I am not able to find as there are overlappings. Please help me how to calculate in the corner rows.”I am not sure what you mean by “corner rows”. What I would be thinking about next is the non-corner edge squares — that is, squares that are on one of the four edges but not in one of the corners. How many such squares are there, on a standard 8×8 chess board?

I also don’t know what you mean by overlappings. Can you explain that? Again, what we care about is the number of

safepositions for the second king if the first king is inany oneof these non-corner edge squares.

My guess is that the student is talking about places along the edges other than the corners (as Doctor Rick is), and is thinking that there are two many of these squares to count all at once as he has been trying to do. If so, then the advice to emphasize the word “each” may be all he needs.

There are 6 non-corner edge squares on each side, for a total of 24 such squares; for each of these, there are 6 squares where the second king can’t be:

The student then wrote twice, ten minutes apart, with updates on his thinking. First:

Ok, I got that I had to calculate the number of places where other king can be.

So if king A is at the corner then king B is anywhere in the rest (16 – 4)×4=36 places. Then there are 6 places in corner edges (taking last two edges) where king A can be. For any of King A’s position there will be (6×4)=24 places left for king B. Now in the middle there are 24 places left where king A can be. For each of king A’s position there are 24 – 9=15 places left for king B. Have I done correct till here? What should I do next?

It appears that the student is either thinking of only a quarter of the board at a time (16 squares), or is still thinking of all four corners being occupied at once rather than one at a time. I can’t quite see where many of these numbers come from, though.

But then he wrote again:

If the white king is in a corner (4 possibilities), the black king can be at any of 64 − 4 positions.

If the white king is at the boundary, but not in a corner (24 possibilities), the black king can be at any of 64 − 6 positions.

In all other cases for the white king (36 possibilities), the black king can be at any of 64 − 9 positions.

In total: 4⋅60 + 24⋅58 + 36⋅55.

Is this correct now?

This is looking good. Sometimes just explaining your thinking to someone else (or to yourself!) can help you see where it is wrong, so you can correct yourself without additional input. We have now included interior squares for the first king, which each exclude 9 for the second:

Doctor Rick answered:

“If the white king is in a corner (4 possibilities), the black king can be at any of 64 − 4 positions.”Correct: king B can’t be in the same corner or any of the three positions adjacent to it.

“If the white king is at the boundary, but not in a corner (24 possibilities), the black king can be at any of 64 − 6 positions”Correct: there are 24 non-corner boundary positions, and six positions for king B are ruled out in each case.

“In all other cases for the white king (36 possibilities), the black king can be at any of 64 − 9 positions.”Correct again.

“In total: 4⋅60 + 24⋅58 + 36⋅55.

Is this correct now.”There is only one more thing we need to consider. The problem asked, “How many ways are there to

place two kingson a chessboard so that they are safe from each other?” It isn’t clear whether the two kings are to be considereddistinguishable or not. I would guess, importing basic knowledge of chess into the problem, that the two kings are of different colors (one white and one black). In that case, the configuration with the white king in the upper left corner and the black king two places to its right, for example, isdifferentfrom the configuration with the black king in the upper left corner and the white king two places to its right. If so, you’ve got it, I believe. But if the kings are indistinguishable, we need to divide your result by 2 to account for each pair of configurations counting as a single “way to place two kings”.

It is important in counting different ways to do something, to consider not only what the “something” is that we are to do, but also what counts as “different”. It seems more or less clear what would be intended here, but another interpretation is conceivable.

So our final answer is that there are 240 ways to place the kings if the white king is in a corner; 1392 ways to place them if the white king is along an edge; and 1980 ways if the white king is in the interior, for a total of 3612 ways. And if all that mattered was which squares they are on, not which king is where, then we would divide that by 2 (since we would have counted each arrangement twice) to get 1806.

What did we have to do in order to carry out this count? We didn’t list all the ways, which would have been very difficult; instead, we used a “divide and conquer” strategy, breaking the possibilities into three groups based on how many squares are around each king; and we used multiplication to handle the “for each” relationship. Within each category, we used multiplication and subtraction to find a total excluding some. But in order to carry all this out, we first had to clearly understand the problem — and clearly communicate our own meanings to one another. Counting may sound like baby stuff, but it involves many skills that are useful throughout mathematics.

Now, I have found that almost any question we get, something close to it will have been discussed here before! After writing the above, I decided to check, and I found this question from 15 years ago:

Chess King Positions On a regular 8x8 board with 64 squares, the total possible positions is 3,612. If the board is then transformed to a 13x9 with 117 squares, how does one go about figuring this out? I have tried simple cross multiplication but that cannot be correct.

So this student had seen our problem and knew the answer, but wanted help solving a larger version with a rectangular board. Doctor Jeremiah answered, first showing the work for the already-known problem:

The two kings cannot be in squares right beside each other (because one of them would be in check) so for each position of the first king, the second king cannot be in every remaining square.If the first king is in a corner, then the other king cannot be on that square or the three surrounding it. For an 8x8 board that means the number of possibilities for the second king is 60 squares when the first king is in one of the four corners.If the first king is on a side, then the other king cannot be on that square or the five surrounding it. For an 8x8 board that means the number of possibilities for the second king is 58 squares when the first king is on one of the 24 side squares.If the first king is in the centersomewhere, then the other kingcannot be on that square or the eight surrounding it. For an 8x8 board that means the number of possibilities for the second king is 55 squares when the first king is on one of the 36 inner squares. Total possibilities = 60 squares for each of four corners plus 58 squares for each of 24 side squares plus 55 squares for each of 36 inner squares: 4x60+24x58+36x55 =3612 possibilitiesNow, how would you do a 13x9 board?

