# How many categories can we can put numbers into?

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.

• What exactly do you mean by "number" in this context? There are lots of different types of number out there ... (E.g. "if I know a varible x is not an odd number does that necessarily mean the varible x MUST be an even number?" - do you also know that the variable x is an integer?) Jul 30, 2021 at 4:21
• @Noah Schweber, by number I mean the entire set of numbers there is be it rational, irrational, integers, decimals, fractions, real numbers, natural numbers and so on. I figure if I said integers alone the answer would be yes there are only even or odd numbers. I wanted to k ow if even and odd are only integers or can other numbers be included. That is can a non integer be classified as even or odd? I ask because if variable x is not odd then what can I know about x other than that? What can I derive from being given just x is not odd? That is my direction I guess. Jul 30, 2021 at 4:33
• Natural numbers are even or odd. Irrational numbers are not usually described as even or odd Jul 30, 2021 at 14:30

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.

• Thank you for your awsome and timely answer! What I am getting at is if all I am given is "varible x is not odd" what else can I derive must be true? Your answer suggests x may be a integer that is even or x can be a non integer. I suppose to ask if the square root of pi is even or odd would be a nonsensical statement? Or would it be the square root of pi is an even number simply be false? From your answer I derive that if I know x is neither even or odd then x can't be an integer. We would need more details to derive any further information from "variable x is not odd" to give certainty? Jul 30, 2021 at 4:43
• Yes: $x$ is either even or odd if and only if $x$ is an integer. Like you mentioned, one way is proven above. For the other direction, observe that if $x$ is even or odd, then it is a sum of integers and therefore an integer. In general, we do not ask questions like, "What can be inferred about the variable $x$?" By definition, a variable has no intrinsic properties of its own. This is why it is common to refer to the variable $x$ as "indeterminate." Jul 30, 2021 at 4:48
• nice answer .... +1 Apr 30 at 21:01

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.