is there a mathematical definition for an integer?

Question

are there any mathematical definitions for any number n being an interger. You know, sort of like the definition of divisibility: a is divisible by 5 if and only if a=5(b) where b is some integer

Answer 1

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Edit: This is a new one. I originally typed up an answer to this question that went into the axiomatic definition of the real numbers, only to have the post not appear because it was too long for the system to handle. Since I couldn't bring it down to an acceptable length without gutting it, I've broken it into two parts and posted the second using an alternate account. To the best of my knowledge, using multiple accounts is not forbidden in the community guidelines provided that it is not for the purpose of misrepresentation or point-gaming.
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Yes, there is, but it's complicated. The main thing that's complicated about it is that, outside of attempts to explicitly construct the real numbers from sets, you generally want to identify an integer as "a real number such that...", in which case you have to first figure out what you mean by a real number. Defining the real numbers, even axiomatically, is complex, and since you seem to be mainly interested in integers, I've moved the entire exposition about them into my next post.

The first step in identifying the integers is to identify the natural numbers. These are the numbers 1, 2, 3, 4... and so on that we use to count. Now, this is hardly a precise mathematical definition, and we need to turn it into one. First, let me point out that since they are counting numbers, each natural number other than 1 can be obtained by adding 1 to the previous number. For instance:

2 = 1 + 1
3 = 2 + 1
4 = 3 + 1
5 = 4 + 1
...

And so on. This gives rise to a set of rules that we can use to identify natural numbers:

Rule 1: 1 is a natural number
Rule 2: If n is a natural number, then n+1 is a natural number
Rule 3: Nothing else is a natural number.

Rule 3 requires some explanation -- what do I mean by nothing else? What I mean is, that anything that is not required to be a natural number under rules 1 and 2 isn't one. Now, some things are required to be natural numbers under rules 1 and 2. For instance, 1∈ℕ by rule 1. And by that fact and rule 2, 1+1=2∈ℕ. And applying rule 2 to that, 2+1=3∈ℕ. And similarly, 3+1=4∈ℕ, 4+1=5∈ℕ, and so on. Clearly, anything which we would naturally want to consider a natural number can be proven to be such under these rules simply by repeating rule 2 enough times. And just as clearly, nothing we
don't want to be a natural number can't be proven to be such, and so by rule 3, isn't one. So, if we can find a way to formalize rule 3, we can define the natural numbers. Consider the following - we can say that if n is a natural number, and S is any subset of the real numbers numbers that obeys rules 1 and 2, then n has to be in S. And it turns out that saying this is enough to exclude everything that isn't a natural number. So we have a formal definition of ℕ:

ℕ = {n∈ℝ: ∀S⊆ℝ, (1∈S ∧ (m∈S → m+1∈S)) ⇒ n∈S}

This is kind of complex definition, even in symbols, but it says exactly what we said above -- n is a natural number if and only if, whenever S is any set of real numbers such that [1 is in S, and if m is in S then m+1 is in S], then n has to be in S (brackets added for clarity).

Of course, once ℕ is defined, ℤ is fairly easy -- simply remember that any integer can be written as the difference of two natural numbers. This is in fact the definition of ℤ:

ℤ = {a-b: a∈ℕ, b∈ℕ}

As you can see, once we've identified the natural numbers in ℝ, defining the other major subsets of ℝ is easy. I'm sure you can guess what the definition of ℚ is at this point:

ℚ = {p/q: p∈ℤ, q∈ℤ\{0}}

So that's how these sets are defined. Of course, as I said, most of the complexity is in defining the notion of real number in the first place, which I have moved into my next post.

Answer 2

The integers are often denoted by the above symbol.The integers (Latin, integer, literally, untouched, whole, entire, i.e., a whole number) are the set of numbers including the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...), and the number zero. In non-mathematical terms, they are numbers that can be written without a fractional or decimal component, and fall within the set {...-2,-1,0,1,2...}. For example, 65, 7, and -756 are integers; 1.6 is not an integer.