Proving that sum of Rational #s is Rational

[rs21867]

1. Let a and b be rational numbers. Prove or provide counterexample that

A) a+b is a rational number.
B) Is ab necessarily a rational number?



2. How can you prove that the sume of two rational number is rational? Well I am not really good at math



3. This is what I've tryed to do for part a. But I'm stuck proving the sum (product) of two integer is an integer.

Since a and b are rational numbers they can be written as a=x/y and b=w/z where x,y,w,z are all integers. Then a+b= x/y+w/z = (xz+wy)/(yz)

Now I'm stuck, and I have no clue how to do part B...

 

[jeffreydk]

You might want to look at this thread

https://www.physicsforums.com/showthread.php?t=166106

it's similar

 

[HallsofIvy]

1. Let a and b be rational numbers. Prove or provide counterexample that

A) a+b is a rational number.
B) Is ab necessarily a rational number?



2. How can you prove that the sume of two rational number is rational? Well I am not really good at math



3. This is what I've tryed to do for part a. But I'm stuck proving the sum (product) of two integer is an integer.

Since a and b are rational numbers they can be written as a=x/y and b=w/z where x,y,w,z are all integers. Then a+b= x/y+w/z = (xz+wy)/(yz)

Now I'm stuck, and I have no clue how to do part B...

You are exactly right for (a): you have shown that a+ b= (xz+ wy)/(yz) and so is the ratio of two integers (and the denominator yz is not 0 because neither y nor z is 0).

To do B, do exactly the same thing! If a= x/y and b= w/z, what is ab? Is it a ratio of two integers? Could the denominator be 0? If anything, B is easier than A because it is easier to multiply two fractions than to add!