Archive for September 6th, 2020

Proof Of Existence Of Square Root Of Two For Ten Year Old (Take That, Euclid!)

September 6, 2020

Yesterday my ten year old daughter cried like I have not seen her cry since her grandmother died. Tears were running down her cheeks:”Cela n’a aucun sens!” It makes no sense! It was a telling choice of words: the Greek mathematicians made the exact same observation, they called such numbers “irrational”. I reassured her: she was brutally confronted by what drove ancient Greek mathematicians crazy: the square root of two.

How to define square roots algebraically is simple: x is the square root of a if and only if: xx = a… No problem if a is one, four, nine… But put a = 2, and a mystery arises: one can write down numbers ever closer to x = square root of two, but then what? Their squares are never quite two… Worse: ancient Greek mathematicians knew (positive) integers, and their ratios, say m/n, where both m and n are integers. They could demonstrate, in a few lines, that square root of two was not a fraction. 

Mathematics is the way of thinking that is most powerful in some situations. For example, as I wrote: xx = 2, I used no less than three mathematical notions that the ancient Greeks mathematicians did not have: the notion of equation, the equal sign, and the notion of unknown x.  

When the Greeks were confronted to this, they realized that some things existed that should not exist, according to their (mathematical) system of thought. Obviously, their mathematics came short. They discovered “irrational” numbers, the hard way. Further reasonings were halted by the rise of fascism in the Hellenistic and then Roman world. Do we have similar situations nowadays? Obviously yes. Nonlocality in Quantum Physics is an obvious example.

There is a mysterious relationship between numbers and geometry. Draw a line: that’s the x axis. Some distance mark one (1); it is the unit of distance. It does more: it represents the number one. Draw the perpendicular axis, traditionally called the y axis. There too, mark a one. Use the proverbial compass to make the units on the x and y axis the same length. Now one is facing a two dimensional plane. It naturally defines a unit of area, the square of sides equal to one. Cut that square in two to get the area of surface ½, etc… A bit of playing around shows that any square of side s has area ss…

Consider the rectangular triangle from the origin with sides one. The longest side can be used as the base of a square. A quick look shows that this square has area two (2). Thus its side is square root of two. 

This demonstrates that the square root of two exists. 

Ancient Greek mathematicians went that far. And that drove them nuts, as, for them, any “number” had to be of the form m/n, with m and n integers… And the square root of two was not such. 

The reasoning above is mine: it was designed to be understood by a very curious ten year old who is not fanatical about mathematics. So the Pythagorean theorem is demonstrated, in a particular case (the general proof is a somewhat confusing generalization; similar ad hoc geometrical proofs work for other numbers, for example square root of 5). 

Patrice Ayme

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Why the Greeks knew square root of two was no fraction (= “rational” number):

If (m/n)^2 = 2, with xx = x^2 and m and n having no common factors… Then:

mm = 2 nn… Thus m is even (if m is odd, mm is always odd). So m = 2u. This forces n to be odd, because by hypothesis, m and n don’t have common factors, so can’t both be divided by 2…

Hence, n = 2v + 1, for some integer v… Plugging back in the initial equation mm = 2 nn we get: 2u2u = 2 (2v+ 1) (2v + 1). Dividing by 2, we get:

2uu = (2v + 1) (2v +1) = 4vv + 4v + 1…

Now that latter equation is impossible: the left side is even, and the right side is odd… Thus the initial hypothesis, (m/n) (m/n) = 2 is impossible…


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