please help me make a proof?

Question

the product of a non zero rational number with an irrational number is always irrational or disprove pls../././

Answer 1

Yes, an irrational times a rational is always irrational.

The proof is by contradiction.
By definition, an irrational number can -not- be expressed as a fraction a/b (where a and b are integers)and a rational number is -always- expressable as an integer fraction. Assume you have an irrational number (call it Q) and you write
Q*(a/b) = c/d (that is, the product is rational), then
Q = (c/d)/(a/b) = cb/ad and, since the product of 2 integers is an integer, Q is shown to be rational which contradicts the assumption of Q being irrational.
Q.E.D.

Doug

Answer 2

Suppose a is a nonzero rational number and b is an irrational number, but suppose ab is rational. Let c=ab. Since a is rational, it is of the form m/n, where gcd(m,n)=1. So, c= m/n *b. Thus c* n/m = b is rational (since the product of two rational numbers is rational). Hence b is rational. This is a contradiction. Hence ab must be irrational.

Answer 3

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