What’s the difference between evens and odds?  When you’re a math tutor, you might need more than the obvious answer.

Everyone knows that 0, 2, 4, 6….are even, whereas 1, 3, 5, 7, 9….are odd.  Negative numbers can also be even or odd:  -8 is even, whereas -7 is odd.  Formally, the mathematical definition of “even” is as follows:

2p, p is any integer.   The integers are  {….-3,-2,-1,0,1,2,3….}.

The definition of odds:

2q+1, q is any integer.

Therefore, 2(-11) = -22 is even.  On the other hand, 2(-8) + 1 = -15 is odd.

An even can’t divide (without a remainder) into an odd:  every even number has 2 as a factor, and 2 won’t divide into an odd number (by definition).

On the other hand, an odd can divide into an even.  3, for instance, divides into 12.

Here’s a fun fact:  the square of an odd is odd.

Proof:  assume the odd is 2t + 1.  Then its square is (2t + 1)².  Multiplying by the foil method (see my post on foil here):

(2t + 1)²=(2t + 1)(2t + 1)=4t² + 4t + 1.

Notice:

4t²+ 4t + 1 = 2(2t² + 2t) + 1.

By definition, 2(2t²+ 2t) + 1 is an odd number:  it is written in the form 2(integer) + 1.

The nuances of even and odd can reveal some surprising discoveries, as we’ll see in future posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.