John had an answer, presumably having been inspired by a clear presentation of the solution:

The answer should be 12,764 positions on a 13x9 board. Corner 113x4 = 452; Sides 111x36 = 3996; and Center 108x77 = 8316.

Doctor Jeremiah confirmed that result, showing the details of the various sub-counts:

]]>If the first king is in a corner, then the other king cannot be on that square or the three surrounding it. For a 13x9 board that means the number of possibilities for the second king is (13x9-4) squares when the first king is in one of the four corners. For all four corners the total combinations = 4 x (13x9-4) = 452.If the first king is on a side, then the other king cannot be on that square or the five surrounding it. For a 13x9 board that means the number of possibilities for the second king is (13x9-6) squares when the first king is on one of the side squares. The total number of side squares is (13-2)+(13-2)+(9-2)+(9-2) = 36. For all the side squares the total combinations = 36 x (13x9-6) = 3996.If the first king is in the center somewhere, then the other king cannot be on that square or the eight surrounding it. For a 13x9 board that means the number of possibilities for the second king is (13x9-9) squares when the first king is on one of the inner squares. The total number of inner squares is the board size minus the side squares minus the corners of 13x9 - 36 - 4 = 77. For all the inner squares the total combinations = 77 x (13x9-9) = 8316. Total possibilities = (13x9-4) squares for each of four corners plus (13x9-6) squares for each of 36 side squares plus (13x9-8) squares for each of 77 inner squares: 4 x (13x9-4) + 36 x (13x9-6) + 77 x (13x9-9) = 452+3996+8316 Which is exactly what you got!

A good way to introduce it will be with a question about how to introduce it, from 2001:

Euler's Formula I have to find Euler's formula for two-dimensional figures and explain it at a university level and at an elementary-school level.

Note that this question is about two-dimensional (flat) figures, not three-dimensional polyhedra. The formula applies to both; if you start with a polyhedron and imagine punching a hole in one face and stretching that hole infinitely, the polyhedron becomes a flat figure (a “graph” or “network”) with one “face” being the region outside the figure itself. Here is an example:

For both the dodecahedron and the graph, V = 20, E = 30, and F =12 (including the “outside”), so V – E + F = 20 – 30 + 12 = 2.

In fact, the network version of the theorem is more general; a graph does not have to correspond to a polyhedron for it to work. Wikipedia states it this way:

Euler’s formulastates that if a finite, connected, planar graph is drawn in the plane without any edge intersections, andvis the number of vertices,eis the number of edges andfis the number of faces (regions bounded by edges, including the outer, infinitely large region), then\(\displaystyle v-e+f=2.\)

Doctor Tom focused on how to help children discover the formula, before giving an informal proof:

Here's what I'd do: First, go through a bunch ofsimple examples where the kids can count vertices, edges, and faces, and verify that V - E + F = 2. Or maybe start with four or five examples where you don't even calculate V - E + F, but just make a table of those values. Then have the kidslook at the table and look for patterns. You should be able to lead them to show that the above sum/difference is two by noticing things like the fact thatif V or F goes up, so does E. Next, after they've guessed the formula (with or without your help), try making some more drawings to test the formula. What I would do here would be to draw a new configuration, count the items, and check it. Then make the item a bit more complicated by adding vertices in the middles of edges and by adding edges that connect two existing vertices (or make a loop from a vertex to itself). You're doing this tosecretly convince the kids that arbitrarily complex connected configurations can be made from simple onesby adding vertices to edges or edges connecting existing vertices.

So far, they have seen that there is a pattern that can be guessed, and that it passes any test they give it; but that doesn’t mean it is always true. Now Doctor Tom describes how to turn that into a proof (though still keeping it elementary):

Finally, for the proof, show thatit's true for a single vertexin the plane (V=1, F=1, and E=0). Next show that if you have ANY configuration,adding a vertex to an edgeincreases V by 1 and E by 1, leaving V - E + F the same, and thatadding an edge between two verticesincreases F by 1 and E by 1, again preserving the Euler characteristic. So you have a trivial situation where the formula holds, and two operations that are guaranteed to preserve the characteristic. Finally, begin with the dot on the plane andshow how to construct a few of the examples you've already done one step at a time.

A university-level proof, of course, would use terms like “induction”, and would prove that any appropriate graph can be produced this way.

Perhaps this isn't a rock-solid mathematical proof, but it should certainly be enough to convince the kids that the theorem is true, and shows them (in a secret sort of way) the ideas of mathematical induction and the idea of using an invariant for a proof. I might even end by showing that exactly the same formula holds for a 3-D cube and an assertion thatthe formula is also true in 3-D, to give some of the brighter kids something to think about and play with.

