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