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Square root of 2 is an irrational number.
Square root of 2 is an irrational number.
The scholars of Pythagoras excited by the discovery of the rule the square on a hypotenuse of a right angle triangle is equal to the sum of the squares on the other two sides were dismayed by then discovering that there were no rational number to express the length of a hypotenuse of a right angle triangle which has two equal length sides. Not a problem for geometry as they could just draw it, but a major blow for arithmetic.
Assume square root of 2 is a/b,
a & b both rational numbers and have no common divisors.
a2 / b2 = 2 or a2 = 2b2
Therefore a2 is an even number which means a is an even number because
even numbers have even squares and odd numbers have odd squares.
So we can write
a = 2c and so 4c2 = 2b2
Divide both side by 2
2c2 = b2
By the same argument you can show b = 2d
We have shown that both a and b are even and
a/b = 2
can be written as
2c/2d = 2 or c/d = 1
which is contradictory to original premise, hence showing you cannot express square root 2 as a rational number
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