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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

Andy wants to prove that the sum of a rational number and irrational number is irrational. He does this by the following.

Given

$x$ is a rational number and

$y$ is an irrational number. Assume that

$x+y$ is rational, and let the sum be represented by

$w$ . Since any rational number can be represented as the division of two integers, let

$x=a/b$ and

$w=c/d$ , where

$a,b,c,d$ are all integers. Therefore,

$x+y=w$ can be written as

$a/b + y = c/d$ . By the subtraction rule of equality,

$y = c/d - a/b$ . Rewriting the difference of the two fractions (using the multiplicative identity and distributive property), this becomes

$y = (cb - ad)/(db)$ . Since integers are closed under subtraction and multiplication,

$cb-ad$ is an integer, and so is

$db$ . Therefore, this would make

$y$ a rational number, by definition of rational numbers. But this contradicts our given, that

$y$ is irrational. Therefore,

$x+y$ must be irrational.

Is this proof correct? If not, why?

Yes, this proof is correct. It is called a proof by contradiction.
No, it is not correct. Andy has to use a two-table format for this to be a valid proof.
No, it is not correct. This only proves that $x+y$ is not rational, but that doesn't necessarily mean it is irrational.
No, it is not correct. Andy cannot prove something by contradiction; he must directly prove it.