# The Axioms of Arithmetic

• Feb 20th 2007, 03:20 PM
Aryth
The Axioms of Arithmetic
I'm typing the ones I know... I was wondering if there are any others... I do not know that many...

1. For any numbers m, n
m + n = n + m and mn = nm.

2. For any numbers m, n, k,
(m+n) + k = m + (n+k) and (mn)k = m(nk).

3. For any numbers m, n, k
m(n+k) = mn + mk.

4. There is a number 0 which has the property that, for any number n,
n + 0 = n.

5. There is a number 1 which has the property that, for any number n,
n x 1 = n.

6. For every number n, there is another number k such that
n + k = 0.

7. For any numbers m, n, k,
if k cannot equal 0 and kn = km, then m = n.

Any additions would be appreciated, I'm interested in reading as many as I can.
• Feb 20th 2007, 08:36 PM
CaptainBlack
You might consider the Peano Aximoms.

RonL
• Feb 21st 2007, 08:47 AM
ThePerfectHacker
A lot of these are not axioms. Rather proofs, careful!
• Feb 21st 2007, 09:07 AM
Aryth
Quote:

Originally Posted by ThePerfectHacker
A lot of these are not axioms. Rather proofs, careful!

I was reading them in the book Fermat's Enigma by Simon Singh and it had these in the appendices, and called them Arithmetic Axioms... Which ones are proofs?
• Feb 21st 2007, 12:26 PM
ThePerfectHacker
Quote:

Originally Posted by Aryth
I was reading them in the book Fermat's Enigma by Simon Singh and it had these in the appendices, and called them Arithmetic Axioms... Which ones are proofs?

In high school, the textbooks have an entire list of the axioms of arithmetic in the cover. But again! They are not axioms they are theorems. I am just not sure why books do that. Anyway, the standard axioms are 5, called Peano Axioms in the link CaptainBlank gave thee.
(They might look strange at first).