For an elaboration of the proof, starting with the planar graph, then moving to polyhedra with “genus zero” (i.e., equivalent to a sphere), and then finally adding “holes” (which we’ll see below), see here:

Euler's Formula for Polyhedra

Now let’s move from the plane and polyhedra, to curved surfaces, which cause some trouble:

Cylinders and Euler's Rule How and why does Euler's rule work for cylinders?

A cylinder is not a polyhedron, but if you “blow it up” you will still get a spherical form, so the formula ought to work. Yet a naive approach to that fails. Students often see the cylinder as having three faces, two edges, and no vertices. (I’ll be discussing how these terms apply next week!) But taken that way, V – E + F = 0 – 2 + 3 = 1, which doesn’t work. I’ll be getting to that later.

Doctor Mike took this:

I suppose you mean the formula V + F - E = 2. A simple example is a cube, which has 8 vertex points, 6 faces, and 12 edges, so 8 + 6 - 12 = 14 - 12 = 2. The faces of a cube are flat, butthis would also work if the faces or edges were somewhat curved, just so long as they don't intersect each other. The reason I mention this is that in the case of a cylinder, one of the faces will be the curved part of the cylinder. Let's concentrate first on that curved face, because that is the "secret" about interpreting Euler's Law for a cylinder. If you examine a 'tin can', common as a food container, you will see that there is a 'seam' from bottom to top.That seam counts as an edge for the purposes of the formula.Also, that edge has a vertex at each of its ends.

We’ll see below *why* the seam is needed!

When we close up the cylinder 'tin can', the top is a circular face, with an edge around its circumference, and the same for the bottom face. Here is a list of all the faces, edges and vertices. Face 1 = the curved surface around the cylinder. Face 2 = the top, which is flat Face 3 = the bottom, which is also flat Edge 1 = the seam up the side of the curved face Edge 2 = the circle around the top face Edge 3 = the circle around the bottom face. Vertex 1 = the point at the top of the seam Vertex 2 = the point at the bottom of the seam Note that the two vertices also serve double-duty as the points where the circular edges start and stop. So, V + F - E = 2 + 3 - 3 = 2 + 0 = 2

Here is the seamed can:

The following year we had a similar question:

Faces, Vertices, and Edges of Cylinders, Cones, and Spheres I need to know how many faces, vertices, and edges do cylinders, cones, and spheres have? Logically I would say that aspherehas 1 face, 0 vertices and 0 edges. Problem: a face is flat, sphere is not flat. Secondly thisdoes not satisfy Euler's formulav - e + f = 2. I would say aconehas 2 faces, 1 edge, and 1 vertex. Problem: while this does satisfy Euler, itdoes not satisfy the definitions.

These are similar to the issue I mentioned above for cylinders. What is going on? I answered:

Properly speaking,Euler's formula does not apply to a surface, but to a network on a surface, which must meet certain criteria. The "natural" faces and edges for these surfaces, or those determined by applying the definitions used for polyhedra, do not meet these criteria.

This is the most important point. Note that what matters about a face is not whether it is **flat** (as on a polyhedron), but whether it is part of a **properly formed graph** on the surface. It is not the nature of the surface itself that matters, but of the graph (network). Now I examined each case he asked about. First, the cone:

Just taking thenatural parts of a cone, as you say, it has one presumed vertex, the apex; one edge, the circle at the base; and two faces, one flat and one curved. (I say "presumed" because the apex is not really a vertex in the usual sense of a place where two or more edges meet, but it is a point that stands out.) This gives 1 - 1 + 2 = 2 Soit does fit the formula; but there is no reason it should, really, because it doesn't fit the requirements for the theorem, namely thatthe graph should be equivalent to a polyhedron.Each face must besimply connected(able to shrink to a disk, with no "holes" in it), and likewiseeach edge must be like a segment(not a circle). One of our "natural" faces has a "vertex" in the middle of it, so it is not simply connected; and the "edge" has no ends, so it doesn't fit either. These errors just happen to cancel one another out.

Next, the cylinder:

As another example, take a cylinder, which in its natural state has no vertices, two "edges", and three "faces": 0 - 2 + 3 = 1 It doesn't work, and the theorem doesn't claim it should. In each case you can"fix" the graph by adding one segment from top to bottom. In the cone, this gives one extra vertex (on the base), and one extra edge, so the formula still holds. In the cylinder, it gives two new vertices and one extra edge, and the formula becomes correct.

Here is the cone with a “seam”:

Here V = 2, E = 2, F = 2, so V – E + F = 2 – 2 + 2 = 2.

We saw Doctor Mike’s “seam” above, where V = 2, E = 3, F = 3, and V – E + F = 2 – 3 + 3 = 2.

What do you have to do to "fix" thesphere?

I left this to Cara. The answer is that you can draw any polyhedron you like on the sphere; or, to keep it minimal, you can just make an “equator” and put one dot on it. Now we have V = 1, E = 1, F = 2, and 1 – 1 + 2 = 2.

I referred Cara to the following discussion for a more detailed explanation of the rules for an appropriate network on a surface:

Euler's Formula Applied to a Torus Can you explain why Euler's characteristic is zero for a torus? If, for example, I drew an arc with two vertices on top of the torus and connected another arc to it to form a circle, wouldn't V=2, E=2, and F=1, so that V-E+F=1? What I am I missing? Isn't this an admissible graph?

