History of Math: Irrational Numbers

I normally end each month with a post on a mathematical story from history. Since I've spent the last month discussing the Pythagorean Theorem, I thought I'd finish it off with a story from the life of Pythagoras.

Pythagoras was born in 570 BC in Samos (which is now in Vathy, Greece). Throughout his life, he dabbled in philosophy, mathematics, music, and religion (he actually was the founder of the religion known as "Pythagoreanism"). He founded an organization (sort of like a school) where lots of ideas were developed. It is unclear if many mathematical and philosophical ideas of the time were directly from him or from one of his students.

The Pythagorean Theorem was used by Babylonian and Indian mathematicians who preceded him. However, he or one of his students was the first to determine a proof of it. Prior to that, the theorem would just be a conjecture (for an idea to be considered a theorem, it must be proven; otherwise, it is a conjecture).

Pythagoras had a belief that all numbers could be expressed as a fraction or terminating/repeating decimal. This held true until Hippasus, one of his students, tried solving the Pythagorean Theorem with legs 1 and 1. This yielded a hypotenuse of length √2.

1² + 1² = c²
1 + 1 = c²
2 = c²
√2 = c

Hippasus then tried to write this number as a fraction. During his attempts to do this, he ended up proving that the number can not be written in this way (this might not be Hippasus's exact proof, but click here to see a proof of this).

After Pythagoras found out, he was outraged. He did not want to accept the fact that irrational numbers exist. He tried proving Hippasus wrong, but could not do it. Because of this, Hippasus got to take a "boat ride" in the lake, where he was thrown overboard and drowned.

I found it interesting that in that day in age, people reacted this way to the discovery of irrational numbers. Though nobody got killed when complex numbers were found in 1539, there was lots of discomfort. Newton and Euler are both recorded as calling them "impossible" numbers, and the letter i stands for imaginary, since this number could not possibly exist. Gauss was the first to recognize complex numbers as a real thing.

Fun Puzzle with the Pythagorean Theorem

Normally, I spend the third week of the month discussing a practical application of a cool mathematical concept. However, my post from last week fit that standard perfectly. So, I chose to give you a practical use of the Pythagorean Theorem, but less on a applicable level and more on a personal level. This is an identity that can be turned into a puzzle for your friends. It can also be used as a proof of the Pythagorean Theorem. Enjoy!

Math in the News: The Pythagorean Theorem in Football

I mentioned last week that I wanted to focus this month on the famous Pythagorean Theorem. Since I usually reserve the second week for a "Math in the News" segment, I was looking for an example of the Pythagorean Theorem in the news.

Though I couldn't find a controversial article about it, I did find a fantastic video that showed its applications to football. And it was done by the National Science Foundation in their news section, so I think it counts. Click here to see the video.

My friends were just joking with me a few weeks ago about how in a sports game, players wouldn't actually be thinking about the math behind what they are doing. It's true that they probably aren't solving for the hypotenuse of a triangle in this dynamic moment. However, they actually do need to take an educated guess as to what angle and speed they should be running at. This does require number sense, which is definitely mathematics. The actual math might not be applied directly, but it is definitely present.

When I talked about connecting game theory with penalty kicks in soccer, this example actually uses the mathematics. Teams do hire statisticians to analyze the skills of the players on both sides, and will give their players advice on what to do. I'm sure coaches of football and other sports do this as well. Why else would they watch the opposing team's previous games to prepare themselves?

I had never thought the Pythagorean Theorem could be applied that way, and it is another reminder of the influence that mathematics has on our lives.

Proof of the Pythagorean Theorem

This month, I decided I would focus on one of the most famous theorems in all of mathematics: The Pythagorean Theorem. Though it sounds kind of dull on the face, it has a lot of extremely interesting things within it.

First, let me go over what it is. If you take a right triangle, and label the measure of its shortest side a, its middle side b, and its longest side c, then you can count on the fact that:

a² + b² = c² 

This can be used in geometry problems, like to find the longest side, or the hypotenuse, of a triangle with shorter sides, or legs, 3 and 4. You would simply plug 3 in for a, 4 in for b, and solve for c.

(3)² + (4)² = c² 

9 + 16 = c² 

25 = c² 

5 = c

The first thing I wondered when seeing this theorem is why it is true. It seems odd that any right triangle’s measurements must fit this criteria. And I never found a proof I liked until extremely recently. So, I would like to share it.

Take the following diagram:

We are going to try to figure out the area of this square. The simplest way to do it would probably be to square the square’s side. This length would be a + b.

(a + b)²

a² + 2ab + b²

Another way we could get this result is to find the area of the inside square, and then the area of the four triangles surrounding it. The inside square has length c, and the outer triangles have a base of b and height of a. So, we would get:

c² + 4(ab/2)

c² + 2ab

Since these two quantities are both the area of the same square, we can set them equal to each other. After simplification, we would get:

a² + 2ab + b² = c² + 2ab

a² + b² = c²

And we end up with our Pythagorean Theorem. I was so intrigued to see that this proof was so quick and simple, as well as pretty interesting.