Page 4 - Science Focus (Issue 016)

The

Square Root of Two

– at the Cost of a Life

Our life is surrounded by numbers, to be more mathematicians believe that he actually succeeded. The
specific, integers. “I have 3 courses today.” “Apples cost 15 proof itself was not too complicated as any undergraduate
dollars per 2 kilograms.” In the latter case, 1 kg of apples student in math can proof it nowadays. But for Pythagoras
costs 7.5, or 15/2 dollars. This kind of number expressed as and his followers, the result was unacceptable. This
ratio of two integers are called rational numbers. Research discovery did not only prove that Pythagoras’ concept
shows that integers and rational numbers actually came of “all is (rational) number” was wrong mathematically,
from daily life’s counting. They were just symbols invented it shook the philosophy behind his entire explanation of
by our ancestors to make records of the amounts of the world. Pythagoras thus ordered Hippasus to keep this
objects thousand years ago. result as a secret. But Hippasus was too excited to be quiet
Around fifth century BC, ancient Greece. Pythagoras – imagine a physics student rigorously proved that Einstein
of Samos, a famous mathematician and philosopher, set was wrong (in math nothing counts but rigorous
up the so-called Pythagorean School. Influenced by proof), it must be impossible for him to not
religious and daily experience, his theory stated that tell anyone about his breakthrough. The
everything in the universe is made out of natural wrath of Pythagoras burst out when
numbers. And ratio can be used to express he knew Hippasus “revealed” the
the relationship between any two things — result to some of his close friends. It
from the rules of planetary motion to the is generally believed that this young
arrangement of musical notes [1]. Scientists now student was then condemned to
believe that the Pythagorean scholars arrived death by being thrown overboard
at this thought based on their belief that there are during a sea voyage [1-4].
infinite number of rational numbers. Accordingly, it The murder did not settle things
is reasonable for Pythagoras to make the assumption down. Not long after Pythagoras’ demise,
that an infinite amount of numbers is enough to describe the idea of irrational numbers quickly took its
the whole world [2]. position in the world of mathematics. Not only in Greece,
mathematicians in India also discovered and announced
A question about this was quickly raised by Hippasus
of Metapotum, one of Pythagoras’ students. He noticed the existence of irrational numbers decades later. Euclid,
that the length of diagonal of a square, of which one of the most important mathematicians in human
rd
each side has the length of 1, might history, proved √2 is irrational in 3 century BC, which was
not be represented by ratios recognized as the first formal proof of this result by modern
of two integers. According to mathematics [1]. More complex irrational numbers were
Pythagoras’ own theory, this discovered and carefully studied since then. Important
number should be equal to irrational numbers such as π and e, nowadays play an
the square root of 2. Hippasus important role in mathematics study and research.
tried hard to express this Although the details of the work done by Hippasus
number as a ratio of integers, is not clear today, mathematicians still consider him as
but in vain [3]. It was the first the first one who discovered irrational numbers. He was
time that someone noticed that admired not only for his work, but also for his spirits of
Pythagoras might be wrong. curiosity, perseverance and questioning the authority,
Hippasus then took a bold which is precisely the soul of math. As for Pythagoras,
move. Instead of giving up on on one hand, mathematicians are grateful for his job of
this annoying number, he tried to clarifying the system of rational numbers and some famous
prove that it didn’t equal to the ratio of results in various disciplines of math. On the other hand,
any two integers. his arrogance, refusal to new ideas and being terrified
to admit making mistakes are considered to be the most
A l t ho u g h w e don ’ t k no w
much about his method, most of harmful characteristics for not just mathematicians, but
scientists in every subject.