Expert: Paul Klarreich - 10/17/2006

Question
prove by contradiction that the sum of a rational and irrational number is irrational.

Answer
Kevin Asks in Category Advanced Math ... Subject: proof by contradiction Private: no
Question: prove by contradiction that the sum of a rational and irrational number is irrational
..........................................
Hi, Kevin,

To do a P by C, you:

Assume the statement true.
State that assumption in the form of the problem.
Derive a contradiction.

So your theorem is:
The sum of a rational and an irrational is irrational.

Let  R be rational, and I be irrational.
Then R = a/b for some a,b, and there is no pair c,d, such that c/d = I.

Assumption: R + I is rational.

Then  There exist  e,f such that  e/f = R + I

Then  I = (R + I) - R  = e/f - a/b
  e     a
= --- - ---
  f     b
  eb - af
= ---------
    bf

Set  c = eb - af,  and  d = bf, and we have I is rational.

That's your contradiction.