Eleventh Grade (Grade 11) Rational and Irrational Numbers Questions

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

Chose the correct missing reason in step 6 of the following proof.

Prove that for rational numbers

$a,b !=0$ , their product is also a rational number.

$" Statement "$	$" Reason "$
$1. a,b in QQ$	$1. "Given"$
$2. a = p/q, b = m/n, \ \ \ " where " p, q, m, n in ZZ$	$2. "Defition of rational numbers"$
$3. ab = (p/q)(m/n)$	$3. "Multiplication Property of Equality"$
$4.ab = (pm)/(q*n)$	$4. "Associative Property of Multiplication"$
$5. pm, \ qn in ZZ$	$5. "Product of two integers is always an integer"$
$6. ab = (pm)/(q*n) in QQ$	$6. ""$

Substitution Property of Equality
An integer divided by an integer is still an integer
Division Property of Equality
Definition of rational numbers

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

Which of the following disprove(s) the statement that the product of two irrational numbers is always irrational? Choose all that apply.

$sqrt(5) * (1/sqrt(5)) = 1$
$sqrt(9) * 1/root{3}{27} = 1$
$1/sqrt(2) * sqrt(8) = 2$
None of the above, since this statement is true.

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

Which of the following disproves the statement that the sum of two irrational numbers is always irrational?

$sqrt(3) + sqrt(3) = 6$
$sqrt(2) + (-sqrt(2)) = 0$
$sqrt(16) + sqrt(4) = 6$
None of the above, since this is a true statement.

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.

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.

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

This question is a part of a group with common instructions. View group »

What is the missing reason in step 2?

Given
Division Property of Equality
Definition of rational numbers
Any number can be rewritten in another form

Grade 11 Rational and Irrational Numbers CCSS: HSN-RN.A.2

This question is a part of a group with common instructions. View group »

What is the missing reason in step 6?

Integers are closed under multiplication
Multiplication Property of Equality
Multiplying two real numbers always results in an integer
Fundamental Theorem of Arithmetic

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Eleventh Grade (Grade 11) Rational and Irrational Numbers Questions