This deals with an extension of the formula to surfaces other than planes and spheres. Any surface has an “Euler characteristic”, which is the number on the right of the formula, replacing 2. In particular, for a torus (a donut shape), the formula is V – E + F = 0. I think his figure looked like this:

Jim, like the others, has drawn a simple figure on the torus, and found that the result is 1 rather than 0. To see why, we need a more detailed explanation of what vertices, edges, and faces have to be like for the formula to apply. I responded, asking for confirmation of what he had done (though I now think that was clear enough):

Your mistake is that one of the "faces" includes the "hole" of the torus, and therefore is not a valid face.A face must be topologically equivalent to a disk; you should be able to flatten it out into a plane. If this doesn't clear it up, please write back and give me the definition you are using of an "admissible graph," so I can use the same terms you are familiar with - there are several ways to describe this.

Jim answered, adding a second example graph:

Thanks for the reply. Actually, the term "admissible graph" comes from Steven Krantz's book, _Techniques in Problem Solving_. He defines it simply asa connected configuration of arcsand his example focuses on a sphere. He alsodefines a face as any two-dimensional region, without holes, that is bordered by edges and vertices. One of his problems is to determine the Euler characteristic for a torus and to show that it will work for any admissible graph on the torus. So let's say you drew two arcs on top of the torus so they formed a circle around the torus (it would look like three concentric circles if you viewed the torus from the top with the hole looking like one of the circles). What is F for this configuration? Also,if you drew one arc from the outer "edge" of the torus to the hole and back up and around(this would sever the torus if a cut was applied along the arc), what is F for this configuration? Is F always zero for a torus? If so, why? I guess my real problem is that I don't understand how Euler's formula applies to a torus. I understand how it applies to figures with pointy edges like a cube, pyramid, etc. Maybe a good comparison to a torus is a nut (square with a hole in the middle). In this case, would F=4 because two of the six sides have a hole and therefore do not count as valid faces?

The new graph looks like this (for which we would still have V – E + F = 1 – 1 + 1 = 1 (assuming he put only one vertex on the circle):

I tried to find a good resource that would explain all the details, and had trouble doing so at the time:

I've found that just about every place I look for a definition of the Euler characteristic and related concepts is eithertoo vague(as yours is), or too deeply embedded in topology (anddependent on definitions given elsewhere, or perhaps never clearly stated) to make a good reference to answer questions like this. The general idea is simply that either we are making an actual polyhedron that is topologically equivalent to, say, a torus, or we are making a graph on the surface that is "polyhedral" in a topological sense. But exactly what this means is seldom stated.

Part of the problem with finding good explanations is that this concept is simple enough to be taught in many sources aimed at an elementary level (where precision is avoided), but also complex enough to be properly explained only to those who have mastered deep ideas in topology. A good explanation is likely to be found only in a source too advanced for most readers. (Here is one good source I have found recently: Euler’s Polyhedral Formula, by Joe Malkevitch.)

Here is one answer I found to a similar question, which is better than most in stating the requirements fairly carefully: How many edges (lines) are in a cylinder? - Final Answers, Geometry and Topology - Gerard P. Michon http://home.att.net/~numericana/answer/geometry.htm#edges In discussing V-E+F=2 for a cylinder with no vertices, two "edges," and three "faces," this says: Nothing is wrong if things are precisely stated. Edges and faces are allowed to be curved, but theDescartes-Euler formula has 3 restrictions, namely: 1. It only applies to a (polyhedral) surface which istopologically "like" a sphere(imagine making the polyhedron out of flexible plastic and blowing air into it, and you'll see what I mean). Your cylinder does qualify (a torus would not). 2. It only applies ifall faces are "like" an open disk. The top and bottom faces of your cylinder do qualify, but the lateral face does not. 3. It only applies ifall edges are "like" an open line segment. Neither of your circular edges qualifies. This is good enough to answer your specific question. Your edges are valid; butyour single "face" is not equivalent to a disk; if you cut along the edges and spread it out flat, it becomes an annulus. That's the problem.

This is true of both of Jim’s attempts; they can also be described as “tubes” (the lateral face of a cylinder); the second especially is best seen that way.

There are several ways in which a "face" may fail the test. One kind of "hole" is that in your example, wherethe "face" is like an annulus or cylinder; its set of edges is not connected. In your example, the inner and outer edges of the annulus are glued together when you put it on the torus, but they are distinct when you view the "face" by itself. It is easy to miss this! You must picture taking the "face" off the surface, so you can see what it really is.

So neither of the figures above is an admissible graph, when defined properly. Instead, we need something like this, which produces one face that is simply connected:

Here, V = 1, E = 2, and F = 1, so V – E + F = 0. (Count carefully!)

Another way is for the "face" to have only one edge, but havea hole in the same sense that the torus has a hole; this is what happens if you simply draw a circle (with one vertex to make it a valid edge) on the side of the torus, so that the inner face is a valid disk, but the outer "face" is all the rest of the torus, including the "hole," which you can also picture as a "handle." This can't be flattened out at all.

Here is a picture of such a graph:

Here V = 1, E = 1, F = 2, so V – E + F = 2, rather than 0; because the second “face” contains the “hole” of the torus, the graph could just as well be on a sphere!

