Applications of Pythagoras’ theorem

In F.2 as part of a class assignment, I tried to present the applications of Pythagoras’ Theorem. If you try to do something, you fail. In my case I simply copied examples from the textbook under the heading “Applications of the Pythagoras’ Theorem”, which meant menial calculations.

But unlike Elsa, I didn’t let it go: (Mum plait my hair yesterday though!)

I still thought hard about the downer I gave back then, and I think I’m ready for another shot.

Firstly, everyone loves to measure these days. From the length of a fiddly string to peak salesmen performance, we measure like mad.

Now, in kindergarten (to regurgitate a lame joke by a professor I know), we know how to stretch a piece of string along a ruler. Sometimes you can’t. So you mark off points on the rigid curve, and then measure each part between the markings.

You also know how to use a computer. Whoa, Windows Paint. Whoa, I can colour any pixel. Whoa, lines are jagged. But how does a computer know how long that fiddly piece of string is? It can only measure in units of pixels – unlike humans who can rotate the rule any way we like!

So we have Pythagoras’ Theorem: in a right-angled triangle, adding up the squares of the two shortest sides gives the square of the longest one. Terence Tao posted an intuitive proof here.

We use Pythagoras’ Theorem with a square root though. That is, for any right-angled triangle, the longest side is the square root of the sum of the squares of the shorter sides. By cutting any curve up to “almost zero” length, each tiny segment becomes almost a straight line. Summing the lengths of all segments gives us the length of the curve. Just like we did in kindergarten.

How about a curve in 3D space? Let’s think of a cube of length 1, with three of its sides on the x-, y- and z-axes, and find the length of the longest diagonal. But here comes the power of the “square” in the theorem: when you find the diagonal of any square face (it is √2 – more on this later), don’t apply the square root yet. Add the square of 1, which is 1, to it, and you get 3. Therefore the length of the longest diagonal in a cube of length 1 is √3. Amazing, isn’t it?

Then we wonder: how about replacing the square (“to the power 2”) with cube (“to the power 3”), 4th, 5th and higher powers, as well as to the power 1? We get what is called the Lp-norm for p = 1, …, ∞. (p is a parameter.) Yes, there is a norm for infinity – it is taking the maximum value of all the numbers you have in your lap. In the cube above, the L1 norm is 1 + 1 + 1 = 3, and the L norm is max{1,1,1} = 1. These norms of various p‘s pop up in various technical measurements.

Back to √2. What is so special about it? In fact it is not the ratio of any two integers, positive or negative. It shocked the Greeks and extended their number system from integers (without the zero the Indians invented) and ratios of integers (“rational numbers”) to “irrational numbers”. The equation that corresponds to √2 is x2 = 2. If we replace 2 on the right by -1, oops. Is there any square on each with an area of -1? -2? -3? … Can’t we take square roots of negative numbers?

That depends on whether you believe √-1 is a valid number. Somewhere along the line, someone said yes, and complex numbers were born. Now all rational and irrational numbers form real numbers, which are numbers you can mark off a ruler. Complex numbers are real numbers plus any real multiple of i := √-1. (:= means “defined as” and it is also used in some programming languages for assigning values to variables). That means that real numbers are part of the complex number system. If we have an infinitely large sheet of paper and mark a straight line across it as the real number line Re, then intersect it at just one point (the origin, zero) with another straight line Im – Im stands for “imaginary” because we invented √-1 but can’t mark it on a ruler – for the real number line times i, then any point z on the entire sheet of paper is a complex number. For simplicity, make both straight lines at right angles to each other. Project the point z on Re and Im. Then the number can be represented as Re(z) + iIm(z), where Re(z) and Im(z) are called the real and imaginary parts of z respectively. Of course, Pythagoras’ Theorem gives us the distance of z from the origin – it’s called the “modulus”.

Once we venture into forbidden territory, we gain new ground. In fact complex numbers drive the oscillations we see in a pendulum or spring. We also need complex numbers to operate electronic circuits. Complex numbers also simplfiy geometrical operations, because multiplying a number by i is the same as rotating it in the infinitely long sheet of paper by 90 degrees – a right angle. Complex numbers also generate fractals, the beautiful structures you can zoom in indefinitely and still have fine details, unlike pixelated images. People are trying to use it as a new way to model images and structures.

We know that 32 + 42 = 52 and 52 + 122 = 132 – (3, 4, 5) and (5, 12, 13) belong to the class of so-called “Pythagorean triples”. But how about replacing the power 2 in Pythagoras’ Theorem by 3, 4, 5 and beyond? Are there integer (not merely real) solutions to the equation xn + yn = zn for n > 2? That it was impossible was Fermat’s Last Theorem, and Andrew Wiles proved it in 1994. And we only started with the squares.

That’s the power of Pythagoras’ Theorem.