**Definitions:**

- An integer is called
*even*if and only if it is divisible by 2, that is some integer is even if for some other integer the statement is true. - An integer is called
*odd*if and only if it is not divisible by 2, that is some integer is odd if for some other integer the statement is true. - A number is called
*rational*if it is in the form of where and are integers. Further, it is said to be in lowest terms if and are not commonly divisible by any integer save the identity 1. A number that is not rational is called*irrational*. - A number is called a
*square root*if it satisfies for some number where for our purposes is a positive integer. “The square root of ” is denoted by .

**Theorem 1** The product of two even integers is even.

**Proof** For any two even integers and , we have

which is even. QED.

**Theorem 2** The product of two odd integers is odd.

**Proof** For any two even integers and , we have

which is odd. QED.

**Corollary 1** If the square of a number is even, then the number itself is even. If the square of the number is odd, the number is likewise odd.

**Theorem 3** is irrational.

**Proof** Assume that is rational and in lowest terms, such that

or

that is is even. Then, by Corollary 1, is even as well. That is,

for some integer . Therefore,

which implies

and thus is even and therefore is even, again by Corollary 1. Therefore, both and are even which contradicts our initial assumption that was rational. Therefore, the assumption was false and is irrational via proof by contradiction. QED.