In either case,]]>the "face" is not simply connected; you can draw a circle in it that can't be shrunk to a point. The problem in the definition you are using seems to be thathe defines an admissible graph without reference to the faces, which really are the determining feature in this context; and perhaps also he has not clearly definedwhat he means by a "hole" in a face. If you remember that this can mean either a hole with an edge, or a "handle," it might be clearer.

Let’s first look at this question from 2004:

Defining and Counting Faces, Edges, and Vertices of Shapes How do you find the faces, edges, and vertices of any shape? I just go blank, because I really don't understand it. When I see a question that asks what the faces, edges, and vertices of a shape are I get really confused and then I stay on that question for the rest of the time we have to do the test. I really hope you can help me!

I answered by first defining the terms (visually, as well as I could with our rudimentary style of the time):

You just have to know what the terms mean, and then count them. Afaceis a flat surface, like the front or top of a box. Anedgeis a line along which two faces come together. Avertexis a "point" on a shape, a corner where several faces (and edges) come together. They look like this: +-----------+ / / | / / | +-----------+ |<--- edge | | | | | | | face | +<-- vertex | | / | | / +-----------+

Many students struggle with the word “vertex”, a Latin word whose plural is “vertices”, pronounced and spelled such that students think that the singular is “vertice”, or spell it “verticies”.

Now, if you actually have the object in front of you, you can just count them all, making sure you count each thing once. On my box, you could count the four vertices on top, then turn it over and count four more vertices on the bottom, making a total of 8 vertices.

You might have a physical object you can mark, to avoid over- or under-counting; or just hold it in such a way that you can tell what has been counted. Once you are really familiar with a shape, you can just imagine turning it around like this. But that can get hard for large objects, and is certainly hard when the shape is new to you.

If you just have apicture, you have to be able to imagine seeing all of it, front and back, top and bottom. And if you just have adescription(such as "square pyramid"), you have to know what the description means, imagine what it looks like, and count in your head. That can get a little hard. But even then,you can count in an orderly way, not trying to visualize the details, but just to think about different parts of the shape. For example, a square pyramid has asquare base, and atrianglegoing up from each side of the base to anapex. Knowing that, we can do our counting: Faces: one square base and four triangles = 5 Edges: four sides of the base, and four lines to the apex = 8 Vertices: four corners on the base, and one apex = 5 So by thinking about the parts that go into making the shape, I canbreak down the count into manageable partswithout having to see them.

Just don’t forget that the *bottom* is a face, too!

Speaking of vocabulary, I also referred LaKesha to the following 2002 question, from a teacher:

Edges vs. Corners I am currently teaching my 3rd graders about geometric figures. Every time we begin using the wordsedges, corners, and surfacesto describe geometric figures, my students get confused on how to determine an edge versus a corner. They also have difficultycounting the number of edgeson a geometric figure, say a cube. Is there some good "not too over the head" vocabulary to help my students determine an edge versus a corner? Can you explain how to calculate the correct number of edges on a cube? Thanks for your help.

It can be tricky to decide when to introduce advanced vocabulary, and how precise to be with young children. In my mind, it’s never too early to be correct:

Part of the problem is thatthe word "corner" is ambiguous; we don't use it in math, and kids use it in too many ways to get a clear picture. The proper terms arevertex, edge, and face, where avertexis a POINT where different faces come together, and anedgeis a LINE (segment) where different faces come together. I can easily picture calling an edge a corner in everyday language, so I'm not surprised they get it wrong. Probably you have been given the terms you are using because "vertex" (with its plural "vertices") seems too hard for young children to learn, but sometimes there's a reason for introducing new words. ("Side" is even more ambiguous, so I'm glad you're using "surface" or "face.")

A “corner” of a wall is an edge, while a “corner” of a table is a vertex. So while I appreciate the curriculum author’s concern not to overwhelm students, there really is no non-technical term that unambiguously says what we mean here. A teacher, required to teach poorly-chosen terms, just has to make the best of it, and I didn’t try to deal with that pedagogical question. If I were teaching that level, I think I might add words, calling a vertex a “corner point” and an edge an “edge line” for a while, then wean the students away from the extra “point” and “line” once the meanings were clear.

I’ll be looking at possible reasons for the use of “surface” instead of “face” next week.

I moved on to the counting question:

Now,how can we count the edges of a cube?It's just a matter of finding anorderly way to keep track of them. You could make a cube out of paper, andmark each edgeas you count it; or use erasable markers on a plastic one. Or you can set the cube on a table and count onegroup of edgesat a time: there are four edges on the table, four on the top, and four standing vertically, making 12 in all. The fun way is to use the kind of tricks mathematicians like, which save work for large problems, while making use of the orderliness of a problem. You can look at each corner (excuse me, vertex) and count the number of edges there:3 come together at each vertex. There are 8 vertices in all, and 3 times 8 gives 24. But that didn't count the number of edges, becausewe counted each edge twice- we really counted the ENDS of edges, and each edge has two ends. So we divide 24 by 2, and get the right answer. You can insteadcount the number of edges on each face; again, each edge will be counted twice, since it belongs to two faces. This kind of thinking is extremely useful, and seeing that all the different methods come out the same can be exciting, maybe even suspenseful!

