By Jack Long 龍眾

Our life is surrounded by numbers, to be more specific, integers. “I have 3 courses today.” “Apples cost 15 dollars per 2 kilograms.” In the latter case, 1 kg of apples costs 7.5, or 15/2 dollars. This kind of number expressed as ratio of two integers are called rational numbers. Research shows that integers and rational numbers actually came from daily life’s counting. They were just symbols invented by our ancestors to make records of the amounts of objects thousand years ago.

Around fifth century BC, ancient Greece. Pythagoras of Samos, a famous mathematician and philosopher, set up the so-called Pythagorean School. Influenced by religious and daily experience, his theory stated that everything in the universe is made out of natural numbers. And ratio can be used to express the relationship between any two things — from the rules of planetary motion to the arrangement of musical notes [1]. Scientists now believe that the Pythagorean scholars arrived at this thought based on their belief that there are infinite number of rational numbers. Accordingly, it is reasonable for Pythagoras to make the assumption that an infinite amount of numbers is enough to describe the whole world [2].

A question about this was quickly raised by Hippasus of Metapotum, one of Pythagoras’ students. He noticed that the length of diagonal of a square, of which each side has the length of 1, might not be represented by ratios of two integers. According to Pythagoras’ own theory, this number should be equal to the square root of 2. Hippasus tried hard to express this number as a ratio of integers, but in vain [3]. It was the first time that someone noticed that Pythagoras might be wrong. Hippasus then took a bold move. Instead of giving up on this annoying number, he tried to prove that it didn’t equal to the ratio of any two integers.

Although we don’t know much about his method, most of mathematicians believe that he actually succeeded. The proof itself was not too complicated as any undergraduate student in math can proof it nowadays. But for Pythagoras and his followers, the result was unacceptable. This discovery did not only prove that Pythagoras’ concept of “all is (rational) number” was wrong mathematically, it shook the philosophy behind his entire explanation of the world. Pythagoras thus ordered Hippasus to keep this result as a secret. But Hippasus was too excited to be quiet – imagine a physics student rigorously proved that Einstein was wrong (in math nothing counts but rigorous proof), it must be impossible for him to not tell anyone about his breakthrough. The wrath of Pythagoras burst out when he knew Hippasus “revealed” the result to some of his close friends. It is generally believed that this young student was then condemned to death by being thrown overboard during a sea voyage [1-4].

The murder did not settle things down. Not long after Pythagoras’ demise, the idea of irrational numbers quickly took its position in the world of mathematics. Not only in Greece, mathematicians in India also discovered and announced the existence of irrational numbers decades later. Euclid, one of the most important mathematicians in human history, proved √2 is irrational in 3^{rd} century BC, which was recognized as the first formal proof of this result by modern mathematics [1]. More complex irrational numbers were discovered and carefully studied since then. Important irrational numbers such as π and *e*, nowadays play an important role in mathematics study and research.

Although the details of the work done by Hippasus is not clear today, mathematicians still consider him as the first one who discovered irrational numbers. He was admired not only for his work, but also for his spirits of curiosity, perseverance and questioning the authority, which is precisely the soul of math. As for Pythagoras, on one hand, mathematicians are grateful for his job of clarifying the system of rational numbers and some famous results in various disciplines of math. On the other hand, his arrogance, refusal to new ideas and being terrified to admit making mistakes are considered to be the most harmful characteristics for not just mathematicians, but scientists in every subject.

**References:**

[1] Gaskin, D. (1999). *An Ancient Mathematical Crisis.* Retrieved from https://denisegaskins.com/2008/05/27/ancient-mathematical-crisis/

[2] Taylor, P. (2014). *Discovery of Irrational Numbers. *Retrieved from

https://brilliant.org/discussions/thread/discovery-of-irrational-numbers/

[3] Gupta, H. (2017). *How Were Irrational Numbers Discovered?* Retrieved from

[4] Esoterx. (2014). *Murder by Math: The Irrational Demise of Hippasus.* Retrieved from https://esoterx.com/2014/12/03/murder-by-math-the-irrational-demise-of-hippasus/