Proof: sum & product of two rationals is rational

what I want to do in this video think about whether the product or sums of rational numbers are definitely going to be rational so let's just first think about the rep the product of rational numbers so if I have one rational number and actually let me start off writing out the word rational let me just represent it as the ratio of two integers so I have one rational number right over there I can represent it is a over B and I'm going to multiply it times another rational number and I can represent that as the ratio of two integers m and N and so what is this product going to be well the numerator I'm going to have a I'm going to have a M I'm going to have a times M and in the denominator I'm going to have B times n 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 the 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 over B and that my second or can be represented as a over B and my second rational number can be represented as M over 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 let me multiply this one times and 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 a n a n plus B M plus B M all of that over all of that over B all of that over B times n B times n so B times n we just already talked about this is definitely going to be an integer right over here and then what is what 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 an integer e of 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