I often recommend to students not to stop with finding one way to solve a problem, but to try to find others. This gives additional practice in problem-solving, and also can help a student to discover what kind of thinking works best for her. The last two methods I suggested may not be suitable for everyone (even 12-year-old LaKesha from the first question), but could be good introductions to the ways we later make formulas for such counts.

It can also be helpful to rename what you are counting. In the “vertex-first” method, what we are really counting is “edge-ends”. There are 3 of these at each of 8 vertices, for a total of 24 ends; and two ends make an edge, so there are 12 vertices. In the “face-first” method, we are counting “face-edges”: each of the 6 faces has 4 face-edges, for a total of 24; but two face-edges make an edge of the cube, so again we have 12.

I didn’t talk about some of the really interesting polyhedra, so let’s try that. Take the regular dodecahedron, which consists of 12 pentagons. Obviously it has 12 faces; how many edges would it have? Each of 12 faces has 5 edges, for a total of 60 face-edges; each of those is shared by two faces, so there are 60/2 = 30 edges. As for vertices, we need to know that three edges (and three faces) come together at each vertex. So the number of vertices is 1/3 of the number of edge-ends, which in turn is twice the number of edges: there are \(30\times 2 \div 3 = 20\) vertices.

If you have a picture, you could organize the counting the way I initially suggested. There are 5 edges on top, 5 going down from there, ten around the “equator”, then another 5 going down and 5 on the bottom, for a total of 5 + 5 + 10 + 5 + 5 = 30 again. And there are 5 vertices on top, 10 around the “equator” and 5 on the bottom, for a total of 20.

For similar answers about counting, see

Edges, Vertices, Surfaces Cube Edges

Now what if we reverse the question: given the numbers of vertices, edges, and faces, what shape is it?

Determining Polyhedron Name from Given Description What is the name of the geometric solid having 7 faces, 10 vertices, and 15 edges? Thank you!

This is tricky. I responded:

A question like this will not have a single answer;most polyhedra have no special name, and most of those that do are defined by special features such as parallel edges or congruent faces, so they would not apply to the kind of general situation you are asking about. For example, if aprismhas this number of faces, edges, and vertices, and you stretch the bottom, you can change it first to afrustum of a pyramid, and then to anameless distorted prism, without changing the fact that it fits your description!

For a table showing how many “topologically distinct” polyhedra there are for various combinations of vertices and faces, see Numericana. That shows that our example here has five different answers, ignoring the difference I mentioned between a prism and a frustum, for example. Here are a prism and frustum, so you can see that the latter is essentially a squeezed or stretched prism:

But you can look for _a_ familiar polyhedron that fits, rather than a name that applies to _every_ such polyhedron. To do that, you can start by looking for properties of familiar polyhedra in terms of their faces, vertices, and edges. For example, suppose you have aprismwhose base is an n-gon. There are n lateral faces and 2 top and bottom faces; n vertices each on the top and bottom; and n edges each on the top, bottom, and sides. So you have F = n + 2 V = 2n E = 3n Do the same for several other kinds of shapes (pyramids, perhaps regular polyhedra and some others), and see whether what you are given fits any of those.

Alternatively, I could have suggested working backward from the specific numbers given, for each type. If is it a prism, then ignoring the top and bottom bases, there are 7 – 2 = 5 lateral faces, so it would have to be a pentagonal prism. That would, indeed, have 10 vertices (5 on top, 5 on the bottom), and 15 edges (5 on top, 5 on the bottom, and 5 on the sides). So this is the answer.

If that had failed (say, there were 7 faces as before, but 7 vertices and 12 edges), we might next try for a pyramid. That has on face on the bottom and the rest (here, 6) on the sides, making it a hexagonal pyramid. And that would, indeed, have 7 vertices (6 on the base and one at the apex), and 12 edges (6 on the base and 6 on the sides).

But in each case, as I described, there are other, less regular, polyhedra that fit each description. The pentagonal prism and frustum shown above are answers to the original question, as is the following “wedge”, and a couple other shapes I’ve thought of, all of which have 7 faces, 15 edges, and 10 vertices:

Here’s another question without a single right answer — this time, there is no answer at all:

Vertices and Edges of Polyhedra We are doing the Polyhedron Activity. I know the formula V+F-E = 2. Can you help me find the number of vertices and edges for the following polyhedra:heptahedron, nonahedron and decahedron?

This has an entirely different issue: In the last problem, we saw three different heptahedra (polyhedra with 7 faces), showing that the name alone is too general to tell us the rest of the facts. And here’s yet another heptahedron (with different numbers of edges and vertices). So I had to give a negative answer here:

No,there are different kinds of heptahedra, etc., which will have different numbers of edges and vertices. For example, ahexagonal pyramidhas a base and six sides, making it a heptahedron with 12 edges and 7 vertices; but apentagonal prismhas two bases and five sides, making it a heptahedron with 15 edges and 10 vertices. In both cases the formula works: V - E + F = 2 7 - 12 + 7 = 2 10 - 15 + 7 = 2 Polyhedra are very slippery things! The name, and the number of faces it implies, is not enough to fully determine what polyhedron you have.

