The Length of the Hypotenuse

 The Length of the Hypotenuse

By Bobby Neal Winters

It is all about language.

And I guess I ought to expand that “it” a little to better make my point.  “It” is “what we know” and “how we think.”  It is all about getting the right words and the right share experiences to go with those words.

This may sound strange coming from a mathematician, but mathematics is just language.  It is a peculiar language with very strange words.  Some of these words we refer to as numbers.

This revelation hit me about the time my oldest grandson was learning to count.  The counting numbers were words that all just came out in the same order: one, two, three, four, and so on.  Whenever grandson stopped counting, it wasn’t because he didn’t understand the math; it is because he didn’t know the next word.  For some it comes after 29 and for others after 999,999,999,999. 

Language has pressure on it to become more nuanced as our experience of the world gets greater.  One can count out pies as one, two, three, but when it comes to dividing the pie, the world gets a little more complicated.  If you have one pie to go among 5 people, what do you do?  Well, you could cut it into fifths--if you are that good with a knife.  More likely you will cut it into eight pieces so that there will be three pieces left over you can sneak back after later in the evening.  But I digress.  You have to invent fractions.

At that point, we have a subtle shift of our mental context.  We shift from looking outward in an unbounded way to looking inward in an unbounded way.  You can cut a pie in half; you can cut the halve in half to get quarters; you can cut the quarters in half to get the eighths I was talking about.  While in practice you will soon get pieces too small for a hungry stomach to work about, in principle this can go on forever.

The Greeks shifted from thinking about pies (or maybe moussaka) to line segments. They were big on geometry as you recall.  They did--in a way a bit different from us--associate numbers to geometry.  To put it in a modern way, they thought about the lengths of lines.

Then one day they started thinking about the length of the hypotenuse of a right triangle whose legs had length one.  They used the mathematical language differently than we do, but when you translate it, they discovered that you cannot express the length of that hypotenuse as a fraction of whole numbers.  New words had to be invented.

The Greeks and the rest of the world for most of human history suffered under the handicap of not having a good way to write numbers.  It was embarrassingly late that the decimal way of denoting numbers turned up in the West.  The change for math was like when we changed from writing words as hieroglyphs to writing them in the letters of an alphabet.

When we refer to our number as the square root of two, it is not only precise, it is exact, but it is not very useful in many contexts.  We can say 1.4, but that is about 2 one hundredths too small; we can say 1.4142136 that is just a tiny bit too big.  But either of these ways of presenting the number will be more useful in a particular context than just saying the square root of two.

Ultimately the most honest way we can present a number like the square root of two is at an estimate plus or minus a margin of error with the margin of error as small as we can get it.  For example, the square root of two is 1.4142 plus or minus 0.00002,  

It is not exact, but it is true in the sense we are letter people know we are off by a little bit.  We are using our language to point at the truth as precisely as we can while letting the world know where we are uncertain and by how much.

This system of language was created by human beings struggling with Nature in order to determine Truth.  It relies not only on ever more precise words but honesty not only to others but most importantly to oneself.

Bobby Winters, a native of Harden City, Oklahoma, blogs at redneckmath.blogspot.com and okieinexile.blogspot.com. He invites you to “like” the National Association of Lawn Mowers on Facebook. )