How to prove $(-3) \times (-4) = 12$?

Question

How to logically understand the multiplication of two integers ?

Eg:

3 x 4 = 12 (is understandable)

-3 x4 = -12 (is also somewhat understandable)

But , 3 x -4 = -12 (is NOT understandable)

-3 x -4 = 12 (is also NOT understandable)

Now , at this we must assume that commutative property may not be true .

What is the logical explanation for something being multiplied negative times ?

Answer 1

$$(-4)\cdot(-3)=(-1)(4)\cdot(-1)(3)=(-1)^2(3\cdot4)=12$$ Because $(-1)^2=1$

Answer 2

Long comment

$(3 \times 4)=12$ --- agreed.

$3 \times (4-4) = 3 \times 0 = 0$ --- agreed ?

$3 \times (4-4) = (3 \times 4) + (3 \times (-4)) = 12 + ? =0$ --- compute.

And finally : $(-3) \times (4-4) = ((-3) \times 4) + ((-3) \times (-4)) = ? + ?? =0$ --- having computed $?$ above, we are able to compute also $??$

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Rules are part of a system of rules : if we start accepting some of them, we are forced (by "logic") to follow other rules implied by the firs ones in order to ensure the consistency (i.e. proper working) of the system.

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And see the post : Historical roots of the justification for the rule for multiplication of negative numbers.

Answer 3

You can use the distributive law as follows:

$0=(3-3)\times 4=3\times 4+(-3)\times 4=12+(-3)\times 4$ whence $(-3)\times 4=-12$

Then also $0=(-3)\times (4-4)=(-3)\times 4+(-3)\times (-4)=-12+(-3)\times (-4)$ whence $(-3)\times (-4)=12$