Now, Leo here mentioned a formula relating vertices, edges, and faces of any polyhedron. That is called **Euler’s Polyhedral Formula** (among other names), and will be the subject of my next post. Knowing the formula, if you know any two of F, E, and V, you can find the other; but knowing only one (in this question, F) is not enough to find the rest unless you have other information.

A recent question asked about one of our explanations of the limit of *x*^{2} (which we have discussed at least five times). This led to a deeper examination of what was said; and as I have looked through this and other pages, I have realized that it would be worthwhile to look at the details more carefully.

The question referred to this page, from 1998:

A Limit Proof Using Estimation

How do I show that the limit of x^2 as x->(-2) is 4, using the delta-epsilon definition of a limit?

Here is the first part of Doctor Rob’s reply:

The definition says that for any epsilon > 0, no matter how small, you can find a delta > 0 such that:

|x-(-2)| = |x+2| < delta implies that |x^2-4| < epsilon

The idea is tostart with what you want to show, that |x^2-4| < epsilon, and to manipulate this until you can get it into the form |x+2| < some expression in epsilon. Then picking delta to be this expression in epsilon will do, and the proof is to work backwards through the steps of the manipulation. In this case:

|x^2-4| < epsilon

<==> |(x+2)^2 - 4*(x+2)| < epsilon

<== |x+2|^2 + 4*|x+2| < epsilon (by the triangle inequality)

<==> |x+2|^2 + 4*|x+2| - epsilon < 0Now you can use the Quadratic Formula to solve for |x+2|, and thus find an upper bound on |x+2| in terms of epsilon. That will be what you choose for delta. One tricky part is that each step needs an implication arrow in one direction (<==) but not necessarily in the other.

Before I look at the new question, we should read between the lines to see if we can follow this, which is not the method I am most familiar with.

First, note the structure of what he is doing: The goal is to show that, given a fact about delta, we can make a conclusion about epsilon; but first we have to find the appropriate delta. So the proof actually has two parts, which shouldn’t be confused. The search (exploration) part looks as if we were solving an inequality; but because of the ultimate goal, we need each step not to *imply* the next, but to *be implied by it*. Note the arrows, <==> and <==, the latter meaning that the new line will imply the one before.

Now, how does he get \(|x^2-4| < \epsilon \Leftrightarrow |(x+2)^2 – 4(x+2)| < \epsilon\)? This is not obvious from what was said; but we can check it by expanding \((x+2)^2 – 4(x+2)\) as \(x^2 + 4x + 4 – 4x – 8 = x^2 – 4\) as claimed. One way to obtain that result (without just guessing) is to make a substitution: We want an expression in terms of \(x+2\), so we let \(u = x+2\), and replace \(x\) with \(u-2\):

\(x^2 – 4 = (u – 2)^2 – 4 = u^2 – 4u + 4 – 4 = u^2 – 4u = (x+2)^2 – 4(x + 2)\).

The benefit of this is that we can use our knowledge of \(|x+2\) in subsequent steps.

How about the next line, which refers to the triangle inequality? This is the fact that, for any *a* and *b*, \(|a – b| \le |a| + |b|\), just as any side of a triangle is less than the sum of the other two sides. Specifically, it tells us that

\(|(x+2)^2 – 4(x+2)| \le |(x+2)^2| + |4(x+2)|\). As a result, if the right side is less than epsilon, we can conclude that the left side is also less than epsilon (since it is even smaller), which is what we will want to conclude when this chain of statements is reversed to make the final proof. This is why we were looking for an expression* greater* than the expression we had.

Now we have a quadratic inequality, \(|x+2|^2 + 4|x+2| – \epsilon < 0\), that we can solve. Doctor Rob suggests using the quadratic formula to do this, and this is where the recent question came in.

Juares wrote,

In this link, http://mathforum.org/library/drmath/view/53357.html , have example:

\(\displaystyle\lim_{x\rightarrow -2}x^2 = 4\):

\(|x+2|^2 + 4|x+2| < \epsilon\)

\(|x+2|^2 + 4|x+2| – \epsilon < 0\)and solve by quadratic formula for any

\(\displaystyle\frac{-b \pm\sqrt{b^2-4ac}}{2a}\)ε

\(\displaystyle\frac{-4|x+2| \pm\sqrt{(4|x+2|)^2-4(|x+2|)(-\epsilon)}}{2|x+2|}\)

He showed more work, but this is enough to see the error. Doctor Rick responded:

Hi, Juares. It looks like you haven’t understood what Doctor Rob meant when he said:

” |x^2-4| < epsilon

<==> |(x+2)^2 – 4(x+2)| < epsilon

<== |x+2|^2 + 4|x+2| < epsilon (by the triangle inequality)

<==> |x+2|^2 + 4*|x+2| – epsilon < 0

Now you can use the Quadratic Formula to solve for |x+2|, and thus find an upper bound on |x+2| in terms of epsilon.”

In your work, you are taking a = |x + 2|, b = 4|x + 2|, and c = – ε, which implies that you are trying to solve the “quadratic equation” |x + 2|x

^{2}+ 4|x + 2|x – ε = 0. That’s not a quadratic equation!When Doctor Rob said to solve for |x + 2|, he meant that you treat |x + 2| as the variable. If we let u = |x + 2|, then the inequality becomes

u

^{2}+ 4u – ε < 0Now solve this inequality for u. That is, solve the related equation u

^{2}+ 4u – ε = 0 for u, which gives the endpoints of the solution interval(s), and decide whether the solution set for the inequality lies between these points or outside them.

