### Natural Numbers

"Can you do Addition?" the White Queen asked: "What's one and one and one and one and one and one and one and one and one?"

"I don't know" said Alice. "I lost count."

"She can't do Addition," the Red Queen interrupted. "Can you do Subtraction? Take nine from eight."

"Nine from eight I can't you know," Alice replied very readily: "but—"

"She can't do Subtraction," said the White Queen.

The natural numbers, N, are the counting numbers 0,1,2,3 ... When I was a lad, I think 0 had not been allowed to join this particular club, but many mathematicians, though not all, include it these days. From a historical point of view it would probably be better not to include 0, because many cultures had a good understanding of the other natural numbers, without making use of 0. The set of "Whole Numbers" is commonly understood as designating the Natural numbers including 0, though some people also use "Whole Numbers" as a synonym for the Integers. From a mathematical point of view the natural numbers without 0 are much less use than with 0, because 0 is the identity element for addition: for any n in N, n+0=0.

Computer programmers generally find that counting from 0 makes life much easier: for example the elements in an array A of ten numbers would usually be designated like this:

`A[0], A[1], .., A[9]`

and not like this:

`A[1], A[2], .., A[10]`

Note that the natural numbers can be ordered, 0 < 1, 1 < 2, and so on, and that for each number there is a definite next larger number (usually called its successor). In fact, addition, multiplication, and the usual ordering of the natural numbers, can all be defined in terms of the successor function, which maps each number to its successor. The Peano Axioms are often used as the starting point for an axiomatic theory of numbers.

The most famous of these axioms, and certainly the one most often explicitly used in proofs, is called the Principle of (Weak) Induction: if a set contains 0, and whenever it contains n, it also contains n+1, then it contains all the natural numbers.

In any set of natural numbers, whether finite or infinite, there is bound to be a smallest number. While a finite set of natural numbers has a largest member, an infinite set of them definitely won't have (for example there is no largest prime number).

Two properties of addition and multiplication are very important, and are called the Cancellation Laws:

• (Cancellation Law for Addition) If m + x = n + x then m = n
• (Cancellation Law for Multiplication) If m * x = n * x and x<>0 then m = n.

These laws can be proved from the Peano Axioms, which include an axiom that says that no two different numbers have the same successor. Incidentally, the proviso in the second law is very important. In proofs you have to be careful not to cancel a common factor that might be 0. Euclid had an axiom that said that two things that were equal to a third were equal to each other, but he unconsciously assumed the geometric equivalent of the cancellation laws.

Alice could not take nine from eight because she was using the natural numbers. To be able to solve any problem of subtraction we need to expand our number system to the set of Integers.