r and s are in the set of real numbers. If r is irrational and s is rational, then r/s is irrational. I need to prove this by contradiction. I have a working proof, however I don't think my contradiction is in the right format. Thanks!
Yoran
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say r,s are in R , r is irrational and s is rational.
Now assume (in spirit of finding a contradiction), that r/s is rational. If r/s is rational, then there exist p,q in Z such that
r/s = p/q
This means, however, that r = sp/q.
Because s is rational, s = a/b, thus r = ap/qb
This means however, because a,b,p,q are in Z, that r is a rational number. This is a contradiction, because we've assumed that r was irrational.
So r/s is irrational
daru
Extra condition: s <> 0;
Assume the statement is false, hence r/s is a rational number q.
r=qs which is rational since the product of two rational numbers is a rational number (why?)
This is a contradiction with the hypothesis which states that r is irrational.
Thus, the statement is true.