Expert: Sherry Wallin - 4/18/2009

Question
Hi Sherry,

I have a question that I am hoping you can help me with.  My professor assigned me with a special project so that I can reinforce the grade on a first test.  I want to think that I am over analyzing the next question and going the long route or that I am not hitting it from the right corner.  Can you look at it and verify my work is correct?

PROVE OR DISPROVE: The sumn of an integer nd two rational numbers is alwyas rational.
First, we prove the sum of two rational numbers is always rational.  Let x, y belong to rational numbers, where m,n,p,q are integers such that n,p do not equal zero.  Let x = m/n and y = p/q. Then x + y = mp/nq + np/nq.  mq + np is the sum of two integes and is therefore in itself an integer.  We know that n, q do not equal zero and so their product cannot be zero.  This shows x + y is a quotient of two integers and its denominator is not zero.  Thus, x + y is number.  
Moreover, since addition of rational numbers is associative then we let z be a positive integer. Then x + y) + z = x + (y + z).  Hence, the sum is also rational.  We let z be a negative integer.  Then our counter example is: 1 + root 2 + (-root 2)= 1 (which is rational)

There is my problem Sherry.... I even lost myself towards the end.  Any advice???

Thanks a bunch,

Evelyn

Answer
You need to say that m/n and p/q are such that gcd(m,n) = 1 and gcd(p,q) = 1. This is so that you know there aren't any common factors later.

You have:"This shows x + y is a quotient of two integers and its denominator is not zero". And you mean: This shows x + y is a sum of two rational numbers and its denominator is not zero. You also have: "Thus, x + y is number".   and it should be "Thus, x + y is rational".   You don't need associativity to show that when you add an integer to a rational number that you have a rational number. Counter examples are used to *disprove* something, so you've used counter example incorrectly.

Assume z is an integer then it can be written as z/1,add this to (m/n)+(p/q). [Incidentally you've added the rational numbers incorrectly, it should be [mq+pn]/nq]. z/1 + [mq+pn]/nq]= [znq + mq + pn]/nq. Note all the products are composed of integers so their products are integers and the denominator is not zero and is a product of integers. Note: You did not show that nq was not 1. What you could do is at the get go is say that p,q are integers not equal to 1. Then you need to say that pq cannot equal 1 since both p and  q are integers not equal to 1. And finally you get to say that since m,n,p,q were picked so that their gcd is 1 they have no factors in common and thus [znq + mq + pn]/nq is  reduced to it's simplest form [gcd(znq + mq + pn, nq) = 1] and is rational.

There are easier ways to do this but I"ve run out of time. If you need more help, ask again.

Math Prof