Question

Show that the product of a non-zero rational number and an irrational number is an irrational number.

Answer 1

Let x is an irrational number and y is non zero rational number.

Let us assume that xy is rational.

Since y is rational then y = `"a"/"b"` where a and b are integers and b ≠ 0

Since x is irrational so x can be written as fraction form.

∵ xy is rational. Let xy = `"c"/"d"` where c and dare integers and

`=> x xx "a"/"b" = "c"/"d"`

`=> x = "c"/"d" xx "a"/"b" = "ac"/"bd"`

Since a, b,c and d are integers so ac and bd are also integers and bd ≠ 0

⇒ x is rational number.

It contradicts our assumptions.

The product of x and y is irratioanl.