SOLUTION: 5)Which number can be multiplied to a rational number to explain that the product of a rational number and an irrational number is irrational?
A)2
B)-2 1/8
C)7
D)8
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Question 962471: 5)Which number can be multiplied to a rational number to explain that the product of a rational number and an irrational number is irrational?
A)2
B)-2 1/8
C)7
D)8
Answer by hkwu(60) (Show Source): You can put this solution on YOUR website!
Your answers don't make sense, since none of them are irrational. What you need is a general proof. Suppose that x is rational, and is not zero. Also suppose that y is irrational. Assume, for the sake of contradiction, that xy is rational. Then
where z ∈ ℚ. However, if this is true, we would be able to write that
or in other words, that y ∈ ℚ. But we assumed that y was irrational, contradicting ourselves in the process. We must conclude that a rational number times an irrational number is irrational.
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