Everyone knows that 0, 2, 4, 6….are even, whereas 1, 3, 5, 7, 9….are odd.\u00a0 Negative numbers can also be even or odd:\u00a0 -8 is even, whereas -7 is odd.\u00a0 Formally, the mathematical\u00a0definition of “even” is as follows:<\/p>\n
2p, p is any integer.\u00a0\u00a0 The integers are\u00a0 {….-3,-2,-1,0,1,2,3….}.<\/p>\n
The definition of odds:<\/p>\n
2q+1, q is any integer.<\/p>\n
Therefore, 2(-11) = -22 is even.\u00a0 On the other hand, 2(-8) + 1 = -15 is odd.<\/p>\n
An even can’t divide\u00a0(without a remainder)\u00a0into an odd: \u00a0every even number has 2 as a factor, and 2 won’t divide into an odd number (by definition).<\/p>\n
On the other hand, an odd can divide into an even.\u00a0 3, for instance, divides into 12.<\/p>\n
Here’s a fun fact:\u00a0\u00a0the\u00a0square of an odd is odd.<\/p>\n
Proof:\u00a0 assume the odd is 2t + 1.\u00a0 Then its square is (2t + 1)2<\/span><\/sup>.\u00a0 Multiplying by the foil method (see my post on foil here<\/a>):<\/p>\n (2t + 1)2<\/span><\/sup>=(2t + 1)(2t + 1)=4t2<\/span><\/sup> + 4t + 1.<\/p>\n Notice:<\/p>\n 4t2<\/span><\/sup>+ 4t + 1 = 2(2t2<\/span><\/sup> + 2t) + 1.<\/p>\n By definition, 2(2t2<\/span><\/sup>+ 2t) + 1 is an odd number: \u00a0it is written in the form 2(integer) + 1.<\/p>\n The nuances of even and odd can reveal some surprising discoveries, as we’ll see in future posts:)<\/p>\n