Significant Digits: Measurements and Exact Numbers

To conclude this series on significant digits, I want to look at some details of their application. Specifically, we will consider questions about how they related to measured values, and to fixed constants.

Measured values

I’ll start with a question from 2011 about measurement and unit conversion:

Changing Units ... and Significant Figures?

Suppose I measured the length of a table and got 2 feet, which is 60.96 cm.

My teacher says I need to round my result according to the number of significant figures in my measured data. Well, the first measurement has one significant digit. The second has four significant digits! So, how many significant digits are there in the above measurement?

I think that when I change from feet to centimeters, the number of significant figures changes, and I have to re-write my result according to that unit. But I have searched Wikipedia for clarity on significant digits and unit conversion, to no avail.

This called for a reminder of what significant digits are all about, before getting into the question of conversion. I replied:

It looks like you've missed the central concept of significant digits (which is not uncommon, if they are not taught in the right context).

What you've done is just to WRITE the quantities "2 feet" and "60.96 cm" with different numbers of significant digits. They don't actually *have* that many. Essentially, you are "lying" about the measurement. Significant digits are supposed to represent how accurately something was MEASURED.

Significant digits can be introduced just by showing a number and asking how many significant digits it has; most of what we have looked at so far has been in such a context. But here, we are originating the numbers, so we have to decide how to write them in order to represent how accurate (I should say, precise) the measurement is.

When you write "2 feet," with only one significant digit, you are implying that you don't know how many inches; the length might really be anything that rounds to 2, from 1.5 feet to 2.5 feet. I very much doubt that that is what you did. More likely, you measured with a ruler or yardstick, and found that it was 2 feet TO THE NEAREST 1/8TH INCH, say -- that is, if the ruler is marked with eighths of an inch, you couldn't be sure it wasn't 2 feet 0 inches and 1/16th of an inch, but you know it's not more than that. So in order to express this, you should say you measured it as 2 feet, 0.0 inches.

(Fractional measurements like this are really hard to express in terms of significant digits, and in reality people making such measurements would just say, "2 feet, plus or minus 1/16 inch" to express this precision.)

The precision of a measurement depends on the instrument (here, the ruler) that was used. I made a guess here; Amie will have to adjust what I say according to her reality. I would like to have known how the measurement was actually done, because, as I said in the last paragraph, this case is not the easiest one to start with when learning these concepts. We don’t yet have an answer to the question of significant digits. I continued:

Now, when you convert (or do any other calculations), you are supposed to use the same number of significant digits that you had to start with. The fact that your calculator or whatever shows 4 digits doesn't mean they are all valid.

If I were forced to use significant digits, I would do the conversion something like this:

     2 feet 0 inches to the nearest 1/8 inch
   = 2*12 + 0 = 24 inches +/- 1/16,
     that is, between 23.9375 and 24.0625.
   
   Call it 24.0 in, since the hundredths are uncertain
   but the tenths are more or less sure.
   This gives three significant digits.

   24.0 in * 2.54 cm/in = 60.96 cm 
   Round this to three significant digits, giving 61.0 cm.

So I first converted the fractional measurement to decimal in order to estimate significant digits. Once I knew that, I could carry out the conversion and retain the same number of significant digits in the result.

What if we had measured in metric initially?

Now, what I'd really do is to measure in centimeters, and avoid all the trouble with fractions. Most likely, the ruler would be marked in millimeters, so it would be accurate to the nearest mm (tenth of a cm), and I would get exactly the same result: 61.0 cm.

So our guess about having three significant digits seems reasonable.

Now, you may have noticed the number 2.54 in there. Does the fact that it is written with three significant digits affect our result? Not really, as we’ll see in the next question.

Exact numbers

Here is the first part of a long question, from 2005:

Exact Numbers and Conversion Statements

I am currently taking chemistry, and something that keeps coming up is exact numbers, which are, by definition, numbers obtained by counting (as opposed to measuring) as in 8 apples.  Also, an exact number could be part of a definition, like a conversion statement such as 1 inch = 2.54 cm. 

Is there a general rule for determining whether or not a conversion statement is exact?  I believe that fairly recently, the inch was redefined to be EXACTLY 2.54 cm.  Are there others that have been defined as exact?  And are conversion statements from metric to English or vice versa the only conversion statements that might be inexact? 

I would really appreciate your help here, since knowing the exactness of a conversion factor plays a huge role in dimensional analysis and rounding an answer to the correct number of significant digits (because an exact number has no limits on significant digits, but an inexact number does).

