[Sussex] The difference between integers and natural numbers, and the inclusion of "0" ;-)

[Mark Harrison]

I dug out my Uni text books, and found the difference between an "Integer"
and a "Natural Number" in number theory.

- The natural numbers are the numbers that go 0,1,2,3,4 ...

- As you say, it is NOT universally accepted that 0 is a natural number. In
some branches of maths, where we want to assume certain properties about all
natural numbers, we have to exclude 0.

- Integers are the ones that go ... -3, -2, -1, 0, 1, 2, 3...

- Zero IS an integer, however, it may or may not be a natural number.

Natural numbers are (normally) defined by intuition - there is no need in
most circumstances for a formal definition. However, Number Theory is the
branch of mathematics which has responsibility for defining the terms
"Natural Number" and Integer.

The problem comes, obviously, because when we use a term like "integer" as a
computer scientist, we are, almost always, referring not to the number
theory definition, but to a data type defined in a particular language or
context, which will, inter alia, have an upper bound, and may or may not
allow negative numbers, or zero.

READ ON ONLY IF YOU'RE A MATHS GEEK, OR ARE INTERESTED IN WHAT MATHS GEEKS
DO :-)

In number theory, however, the Natural Numbers are formally defined by
Peano's axioms (dating from 1888), which give:

P1: There is a number 0
P2: For each number, n, there is another number n' (the successor of n)
P3: For no number n is n' equal to 0
P4: If m and n are numbers and m'=n', then m=n
P5: If A is a set of numbers which contains 0 and contains n' for every n in
A, then A contains all the (natural) numbers

The integers can be constructed from the Natural Numbers as follows:

Definition: There is a two place function called addition over the natural
numbers, (denoted by the infix operator +), with the properties:

    m + 0 = m for every natural number m
    m + n' = (m+n)' for all natural numbers m, n

Syntax Definition: We use the symbol "1" to represent 0'. (The successor of
zero)

Definition: Let a,b,c,d be Natural Numbers. We say that the ordered pair
(a,b) is related to (c,d) written (a,b) # (c,d) if a+d = b+c. (We can't
write a-b yet, since we haven't defined subtraction, and indeed CAN'T define
subtraction in a consistent manner across only the natural numbers, since
some answers will be undefined.)

Defintion: The INTEGERS are the equivalence class under the relation #. The
intuitive mapping from the ordered pairs (x,y) are equal to the integer x-y.
For example -1 is defined as the set of all all ordered pairs (a,b) where
a,b are both Natural Numbers, and a+1 = b.

At which point, we can now define addition over the integers:

Defintion: There is a two place function called addition over the integers
(again, denoted by the infix operator +), defined as follows:

    a,b,c,d are Natural numbers
    (a,b)+(c,d) = (a+c), (b+d)