Negative times negative is positive?

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

Char. Limit

Gold Member
How rigorous a proof are you looking for?

Bipolarity

As rigorous as rigorous gets :D

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.

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

micromass

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
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

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