# Negative times negative is positive?

1. Mar 3, 2012

### Bipolarity

Probably the stupidest question I have ever asked, but is it possible to prove that the multiplication of two negatives yields a positive? Go easy on me I've asked better questions :D

BiP

2. Mar 3, 2012

### Char. Limit

How rigorous a proof are you looking for?

3. Mar 3, 2012

### Bipolarity

As rigorous as rigorous gets :D

4. Mar 3, 2012

### micromass

What axioms are you accepting??

5. Mar 3, 2012

### Dickfore

These are the usual axioms of real numbers system:

http://www.gap-system.org/~john/analysis/Lectures/L5.html

Specifically, look at the 2nd axiom of order.

EDIT:
Then, prove the following:
$$a < 0 \Rightarrow -a > 0$$
By axiom II.c
$$0 > a \Rightarrow 0 + (-a) > a + (-a) \Leftrightarrow -a > 0$$
Q.E.D.

Then, look at the following:
$$a, b < 0 \Rightarrow a \cdot b = (-a) \cdot (-b)$$
Then you have a product of two positive numbers, which by the quoted axiom is positive.

6. Mar 3, 2012

### Bipolarity

Ahh, so this is only true for ordered fields then? Is there any number system where the field is not ordered (i.e. does not satisfy axiom II) ?

BiP

7. Mar 3, 2012

### micromass

Sure, $\mathbb{Z}_2$ is not an ordered field.

If you don't know what it is: it's just the set {0,1} with

0+0=1+1=0, 1+0=0+1=1
0*0=0*1=1*0=0, 1*1=1