God created the natural numbers. All else is the work of man.
— Leopold Kronecker [1]
The natural numbers is a fancy name for the numbers we use to count things, starting from 1, 2, 3, and so on.
These are the numbers we first learn as children, and are the numbers we are most familiar with to actually be numbers. Have you ever had someone tell you that zero wasn't a number? Or that negative numbers didn't really exist, we just use them as a convenient shorthand for owing people things? The fact that these numbers tend to make sense to our heads without any sort of special mental training means they're called the natural numbers.
There are many ways to define the natural numbers formally. We'll use this one here, because it makes use of only two things:
The set of natural numbers is the set of numbers that can be obtained by starting from and counting up.
The two things that are used here are and the concept of "counting up". [2]
Counting up is exactly what you might think it is — starting from , counting up gives us , , , and so on. We usually write this set as , using the ellipsis to denote that the sequence goes on as you would intuitively expect (continuing with , , etc.) forever and ever.
Let's create some notation around this "counting up" concept that we'll use in the rest of the article. If you have a natural number [3], then will denote "the number you get after you count up from ". So is , is , is , and so on. We can also string together a bunch of s to denote counting up twice, three times, etc., for example: is .
Notice that we defined the natural numbers to start from instead of like we did at the top of the article. Although some people define the natural numbers as starting from , it makes certain later parts of this article easier if we start from instead.
We just mentioned that a set has properties that can be used to infer things. What properties do the natural numbers have?
The first thing we usually notice is that certain numbers are bigger than others. The rule for this is very simple: if you can count up from one number to get to another number, the first number is less than the second number. In letter notation, we'd write: if we can write as for some number of s.
So because , we know that .
We know we can compare two natural numbers — and we can use the particular type of comparison being used to show that we can put the natural numbers in order as well. This requires us to show the following thing:
If and , then .
If this property is true, then we know we can't get any strange cycles where , , and like a rock-paper-scissors game with numbers, and all the numbers will fit neatly onto a line where any number to the left of another number is less than that number.
We can certainly produce specific examples of this happening — for example, and , so . But how can we show that it's true for all these cases?
Here is where properties of natural numbers as a set come in handy — rather than just taking this fact on faith, we can actually prove that this is true. In particular, we'll use the fact that less-than was defined using counting-up.
If , we know this means that for some number of s. Similarly, if , this means that for some other number of s. Knowing this, we can immediately see that which is just for a somewhat bigger number of s.
From here, we will start defining operations on the numbers, which are procedures you can perform on them.
To add two natural numbers, take the number of times the first number was counted up from zero and count the second number up the same number of times. To find out , note that , so .
Saying that natural numbers can be added is to say that this procedure always works (as in it always gives us another natural number). This is important because it allows us to use this operation to define new operations, like the next one.
To multiply two natural numbers, take the number of times the first number was counted up from zero, and add the second number to zero the same number of times. As in the addition example, to find out , note that , so .
This is another way of saying "multiplication is repeated addition", but the important thing is that this procedure depends on addition always working. If addition didn't always work, multiplication might be undefined in some cases.
For example, let's take the set of odd natural numbers . If we try to multiply by , we run into the problem of what we get when we add to in the second operation. The answer is , but this isn't in our set, so addition doesn't work on the odd natural numbers, and so we can't define multiplication this way.
An astute reader might notice that even though we can't define multiplication this way, the odd numbers can still be multiplied if we somehow get to that definition some other way — the product of two odd numbers is always another odd number.
This shows that there are sometimes sets that satisfy some properties and not others, even if the second property seems like it must depend on the first. The particular property this set satisfies makes it a multiplicative semigroup, which is another type of number set. But that's just an interesting fact — again, you don't need to know what that means right now.
Also important is that multiplication always works as well, because we can now use it to define even more operations, and properties on the numbers.
You can subtract a natural number from another natural number if there exists a third natural number such that the sum of the first and third numbers is the second number. Written out in letter notation, a natural number can be subtracted from another natural number if there exists a natural number such that .
For example, to quote a Tom Lehrer song about the New Math, "you can't take from ; is less than ". There is no natural number such that .
This might seem like a lack of a property at first, but having numbers that might not necessarily divide other numbers leads to interesting properties that can be analyzed. For example, this is another way of defining the ordering on the natural numbers. One natural number is less than another if you cannot subtract the second natural number from the first.
If you want numbers to always be subtractable, then you'll need a set bigger than just the natural numbers — you'll need to extend it into the integers, which is what the next article is all about.
A natural number is said to divide another natural number if there exists a third natural number such that the product of the first and third numbers is the second number. Written out in letter notation, a natural number divides another natural number if there exists a natural number such that .
Just like subtraction, this doesn't always work. Right away we notice that you can't divide by — there's no natural number such that .
Having some numbers not able to divide others leads to the existence of prime numbers, which are numbers that that have exactly two natural numbers can divide them — and itself. In a set of numbers where every number can divide into every other number (a field, as we mentioned above), there are no primes.
Technically he said God created the integers ("die ganzen Zahlen" in German), but negative numbers are arguably the work of man as well.
Mathematicians tend to call this the successor_function instead, which is just a fancy way of saying "the next number".
The point of using a letter here is that you can replace it with any number and it will still work.