4th December 2019

artofproblemsolving
14

Is the square root of 2 a rational or irrational number?

Proof that the square root of 2 is irrational. Assume is rational, i.e. it can be expressed as a rational fraction of the form , where and are two relatively prime integers. Now, since , we have , or . Since is even, must be even, and since is even, so is .

Is the square root of 2 A whole number?

A proof that the square root of 2 is irrational. Let's suppose √2 is a rational number. Then we can write it √2 = a/b where a, b are whole numbers, b not zero. We additionally assume that this a/b is simplified to lowest terms, since that can obviously be done with any fraction.
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