The Square Root of Two

The Square Root of Two cannot be expressed as a ratio between two integers; that is it is an irrational number. Proof Let us assume the converse that the Square Root of two can be expressed as the ratio of two numbers a and b. In that case there must be an a and b … Continue reading “The Square Root of Two”

The Square Root of Two cannot be expressed as a ratio between two integers; that is it is an irrational number.

Proof

Let us assume the converse that the Square Root of two can be expressed as the ratio of two numbers a and b. In that case there must be an a and b which are not both even as if they are both even we get the same ratio by dividing each by two until at least one of them is odd. Therefore if

a/b = SQRT (2) then
a2/b2 = 2
a2= 2b2 and
so a2 is even and therefore a is even; so
a=2c for some c . Thus
a2= (2c)2 and
a2 = 4c2
2b2 = 4c2
b2 = 2c2
So b2 is even and therefore b is even.
Therefore a and b are both even which is a contradiction.

Therefore the Square Root of Two cannot be expressed as the ratio between two integers.

QED.

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