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## The Pythagorean Theorem

Pythagoras of Samos

Many people have heard of the Pythagorean Theorem (that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse), and many think of it as a²+ b²= c²

It’s pretty clear that many cultures were aware of this relationship.  The concept appears in Egyptian, Babylonian, Indian and Chinese mathematics.  Pythagoras himself may have learned of this relationship during his time in Egypt or Mesopotamia.

Pythagoras eventually ended up in Crotona, in southern Italy and established a community based on certain principles.  The exact nature of the Pythagoreans community is not well known as they were very secretive about their practices.  It is known that the Pythagoreans studied four interrelated subjects very closely -Music, Geometry, Arithmetic and Trigonometry.

The Pythagoreans are sometimes called number mystics because numerology played an important part in their belief system.  Part of their system of belief involved the idea that all numbers or quantities were rational numbers.  That is, they could be represented as a ratio of whole numbers.  Two lengths like 1/2 and 3/4 that could be represented as the ratio of whole numbers were called commensurable because they could be layed off as distances against each other.  Two lengths of 3/4 would equal three lengths of 1/2.  The Pythagoreans believed that this was true for all numbers.

However, being familiar with Geometry and the Pythagorean Theorem, they were familiar with the diagonal of a square with a side of length 1.  So they knew that this distance was the square root of 2, but they couldn’t figure out how to represent this number as a ratio of whole numbers.  Finally, one of the Pythagoreans named Hippasus of Metapontum showed that square root of 2 is not commensurable and cannot be written as the ratio of whole numbers.  He also showed that this is true of the length of the side of a regular pentagon.

Here’s the general proof that square root 2 is irrational (from Ask Dr. Math)

To prove: The square root of 2 is irrational. In other words, there
is no rational number whose square is 2.

Proof by contradiction: Begin by assuming that the thesis is false,
that is, that there does exist a rational number whose square is 2.

By definition of a rational number, that number can be expressed in
the form c/d, where c and d are integers, and d is not zero.
Moreover, those integers, c and d, have a greatest common divisor,
and by dividing each by that GCD, we obtain an equivalent fraction
a/b that is in lowest terms: a and b are integers, b is not zero,
and a and b are relatively prime (their GCD is 1).

Now we have

[1] (a/b)^2 = 2

Multiplying both sides by b^2, we have

[2] a^2 = 2b^2

The right side is even (2 times an integer), therefore a^2 is even.
But in order for the square of a number to be even, the number
itself must be even. Therefore we can write

[3] a = 2f

Using this to replace a in [2], we obtain

[4] (2f)^2 = 2b^2
[5] 4f^2 = 2b^2
[6] 2f^2 = b^2

The left side is even, therefore b^2 must be even, and by the same
reasoning as before, b must be even. But now we have found that both
a and b are even, contradicting the assumption that a and b are
relatively prime. Therefore the assumption is incorrect, and there
must NOT be a rational number whose square is 2.

It is believed that Hippasus died by drowning shortly after his discovery of irrational numbers.  The exact sequence of events that led to his death are not well known:

They say that the man (Hipassus) who first divulged the nature of commensurability and incommensurability to men who were not worthy of being made part of this knowledge, became so much hated by the other Pythagoreans, that not only they cast him out of the community; they built a shrine for him as if he were dead, he who had once been their friend. Others add that even the god became angry with him who had divulged Pythagoras’ doctrine; that he who showed how the icosagon (that is the dodecahedron, one of the five solid figures) can be inscribed within a sphere, died at sea like an evil man. Others still say that the same misfortune happened on him who spoke to others of irrational numbers and incommensurability. Hyamblicus (or Iamblichus of Chalkis), De vita pythagorica 246-247