The absolute value of a number is its distance from 0. Thus, all absolute values are either positive or 0. That is, if we have a number , then the absolute value of which can be written as is equal to

(a) if is positive

(b) if is negative

(c) if is 0.

Also, for any real number, is positive or 0 (if ). Therefore, is

(a) if is positive

(b) if is negative

(c) if is 0.

If the part is a bit confusing, consider take for example

.

From the discussion above we can conclude that .

Using this fact we are going to prove to arguments in this post.

If and are two real numbers,

(1) the absolute value of their product is equal to the product of their absolute values or

and

(2) the absolute value of their quotient is equal to the quotient of their absolute values or

Proof of (1)

From above, we know that , so

But from the definition above,

Therefore, .

Proof of (2)

Since

Therefore,

That’s all for now, see you in the next post.