How many categories can we can put numbers into?

Question

I am not sure of the answer here (I assume the answer is no) but I will ask the question the best way I can. I have heard of even numbers and odd numbers. Are those categories [even numbers & odd numbers] such that what we can put all numbers into one or the other category? That is, can there be a number in existence that is neither even or odd? For example if I know a varible x is not an odd number does that necessarily mean the varible x MUST be an even number? One definition of odd number is 2k +1 and an even number is defined as 2 times a number 2k. Is there a Z out there that cannot be simplified to 2k or 2k+1? [My thought is no about this: I don't see fractions as either even or odd; I don't think of decimals as even or odd.] I would like some clarity on this please. Thanks in advance.

Answer 1

By number, I assume you mean integer, i.e., a whole number that is positive, negative, or zero. If this is the case, then yes: every integer is either odd or even.

Proof. By definition, an integer is odd if it can be written as $2k + 1$ for some integer $k$ and even if it can be written as $2k$ for some integer $k.$ Now, if $n$ is a positive integer, then we can subtract $2$ from $n$ as many times as possible without the difference becoming negative. Write $q$ for the number of times we may subtract $2$ from $n$ without the difference becoming negative. Observe that $n - 2q$ must be either $0$ or $1.$ Otherwise, we could subtract $2$ from $n$ more than $q$ times. But this is impossible by definition of $q.$ Put another way, we have that $n = 2q$ or $n = 2q + 1.$ On the other hand, if $n$ is negative, then we can make an analogous argument by adding copies of $2$ to $n$ without the sum becoming positive. In this case, if we write $r$ for the number of times we may add $2$ to $n$ without the sum becoming positive, then $n + 2r = 0$ or $n + 2r = -1,$ from which it follows that $n = 2(-r)$ or $n = -2r - 1 = -2r - 2 + 1 = -2(r + 1) + 1.$ Of course, if $n = 0,$ then $n = 2 \cdot 0.$ In any case, it follows that $n$ is even or $n$ is odd. QED.

If by number you also mean to include decimals (both terminating and non-terminating) base ten, then the answer is no. For instance, the rational number $0.5 = \frac 1 2$ is neither odd nor even because there is no integer $k$ such that $\frac 1 2 = 2k$ or $\frac 1 2 = 2k + 1,$ as this would imply that either $4k = 1$ or $4k + 2 = 1,$ neither of which can possibly be true. (Check for yourself what the integers $4k$ and $4k + 2$ look like.)

In general, the integers are a small (in a very precise way that I won't mention) subset of "numbers." Every integer is a rational number is a real number is a complex number, but the reverse inclusions do not hold. E.g., $i = \sqrt{-1}$ is a complex number that is not real; $\sqrt 2$ is a real number that is not rational; and as we have just seen, $\frac 1 2$ is a rational number that is not an integer. Not even all integers are whole numbers: by definition, whole numbers are positive integers. For instance, $-1$ is an integer that is not a whole number.

Answer 2

Any number you want : units: 1,-1 Primes: 2,3,5,7,11,... Composites: 4,6,8,9,10,... Perfect numbers: 6,28,496,... Imperfect numbers: 1,2,3,4,5,7,8,9,... The point is we can create one set by definition and the other by complement in the integers. Or break up into remainder classes. Etc.