# If 'a' is rational and 'b' is irrational is 'ab' neccesarrily irrational?

• Jul 19th 2010, 09:52 AM
mfetch22
If 'a' is rational and 'b' is irrational is 'ab' neccesarrily irrational?
So, this is a question in my textbook. I'm sure it's a very simple answer, that's right under my nose, but that I'm just missing for some reason. Anyway, if we have two numbers $a$ and $b$ and $a$ is rational, but $b$ is irrational, then is $ab$ neccesarily irrational? Please provide a proof with your answer, thanks in advance.
• Jul 19th 2010, 10:01 AM
Also sprach Zarathustra
Suppose a*b=t (rational!)

a rational so a=p/q

t is rational so t=m/n

Hence, ab=m/n ==> b=m/n * q/p ==> b rational! A contradiction!
• Jul 19th 2010, 10:49 AM
Defunkt
Unless a = 0... :)
• Jul 19th 2010, 06:04 PM
HallsofIvy
And a single counterexample is sufficient to prove a general statement false!