A Sample of Ask Dr. Math, Part 2: Questions Outside of School

In the first post, I gave a small sampling of questions we’ve had from students, parents, and teachers, all related to school, and discussed how we like to deal with these. But we also get many questions with no direct relation to school. These may come from people who actually use math in their work (computer programming, plumbing, and anything in between), or from students who are trying to deeply understand what they are studying, or who are just curious. Often, of course, these too are from students or teachers, but they are going beyond that role. These are perhaps the most interesting of all. They are really why we are here.

A real-life question

This question arrived in 2014 from a flooring inspector, Tim, asking about how high a floor would buckle due to a given increase in length if the ends are fixed:

Modeling How Much Wood Expands

Say I place a six foot long yardstick wide side down. One end is butted against a wall (fixed object). When the other end of the yardstick is pushed towards the wall, by say 1/4", the middle area rises -- and it rises much, much higher than 1/4". 

I would like to know if there is a fixed formula for this exponential gain of "flat" vs. "up." 

I come across this daily, seeing floors buckle up and away from substrates because they are a tight fit.

(He misused the word exponential; I suppose he was just thinking that the rise is surprisingly high, as in exponential growth.)

Dr. Rick and I both recognized this problem as related to previous questions we had answered, and referred to those in our replies. His old answer assumed a circular arc for the new shape, and did a sanity check by comparing the surprising answer to one obtained by supposing that the board just bends sharply in the middle. The circular arc version does not lend itself to easy calculation, but the sharp bend could be turned into a formula.

There is a way to calculate the height, but it isn't a formula, as such. 
...
For more on this topic, including a way to do a *rough* approximation with a simple formula, see:

  Railroad Track Expansion
    http://mathforum.org/library/drmath/view/62613.html

I had been thinking while Dr. Rick wrote, and found a different old answer (by Dr. Jeremiah) that didn’t actually come up with a method, just a starting point. But I “played” with numbers using a spreadsheet much as Dr. Rick had for his earlier answer, and saw a pattern that suggested an approximate formula he might try. When I saw what Dr. Rick had written, I wrote to propose this formula.

I see Dr. Rick beat me to most of what I was going to say, except that my link to our archives is not quite as helpful:

  Rail Bend in Hot Weather
    http://mathforum.org/library/drmath/view/61465.html 

But I made a spreadsheet carrying out the calculations for a circular arc; and have found experimentally that the rise seems to be quite accurately approximated by a simple formula. I have to study more to try to justify it, but here goes.

For the actual length of the stick, s, and the (small) decrease in the space available, x,

   h = 0.605 sqrt(sx)

But people who enjoy math can’t be fully satisfied with a guess; I continued pursuing my approximation, hoping to justify it. The patient, Tim, didn’t need a proof, but I did. By the end of the day, I could write up a derivation of the formula and send it along, and then, the next day, apply it to some examples.

I've verified my simple approximation, which turns out to be 

   h = sqrt(6sx)/4

This changes my 0.605 to 0.612, which fits better for very small x but not quite as well for larger ones.

I'm sure you don't need to see this, but for my records, here is the derivation of my approximation: ...

A missing formula for a counter-intuitive observation (a half-inch shift can result in almost a 7-inch buckle) can be a great motivation for some fun with math. And we also have a reminder that in the real world, an approximation can be both good enough, and as good as you’ll get. (Be sure, as always, to read the whole story in the original page!)

A curious question

(Well, that last one arose from curiosity, too, but this is pure-math curiosity.)

Within months of my starting with Ask Dr. Math, I wrote this answer, to a question wondering about place value in decimal numbers:

The Oneths Place

Why is there not a oneths place? My classmates and I were wondering.  It is weird that decimals begin with tenths, hundredths, and so on. We appreciate your help.

I love it when children are wondering, especially when a teacher encourages it. So I was happy to think about this, even though I had probably never considered the question before. (Since then, we have had some form of this question at least a dozen more times! Often, I have answered by saying, “Believe it or not, you’re not the first to ask this; if you search our site for the word ‘oneth’, you’ll find this answer …”)

I’m sure my answer is not all that could be said about it, but I managed not only to give the quick answer (oneths are just another name for ones, since 1/1 = 1, and we don’t want two places with the same value), but to relate it to some other perceived asymmetries in the way we write numbers, and bring in signed exponents (which may be beyond the students’ knowledge, or may be something they were just about to learn). I closed with a general comment about curiosity:

I hope that helps. It's interesting how we tend to expect things to be symmetrical. Mathematicians and scientists often expect it too, and when things don't seem balanced, they try to find out why. So keep wondering about things like this.

A philosophical question

Another question we’ve received a number of times involves how a line segment can have a finite length when it is composed of infinitely many points. As a result, we’ve referred to the following question a dozen times or so, in addition to giving fresh answers from different perspectives:

How Can a Line Have Length?

We have a line, OR a line segment, OR a ray (it really doesn't matter). If this particular linear entity (let us say that it is a line segment, for the sake of hypothetical simplicity) consists only of infinite points, all with zero volume, mass, diameter, cross-sectional area, etc., how can it have length?  Why are we able to take a particular line segment and give it a length of 5?  If a substance can only be composed of volumeless points, how can it have volume in and of itself?

Dr. Daniel’s answer (from 1998) touches on several key ideas in philosophical thinking about this topic, making a concise summary without digging in too deeply. (I try to avoid digging too deeply, because of the possibility of cave-ins … .) When I answer related questions, I often refer to that answer, and also to this one:

Point and Line

I'm in high school, but this problem has really nothing to do with school, it's just been bothering me for a while. 

A point has no dimension (I'm assuming), and a line, which has dimension, is a bunch of points strung together. How does something without dimension create something with dimension? 

Going even farther, a point essentially is nothing, because it has no dimension. My question is, How does a bunch of nothings (a point) create a something (a line)?

Here Dr. Jordi got into a long discussion about the problem, perhaps not very effectively in the end. After referring to these two answers, I generally add something like the following as my own brief perspective:

Neither will be a fully satisfying answer, largely because we humans are finite beings, not able to directly observe infinity. (It's amazing enough that we can THINK about infinity, and get some of the answers right!) But the basic idea is that we don't define a line as something made by putting points together; rather, we start by defining a line, and then note that it consists of points. So we can find points on a line, without having to _make_ the line _out of_ points (or make a cube out of stacked squares). 

Note also that we do not define length in terms of points; since 0, the width of a point, times infinity, the number of points, is an indeterminate quantity (we can't assign any one value to it), that would be useless to attempt. So length is an independent concept, defined only in relation to the location of the endpoints of a segment, not to the points that make it up.

In choosing these examples of “curious questions”, I have chosen to avoid  questions with really interesting answers, which deserve fuller treatment on their own! Keep watching, and I’ll get to some of them.

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