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Type: Multiple-Choice
Category: Rational and Irrational Numbers
Level: Grade 11
Standards: HSN-RN.B.3
Author: nsharp1
Created: 5 years ago

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Rational and Irrational Numbers Question

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Grade 11 Rational and Irrational Numbers CCSS: HSN-RN.B.3

Is the following proof correct? If not, why not?

Prove that

$x*y$ is irrational.
Given

$x$ is rational,

$y$ is irrational. Assume that

$x*y$ is rational, and therefore one can let

$x*y = m/n$ , where

$m,n$ are integers. Since

$x$ is rational, it can be represented as the division of two integers. Let

$x=a/b$ , where

$a,b$ are integers. Therefore,

$(a/b)*y=m/n$ . Applying the multiplicative inverse and associative property, this becomes

$y = (m/n) * (b/a) = (m*b)/(n*a)$ . Since integers are closed under multiplication, this value is rational, by definition of rational numbers, implying that

$y$ is rational. But this contradicts the given. Therefore,

$x*y$ is irrational.

Yes, it is correct.
No, it is not correct. It needs to be in a two-column table format.
No, it is not correct. If $x=0$ , then $x*y$ is rational. At the start, one needs to restrict $x$ to being a non-zero rational number.
No, it is not correct. The proof cannot be done by contradiction. It needs to be shown directly that $x*y$ cannot be rational.