Even and Odd Integer Expressions MathBitsNotebook.com Topical Outline | Algebra 1 Outline | MathBits' Teacher Resources Terms of Use   Contact Person: Donna Roberts

Even and odd integers are a familiar concept in mathematics.

Even Integers: {. . . , -4, -2, 0, 2, 4, 6, 8, . . .}
Yes, 0 is considered an even number.

We say that an integer is an even integer if that integer can be written as twice another integer.

Even integer: "a" is an even integer if there exists another integer, "n", such that a = 2n.

Thus, we think of even integers in relation to the expression "2n".

Even integers are integers that are divisible by 2 (with no remainders).

Odd Integers: {. . . , -3, -1, 1, 3, 5, 7, 9, . . .}

We say that an integer is an odd integer if that integer is not an even integer.

Odd integer: "b" is an odd integer if there exists another integer, "n", such that b = 2n + 1 (or also 2n - 1).

Thus, we think of odd integers in relation to the expressions "2n ± 1".

Odd numbers are numbers that are divisible by 2 with a remainder of 1.

Properties of Even and Odd Integers:

1. Show that an even integer added to an odd integer will be another odd integer.

Solution: Let a and b be integers. Let even integer = 2a . Let odd integer = 2b + 1.
2a + 2b + 1 = 2(a + b) + 1
Since integers are closed under addition, we know that a + b is an integer, call it c.
Let a + b = c.
2(a + b) + 1= 2c + 1 which represents the format for an odd number.

2. Write an expression to represent the sum of two even integers. What conclusion can be made?

Solution: Represent the even integers by 2n and 2m.
2n + 2m = 2(n + m)
Since the integers are closed under addition, we know that n + m is an integer.
Let n + m = p
2(n + m) = 2p
Since 2p is the format for an even integer, the conclusion is that the sum of two even integers is another even integer.

3. Show that the sum of two odd numbers is an even number.

Solution: Let a, b and c be integers. Let odd integers = 2b + 1 and 2c + 1.
2b + 1 + 2c + 1 = 2(b + c) + 2
The integers are closed under addition, so b + c is an integer.
Let b + c = a
2(b + c) + 2 = 2(a) + 2 = 2(a + 1)
Again, since integers are closed under addition, we know that a + 1 is an integer, call it c.
Let a + 1 = c.
2(a + 1) = 2c which represents the format for an even number.