Now Juares correctly applied the quadratic formula to solve the equation:

\(\displaystyle\frac{-4\pm\sqrt{16+4\epsilon}}{2} = -2+\sqrt{4+\epsilon}\)This is δ. Let ε = 0.01. Then \(\displaystyle\delta =-2+\sqrt{4,1} = 0,025\)

This was followed by some long calculations presented as an image, so I will not attempt to reproduce it all. The important thing is that he has found a value for delta in general, and then is observing what happens for a specific value of epsilon, which is an excellent way to get a better feel for what is happening.

Doctor Rick answered,

Hi again, Juares. Let me see whether I understand what you have done here.

We are working on solving the inequality u

^{2}+ 4u – ε < 0, where u = |x + 2|. You apply the Quadratic Formula to solve the related equation u^{2}+ 4u – ε = 0, and this gives you the result-2 – √(4 + ε) < |x + 2| < -2 + √(4 + ε)

Now, the quantity on the left is clearly negative (for any positive value of ε), whereas an absolute value (or modulus, as you probably call it), |x + 2|, must be non-negative, so we can further restrict the possible values for |x + 2|:

0 ≤ |x + 2| < -2 + √(4 + ε)

As Doctor Rob said, “The definition says that for any ε > 0, no matter how small, you can find a delta > 0 such that: |x-(-2)| = |x+2| < δ implies that |x

^{2}– 4| < ε.” We have in fact found that:For any ε > 0, we can choose δ = -2 + √(4 + ε)

and then |x – (-2)| < δ implies that |x

^{2}– 4| < ε.So we have found what is needed for the proof.

The rest of what you have written appears to be a check of this result by testing one possible value for ε, namely, ε = 0.1. In this case, we choose

δ = -2 + √(4 + 0.1) = 0.024845…

0 < |x – 2| < 0.024845…You have rounded that last number to 0.025; I have just kept more digits.

What I find, next, is

2 – 0.025 < x < 2 + 0.025

1.975 < x < 2.025

(I used the idea that |x – a| < b means that the distance between x and a is less than b, that is, x is between a-b and a+b.)We want to demonstrate that, for any x in this interval, |x

^{2}– 4| < 0.1. I suppose that you may be trying to use the fact that |x^{2}– 4| = |x + 2| |x – 2| to prove this. I would prefer to note that f(x) = |x^{2}– 4| is an increasing function on [1.975, 2.025], so that it obtains its greatest value at x = 2.025; and in that case, we findIf x = 2.025 then |x

^{2}– 4| = 0.10065If we hadn’t rounded our δ up, we would find that when x = 2 + δ, |x

^{2}– 4| = 0.1; and for any x between 2 – δ and 2 + δ, |x^{2}– 4| < 0.1. This is what we expected to find, if our formula for δ in terms of ε is good.So you have found a valid proof, and I hope I have helped you see that it is valid.

Note that since delta is an upper bound, we don’t want to increase it! So rather than rounding up, it would have been appropriate to round down, to ensure the required conclusion.

Now I want to continue and look at the last part of Doctor Rob’s answer, where he gives an alternative approach (which is what I am familiar with). It is important to observe that there is not just one valid delta in these proofs; not only can Doctor Rob’s first delta be replaced by anything smaller (since that will still imply the required conclusion), but he hasn’t determined that his delta is the largest or best possible number to use. In the next part, he obtains an entirely different formula for delta:

Another tricky part is that there isn't necessarily a unique answer. In this case, you could have proceeded like this instead:

|x^2-4| < epsilon

<==> |(x+2)*(x-2)| < epsilon

<==> |x+2|*|x-2| < epsilon

<== |x+2|*(4 + |x+2|) < epsilon (by the triangle inequality)

<== 5*|x+2| < epsilon (provided |x+2| <= 1)

and so on. At the end, you pick delta to be the minimum of 1 and the expression involving epsilon, and this ensures the "provided ..." part.

Other expressions for delta in terms of epsilon may also work.

Finishing this work, we see that if \(|x+2| < \epsilon/5\), then \(|x^2-4| < \epsilon\), so we can take delta to be \(\epsilon/5\) or anything smaller.

So this version of the proof takes \(\delta=min(\epsilon/5, 1)\). In Juares’ example with *ε* = 0.01, we got *δ* = 0.024845… . This new version finds \(\delta = min(0.01/5, 1) = 0.002\). This is a much smaller value (and therefore also works). The first method obtained a less restrictive value for delta, but both obtain values that provably lead to the required limit.

This same method (though for *x* approaching 2 rather than -2) is discussed in detail by two Math Doctors here:

Epsilon/Delta Definition of Limits

We also saw Doctor Fenton’s version of the same approach on Monday, as part of this answer:

Formal Definition of a Limit

In the following page, I looked at the same problem with a numerical value of delta (similar to Juares’ check):

Definition of the Limit

Finally, I examined a specific feature of this proof here:

Delta-Epsilon Proofs and Arbitrary Epsilon Choice]]>