In a previous post, the concept of exact numbers was mentioned in passing: When a calculation includes a number that is not obtained by measurement (such as the 2 we divide by for the area of a triangle), we treat it as having an infinite number of significant digits, so that it does not restrict the precision of the result. The same is true of conversion factors such as 1 foot = 12 inches (a definition, which can also be thought of as counting); and as was said here, 2.54, though not an integer, is by decree an exact number, part of the definition of the inch as used today. (It was not always so, because the two systems were originally independent, based on separate standards.) See here for historical details:

Millimeters, Inches, Feet, Miles

Jason continued:

And on a more general note, what other types of numbers are classified as exact (e.g., infinite, infinite repeating, fractions, pi)? I know a lot of times it has to do with the context.

An example of my dilemma is in the following problem:

What is the mass of a troy ounce of gold in grams? 
1 troy ounce = 20 pennyweight (exact)
1 pennyweight = 24 grains (exact)
1 grain = 0.0648 gram (NOT exact)

I used dimensional analysis, assuming that the "one troy ounce" I was starting with was inexact (because it is a measurement), so I assumed the answer should have only ONE significant digit, but the answer key said the answer should have 3 significant digits because of the 0.0648. 

I am really lost now, and I don't know how to distinguish between a measurement that is inexact (which I thought a measurement always is) and one that is exact (which I thought they never were, until this problem).  The only explanation I can think of is that the "one troy ounce" wasn't actually measured, they just wanted us to use it for the sake of converting.  PLEASE HELP!!!!!!!!!!!!!!!  Thanks.

Jason is correct in this final supposition. I replied:

The general rule is that a number is exact if you are told it is exact. There is nothing more general that can be said.

Any formula you are given should make it clear whether the constants in it are exact.  Generally, within a system (such as the 20 and 24 in your example), all numbers are exact integers; between systems, such as your 0.0648, the numbers are usually approximate, and that should be indicated by using an "approximately equal" sign, or saying "to three significant digits", or something like that.  But since it is so unusual for a conversion factor between systems to be exact, you can take it the other way and, in that setting, assume a number is inexact unless it is explicitly stated, as it will be for 2.54.

As far as I know, the conversion between metric and U.S. lengths is the only exact one between different systems, though I could be wrong. I don’t know that I have seen many conversion tables that state that within the table. Apart from this one (we have to also include feet to meters, etc.), if the conversion is given as a decimal, it is probably accurate only to the significant digits shown.

I continued:

You asked about "infinite, infinite repeating, fractions, pi".  You didn't really mean "infinite numbers", but rather "infinite decimals". If a decimal is given to you indicating how it repeats, as with a bar over it, then that is exact because you are being told every digit in the number.  If you are just given some decimal places and are not told how the rest of it behaves, then obviously it can't be considered exact.  That is true of pi, which as an irrational number can't be expressed exactly.  

As for fractions, they are exact unless you are told that the fraction is approximate.  For example, the 5/9 or 9/5 in conversion between Fahrenheit and Celsius is an exact fraction.  This means that in such cases you don't need to consider the number of significant digits.  Other fractions, such as 22/7 for pi, are approximations; to use this with significant digits, you would have to determine HOW accurate it is, by writing it as a decimal and comparing to a good decimal approximation to pi.  A decimal approximation makes a lot more sense in such a context.

It turns out that 22/7 is just about as accurate an approximation of pi as is 3.14.

To see how it happens that Fahrenheit and Celsius are related by a simple fraction, see

Converting Celsius to Fahrenheit and Why It Works

Finally:

Now, in your question about "the mass of a troy ounce in grams", there is no number given!  There is no measurement here!  It is "the mass of an exact troy ounce", not "a mass measured as 1.0 troy ounces".  That means you can't take "1" as having a specific number of significant digits; so you have to take it as exact.  So the only number in your calculation that has a specific precision is the last conversion factor, and only that limits the precision of your answer.

Note that in this case, the precision is not a matter of measurement at all, but just of how accurate a conversion factor is available. The whole problem, in fact, is not about measurement, but about obtaining a new conversion factor from known conversion factors. Typically, these are given with enough precision that in ordinary use it will not interfere — that is, it will be more precise than most measurements are expected to be.

Irrational numbers

Now let’s dig a little deeper into Jason’s question about pi and infinite (non-terminating) decimals, by looking at a similar question from 2005:

Significant Digits and Irrational Numbers

How can you determine the significant digits of a non-terminating or irrational number like pi?

I know the rules for finding significant digits of an answer, but what if your answer doesn't terminate?  How do you express the significant digits?

There is an explicit question here, and another implied by the wording of the second paragraph. I started with the former:

Any exact number (not a measurement, but a known number such as pi or sqrt(2)) is considered to have infinitely many significant digits, in itself, since there is no limit to the number of digits we COULD know.  However, in a specific calculation, we can only use some approximation to it, which means using some chosen number of significant digits, which will restrict our accuracy.  The proper thing to do, then, is to be sure to use at least enough sigdigs in your calculation that it will not affect the accuracy of the result.  For example, if you are finding the area of a circle using a radius given as 1.23 meters, you would want to use at least three digits of pi, say 3.14 or 3.142.  Most likely, these days, you would just use the full accuracy of the constant built into your calculator, and not have to worry! Effectively, then, you have infinitely many sigdigs and can ignore it.

