The Pythagorean Theorem: More Than Just a Math Formula

Let’s start with something everyone has at least seen once: the Pythagorean theorem. The theorem shows that in a right triangle, the square built on the longest side, also known as the hypotenuse, is exactly equal to the sum of the squares built on the other two sides. In other words, $a^2 + b^2 = c^2$.

The sum of the area of two squares whose sides are the two legs (blue and red) is equal to the area of the square whose side is the hypotenuse (purple).

It is an infamous topic in every high school classroom. In addition, it is very common that someone asks “Why does this matter?” or something along the lines of “When will I ever use this in real life?”.

Let’s rewind a bit. The Pythagorean theorem is one of the oldest and most famous ideas in all of mathematics. Actually while doing a lot of science, the Pythagorean theorem is an elementary piece of math. In engineering, it becomes so obvious that you don't even think about it anymore. That's how common it is in real life, after all.

But something is up with it. The Pythagorean theorem was already known long before Pythagoras. The math is over $2500$ years old and wasn’t even discovered by Pythagoras first. The theorem appears in ancient Babylonian tablets dating back to $1800$ BC in addition to The Zhoubi Suanjing, an ancient Chinese work of mathematics. Big man Pythagoras did the proper marketing and got the branding rights.

This brings us towards the key question from the beginning. Why does it matter? Well, ancient civilizations often used the technique for land divisions and constructions. Nowadays, architects use it to measure diagonals and make sure buildings don’t collapse. Video game designers use it to calculate distances on the screen. GPS satellites use it every time you check Google Maps. Often, sometimes without knowing it, you’re relying on this simple relationship of squares and triangles. Even while working on engineering problems, the Pythagorean theorem always may sneak into the setting and it might not even be that obvious. This is the reason why it is a common thing in teaching.

As it is a very famous result, there are hundreds of different proofs of the theorem. As the theorem is such a central idea, many people across time kept finding new ways to see it.

A simple example of a proof starts with a square with length a+b where a square of length c is located inside it.

Basic geometric proof of the Pythagorean Theorem.

The area of one triangle with height $a$ and basis $b$ is $\frac{ab}{2}$. The area of the big square is $(a+b)^{2}$ and the area of the small square is $c^2$. Knowing that the large square equals the area of the four triangles and the small square results in the Pythagorean theorem. The following formulas show it:

$$ A_{large square}=A_{small square}+A_{triangles} $$ $$ (a+b)^2=c^2+4(\frac{ab}{2}) $$ $$ (a+b)^2=a^2+ab+ab+b^2 $$ $$ a^2+2ab+b^2=c^2+2ab $$ $$ a^2+b^2=c^2 $$

Hooray! We found the Pythagorean theorem. Using a trapezoid instead of a square, a similar proof was published by James A. Garfield before becoming U.S. President in $1881$.

Considering the figure, the area of the trapezoid is equal to the area of the three rectangles. This leads to the following proof.

Diagram to explain Garfield's proof of the Pythagorean theorem

$$ A_{trapezoid}=A_{triangles} $$ $$ (a+b)\frac{(a+b)}{2}=\frac{(ab)}{2}+\frac{c^2}{2}+\frac{(ab)}{2} $$ $$ \frac{a^2+2ab+b^2}{2} =\frac{2ab}{2}+\frac{c^2}{2} $$ $$ a^2+b^2=c^2 $$

A more involved proof was constructed by Euclid, also known as the father of geometry. It is known that Abraham Lincoln taught himself the theorem from Euclid as a way to sharpen his reasoning.

Concluding this first blog post, it is clear that the Pythagorean theorem isn’t just a fact about triangles. There is much more behind it. Knowing that a square root ⋅ is the opposite operation of a square means that for a triangle with $a=b=1$ the hypotenuse is $\sqrt{2}$. But then, what on god’s earth is this root of $2$? As roots of whole numbers show often up in geometric problems it is no surprise that ancient civilizations were also struggling with finding values for these roots. It’s ok if this is not really clear to you right now it took our ancient civilization homies also some time to figure out. The square root of $2$ was the first crack in the belief that the universe could be perfectly explained with whole numbers and fractions. It is such a huge and underestimated aspect of math which is maybe is a topic for a future post.