Question

If r is rational (r is not 0) and x is irrational, prove that r+x and rx are irrational.

Answer 1

proof by contradiction:
assume that r + x is rational
r + x = p/q, gcd(p,q) = 1 (this makes the fraction reduced)
then x = p/q - r
p/q - r is a rational number, but we said x is irrational
so basically this statement is that x is irrational and rational, which is a contradiction.
Therefore our initial statement is false, and r + x is irrational

Answer 2

im not great at proofs but if x is irrational then it has an infinite integers after decimal point, if i add a rational number then based on rules of decimal addition a finite number of decimal places will change but the sum will still have an infinite num of integers after decimal

Answer 3

thats much better than my explaination :)

Answer 4

second part, another proof by contradiction:
assume rx is a rational number
rx = p/q, gcd(p,q) = 1
Then x = (p/q)(1/r)
after this is the same argument as before
one side of the equation says x is irrational, the other side says x is rational
contradiction

Answer 5

Very nice

Answer 6

brilliant proof, joe math.