So for such a constant, rather than finding how many significant digits it has, we decide how many to use.

You mention your "answer".  You always determine the number of sigdigs there not from what the answer looks like itself, but from the sigdigs in the given numbers, and then round your answer appropriately.  So it doesn't matter whether the answer you calculate terminates or not--you MAKE it terminate at the appropriate place!

My concern here was that, like our first question above, Alex was thinking that getting four (or 12) digits on his calculator meant something in itself. So it was important to add this reminder.

Fractions

I mentioned fractions like 5/9 and 22/7 above; the next question, from 2001, is unique:

Significant Figures in 36 2/3

How many significant figures are there in the area "36 2/3 Acres"?

This type of notation for acreages is common in old deeds, yet with current standards in most jurisdictions in the U.S.A., areas and sub-areas are to be measured in acres to 4 decimal places (i.e. 36.6667 Acres). Any calculations based on the deeded area figure should take significant figures into account. 

My conclusion: 36 2/3 = 36.67 decimal.

Ultimately, the answer is going to be that if you use fractions, you really aren’t implying anything about precision; but it took some thought to answer:

Interesting question! My first thought was to apply what I said here:

  Fraction or Decimal?
  http://mathforum.org/library/drmath/view/58148.html   

To a mathematician, a fraction represents an exact number - effectively, infinitely many decimal places. A decimal number might represent an approximation, whose precision is implied by the number of significant digits.

But it's also possible to take a fraction as a rough number: "about 2/3." In that case, you can estimate its precision by thinking of it as a quotient; since the numerator and denominator each have one significant digit, it could be taken as having one significant digit, so we would take it as 0.7. Since in addition it is not the number of significant digits, but the number of decimal places that counts, and since the whole part of a mixed number can be seen as a precise number (constrained as it is to be a whole number), your number would then be 36.7, with three effective significant digits and one decimal place.

But legally, I would think the real issue would be to determine the intent of the original deed, rather than to legalistically count digits in the number as written. I doubt that precision was even considered when it was written; so the only reasonable thing I can see to do is to take it at face value as an exact number, and then rewrite it according to modern standards as 36.6667 rounded to four decimal places. Other considerations might have to be taken into account, such as the accuracy of measurements used to calculated that figure, but I can't see any better answer given what you have told me. Is there any law dictating how to interpret old deeds?

Math and law are not the same thing!

Is pi 3.14?

Let’s close with one final question from 1998 about an irrational number:

Does Pi Equal 3.14?

In my math class pi has always been 3.14. We use this to find area, volume, and any other equations using pi. If pi has an infinite number of digits, how can 3.14 be exact? I am very concerned about our society using 3.14 as pi when it has an infinite number of digits. Anyway, I was just wondering about this.

Doctor Rob took this:

You only need to use a value of Pi with as many significant figures as the measurements you used for other values in the same equation.  

For example, if you wanted to measure the area of a circle, and the radius was measured as 36.2 feet (to three significant figures), the actual radius might be anywhere in the interval 36.15 < r < 36.25.  Then the area would be Pi*1306.8225 < A < Pi*1314.0625. Even if we used 15 decimal digits for Pi, all we would know about the area is that

  4105.50396554585 < A < 4128.24909635784.

If we use just 3.14 for Pi, we get 

  4103.42265 < A < 4126.15625.

In either case, we can't tell what the fourth significant digit of A is, and the third digit is probably a 1, but maybe a 0, 2, or 3. The extra digits of Pi don't help unless you make more accurate measurements.  

The rule of thumb is to use the same accuracy for your constants as for your least accurate measurement, and the answer will have the same accuracy. In this example, 3.1400000*36.200000^2 = 4114.7816 (exactly), but if we round off to three significant figures, we get 3.14*36.2^2 = 4110 as the best thing we can write for the area.

Don't worry about society - engineers and mathematicians are well aware of this situation! 

If you want to use more decimals, you can do so, but your answers will still only have the accuracy of the least accurate quantity in the formula. We usually used 22/7 when I was in school, which is about the same accuracy as 3.14. If you want a better value, you can use 3.1416, or 3.14159, or 355/113, or 3.14159265, depending on the accuracy of your inputs.

If the measurements are ideal, and hence exact to infinitely many decimal places, I would leave the answer in the form 100*Pi (if r = 10, say).  The context of the problem should make it clear whether the quantities are exact or only approximate.

Again, we decide how much precision we need, and use that much precision in our constants. In many school situations, you aren’t going to get any better than three significant digits, so why stress your memory? And if you are not allowed to use a calculator, 22/7 might be easier to use, with no loss of precision. But as I said before, with modern calculators, you can just use the \(\pi\) button and never worry about it.

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