# Prove that both the sum and product of two rational numbers are rational

• Oct 4th 2012, 09:13 PM
evthim
Prove that both the sum and product of two rational numbers are rational
I'm new to proofs and I want to make sure that I'm doing them right, so I might be doing a lot of this "here's my proof, is it suitable?" posts.

My proof:
1st statement: sum of two rational numbers is rational
If a number is rational it can then be written in the form a/b.
We are saying that the sum of these two rational numbers (a/b) and (c/d) (where a,b,c,d are integers) is also a rational number.

-let k = (ad+bc) and j = bd
Therefore ((ad+bc)/bd) = k/j where k and j are integers because the product of two integers is an integer and the sum of two integers is an integer.
We've just proven that the sum of two rational numbers is a rational number.

2nd statement: the product of two rational numbers is rational
given (a/b) and (c/d) where a,b,c,d are integers
(a/b)*(c/d) = (ac/bd)
ac is an int because the product of two integers is an integer
bd is an int because the product of two integers is an integer

Therefore (ac/bd) is rational
• Oct 4th 2012, 09:21 PM
Prove It
Re: Prove that both the sum and product of two rational numbers are rational
Quote:

Originally Posted by evthim
I'm new to proofs and I want to make sure that I'm doing them right, so I might be doing a lot of this "here's my proof, is it suitable?" posts.

My proof:
1st statement: sum of two rational numbers is rational
If a number is rational it can then be written in the form a/b.
We are saying that the sum of these two rational numbers (a/b) and (c/d) (where a,b,c,d are integers) is also a rational number.

-let k = (ad+bc) and j = bd
Therefore ((ad+bc)/bd) = k/j where k and j are integers because the product of two integers is an integer and the sum of two integers is an integer.
We've just proven that the sum of two rational numbers is a rational number.

2nd statement: the product of two rational numbers is rational
given (a/b) and (c/d) where a,b,c,d are integers
(a/b)*(c/d) = (ac/bd)
ac is an int because the product of two integers is an integer
bd is an int because the product of two integers is an integer

Therefore (ac/bd) is rational

1. Is acceptable if you have already shown that the sum and product of integers is always an integer.

2. Is acceptable if you have already shown that the product of two integers is always an integer.
• Oct 4th 2012, 10:15 PM
richard1234
Re: Prove that both the sum and product of two rational numbers are rational
Another way to phrase that is that the set of integers is closed under addition and multiplication, that is, the product or the sum of two integers is always an integer.