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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.

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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.