# Why is negative times negative equals positive?

We have learned that the product of a positive number and a negative number is negative. How about the product of two negative numbers? How do we know that it is positive? Below are some illustrations and proof that show why is the product of two numbers positive.

Reason 1 – Pattern

Let us consider the pattern below.

$3 \times - 4 = -12$

$2 \times - 4 = -8$

$1 \times -4 = -4$

$0 \times -4 = 0$

$-1 \times - 4 = ?$

From the patterns, it should be $4$.

Reason 2 – Contradiction If Otherwise

Suppose the product of two negative numbers is negative. Then, (-2)(-3)  = -6. Since the distributive property multiplication  over addition holds,

-2 (-3 + 3) = -2(-3 + 3)

-2 (0) = (-2)(-3) + (-2)(3)

0= -6 + -6

0 = -12.

The proof below requires elementary knowledge in algebra, so I have included it.

The Proof

Let $a$ and $b$ be real numbers.

Define  x = ab + (-a)(b) + (-a)(-b).

We factor out $-a$ from x = ab + (-a)(b) + (-a)(-b):

$x = ab + -a (b + (- b))$

$x = ab + -a(0)$.

$x = ab$. (1)

We now factor out $b$ from x = ab + (-a)(b) + (-a)(-b):

$x = b(a + -a) + (-a)(-b)$

$x = b(0) + (-a)(-b)$

$x = (-a)(-b)$. (2)

Since (1) and (2) are equal, $ab = (-a)(-b)$.

Therefore, the product of two negative numbers is positive.