Essay on the square root of 2

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Although it wasn't Pythagoras himself who discovered the square root of two and the changes it caused to Ancient Greek mathematics as well as the future of mathematics, his follower did and because of this he is mainly accredited. It is not believed that Pythagoras himself who revealed this mathematically changing idea because it went against his philosophy that all things are numbers. It was in reality a Pythagorean philosopher Hippasus who was able to demonstrate the irrationality of the square root of 2. The legend is that after doing so he was killed by other Pythagoreans who were scared and frantic by the thought of an irrational number. Pythagoras' follower most likely used a geometrical proof when he was first discovering the irrationality of the square root of two. This proof uses Pythagoras' theorem that in a right triangle, a2 + b2 = c2 .

If a=1, and b=1 then 2= c2. Then c=√2 and then you must find c. However there is no rational number which satisfies this requirement. The new idea of irrational numbers changed Ancient Greek mathematics because it created two divisions including no longer just numbers (or algebra) but to geometry. It has been called a "scientific event of the highest importance." Geometry deals with distances and magnitudes and algebra focuses more on numbers. There was a crisis caused by this because in was not possible to express the quantities of irrational numbers. This gave way to future mathematicians to be able to use irrational and imaginary numbers to prove problems and discover new theories.

Two was the first irrational number to be discovered but is not the only irrational number there is. Any real number is irrational if it cannot be written as a fraction (a / b) with both a and b as integers and b not being zero. The proof used to prove the irrationality of the square root of two as well as any other number is as follows:

1. Assume that √2 is a rational number, meaning that there exists an integer a and b so that a / b = √2.

2. Then √2 can be written as an irreducible fraction (the fraction is shortened as much as possible) a / b such that a and b are coprime integers and (a / b)2 = 2.

3. It follows that a2 / b2 = 2 and a2 = 2 b2.

4. Therefore a2 is even because it is equal to 2 b2 which is obviously even.

5. It follows that a must be even. (Odd numbers have odd squares and even numbers have even squares.)

6. Because a is even, there exists a k that fulfills: a = 2k.

7. We insert the last equation of (3) in (6): 2b2 = (2k)2 is equivalent to 2b2 = 4k2 is equivalent to b2 = 2k2.

8. Because 2k2 is even it follows that b2 is also even which means that b is even because only even numbers have even squares.

9. By (5) and (8) a and b are both even, which contradicts that a / b is irreducible as stated in (2).

The discovery of the square root of two was a frightening thing to the Pythagoreans however we can now thank them for our wonderful math classes today. They gave new idea to the possibilities of numbers and what kind of numbers are out there.