Expert: Paul Klarreich - 9/14/2009

Question
If 'x' is a rational number not equal to 0, 'y' is an irrational number, and x*y=z, then z is an irrational number. (prove by contradiction)

Answer
Questioner: Abby
Country: United States
Category: Advanced Math
Private: No
Subject: Math Proof by Contradiction
Question: If 'x' is a rational number not equal to 0, 'y' is an irrational number, and x*y=z, then z is an irrational number. (prove by contradiction)
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Hi, Abby,

Proof by contradiction means:

1. Apply the definitions.
2. Assume the conclusion is false.
3. Do some calculation and prove a statement that contradicts something that is known to be true.

A. x is rational.

  That means there are a,b such that  x = a/b, and b /= 0

B. x /= 0

  That means that a /= 0

C. y is irrational.

  That means we CANNOT find integers c,d such that  y = c/d,  d /= 0

D.  xy = z

  (What that means we will say later.)

E. ASSUME  z  is rational.  [Assuming the conclusion to be false.]

  That means there are integers e,f such that  z = e/f, and f /= 0

If xy = z, then

a      e
- y = ---
b      f

Cross-multiply: which is legal since  b, f  both /= 0.
(Not sure you really have to say that)

afy = eb

Divide by af, which is legal since  a, f  both /= 0.
(REALLY SURE you do have to say that.  a/=0 was one of the conditions)

y = eb/af

But eb is an integer, which we can call c.
And af is an integer, which we can call d, and d = af /= 0.

So we have found integers  c,d, such that y = c/d,  d /= 0.

That is our contradiction.  It contradicts statement C.