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# Proof: sum & product of two rationals is rational

Sal proves that the sum, or the product, of any two rational numbers will always be a rational number. Created by Sal Khan.

Video transcript

What I want to do in
this video is think about whether the product or
sums of rational numbers are definitely going
to be rational. So let's just first
think about the product of rational numbers. So if I have one rational
number and-- actually, let me instead of writing
out the word rational, let me just represent it
as a ratio of two integers. So I have one rational
number right over there. I can represent it as a/b. And I'm going to multiply it
times another rational number, and I can represent that as a
ratio of two integers, m and n. And so what is this
product going to be? Well, the numerator,
I'm going to have am. I'm going to have a times m. And in the denominator, I'm
going to have b times n. Well a is an integer,
m is an integer. So you have an integer
in the numerator. And b is an integer
and n is an integer. So you have an integer
in the denominator. So now the product is a ratio
of two integers right over here, so the product is also rational. So this thing is also rational. So if you give me the product
of any two rational numbers, you're going to end up
with a rational number. Let's see if the same thing
is true for the sum of two rational numbers. So let's say my first
rational number is a/b, or can be represented as a/b, and
my second rational number can be represented as m/n. Well, how would I add these two? Well, I can find a
common denominator, and the easiest
one is b times n. So let me multiply
this fraction. We multiply this one times
n in the numerator and n in the denominator. And let me multiply
this one times b in the numerator and
b in the denominator. Now we've written
them so they have a common denominator of bn. And so this is going to
be equal to an plus bm, all of that over b times n. So b times n, we've
just talked about. This is definitely going to
be an integer right over here. And then what do
we have up here? Well, we have a times
n, which is an integer. b times m is another integer. The sum of two integers
is going to be an integer. So you have an integer
over in an integer. You have the ratio
of two integers. So the sum of two
rational numbers is going to give you another. So this one right over
here was rational, and this one is right
over here is rational. So you take the product
of two rational numbers, you get a rational number. You take the sum of
two rational numbers, you get a rational number.