Negative multiplied by negative is positive.

This is usually taught to us in school after our knowledge about numbers
expands to include the negative numbers. Most of us take this fact for
granted. I took it on faith as well.

One of my pastimes is to ponder on these facts that I once learnt by
faith and observable evidence rather than reason and logic. I see
whether I can see these facts in a new light. A few years ago, I thought
over this one as well and proving this fact was easy.

First let me provide an intuitive understanding of why it should be so.
In the discussion to follow, we will assume that we already know some
basic properties like distributive
property, a × (−b) = −(a × b), etc. because
I don't want to prove each and every thing involved here right from the
Dedekind–Peano
axioms.

We know that 7 × 8 = 56. Let us express 7 as (10 − 3) and 8
as (10 − 2).

So, the following must be true.

(10 − 3) × (10 − 2) = 56

Using the distributive property of multiplication over addition, we get,

10 × 10 + (−3) × 10 + 10 × (−2) +
(−3) × (−2) = 56

We already know that −3 × 10 = −30 and 10 ×
−2 = −20. But we do not know what (−3) ×
(−2) is. So, let us see what we get by using what we know.

10 × 10 + (−3) × 10 + 10 × (−2) +
(−3) × (−2) = 56

⇔ 100 + (−30) + (−20) + (−2) × (−3) =
56

⇔ 50 + (−2) × (−3) = 56

⇔ (−2) × (−3) = 6

We see that (10 − 3) × (10 − 2) = 56 is true if and
only if (−2) × (−3) is considered as 6.

However this is not a proper proof. One may ask whether this rule is
true for all such calculations even when the numbers are different. Can
we prove that −a × −b = a × b is absolutely
necessary for all real numbers a and b? So, let me try a proper
algebraic proof.

Let a and b be two positive real numbers. We know that a − a = 0.

Multiplying both sides by −b, we get,

(a − a) × (−b) = 0 × (−b)

Again, using the distributive property of multiplication over addition,
we get,

a × (−b) + (−a) × (−b) = 0.

We know that a × (−b) = −(a × b), but we do not
know what (−a) × (−b) is. Using what we know, we get,

−(a × b) + (−a) * (−b) = 0.

Adding (a × b) to both sides, we get,

(−a) × (−b) = a × b