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Algebra II : Number Theory

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Example Questions

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Example Question #71 : Number Theory

Which of the following is an irrational number?

Possible Answers:

$1.6\%$

$\sqrt{-3}$

3.14159

$\sqrt{11}$

Correct answer:

$\sqrt{11}$

Explanation:

An irrational number is a number that cannot be rewritten as a ratio of two numbers such as $\pi$ or since their decimal numbers are continuous.

Square roots that are not perfect squares are irrational such as:

$\sqrt2, \sqrt5, \sqrt{11}$

Be careful with numbers that are fixed. Fixed numbers are finite and can be rewritten as a fractional term.

Imaginary numbers are neither irrational or real numbers and is defined as the term . The value of $\sqrt{-3}= i\sqrt3$ is an imaginary number. These are called complex numbers.

Percentages can be rewritten as a fraction.

The answer is: $\sqrt{11}$

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Example Question #5121 : Algebra Ii

Try without a calculator.

True or false: the set

$\left \{ \sqrt {15}, \sqrt {16} , \sqrt {17}, \sqrt {18},\sqrt {19} \right \}$

comprises only irrational numbers.

Possible Answers:

True

False

Correct answer:

False

Explanation:

The square root of an integer is rational if and only if the integer is a perfect square of another integer.

$16 = 4 \times 4 = 4 ^{2}$ ,

so 16 is a perfect square, and $\sqrt{16}$ is a rational number. The statement is therefore false.

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Example Question #5122 : Algebra Ii

Try without a calculator.

True or false: the set

$\left \{ 7 \pi , 8 \pi, 9 \pi, 10 \pi, 11 \pi \right \}$

comprises only irrational numbers.

Possible Answers:

True

False

Correct answer:

True

Explanation:

$\pi$ is an irrational number, as is any integer multiple of $\pi$ . All of the elements are integer multiples of $\pi$ , so all of them are irrational.

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Example Question #5123 : Algebra Ii

Try without a calculator.

True or false: the set

$\left \{ 5e, 6e, 7e, 8e, 9e \right \}$

comprises only irrational numbers.

Possible Answers:

True

False

Correct answer:

True

Explanation:

is an irrational number, as is any integer multiple of . All of the elements are integer multiples of , so all of them are irrational.

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Example Question #75 : Number Theory

Which of the following represents an irrational number?

Possible Answers:

$-\sqrt{144}$

$0.\overline{3}$

$-\frac{1}{1000}$

None of the numbers is irrational.

$\sqrt{26}$

Correct answer:

$\sqrt{26}$

Explanation:

First, recall the definition of irrational numbers. Irrational numbers cannot be expressed as a ratio of integers.

The answer $0.\overline{3}$ is incorrect because it is a repeating decimal that can be rewritten as the following fraction:

$\frac{1}{3}$

In a similar vein, the following choice is already written as a simple fraction (negative numbers are not irrational):

$-\frac{1}{1000}$

It is important to note that not all square roots are irrational. For instance, the following square root can be simplified quite easily:

$-\sqrt{144} = -12$

The only answer that cannot be expressed as a simple fraction is:

$\sqrt{26}=5.09901951359278449161$

The answer is:

$\sqrt{26}$

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Example Question #1 : Irrational Numbers

Which of the following is an irrational number?

Possible Answers:

$\frac{21}{5}$

$0.\overline{3333}$

$\sqrt{9}$

$\pi$

5.75

Correct answer:

$\pi$

Explanation:

A rational number can be expressed as a fraction of integers, while an irrational number cannot.

$0.\overline{3333}$ can be written as $\frac{1}{3}$ .

$\sqrt{9}$ is simply , which is a rational number.

The number 5.75 can be rewritten as a fraction of whole numbers, $\frac{23}{4}$ , which makes it a rational number.

$\frac{21}{5}$ is also a rational number because it is a ratio of whole numbers.

The number, $\pi$ , on the other hand, is irrational, since it has an irregular sequence of numbers ( 3.14159 ...) that cannot be written as a fraction.

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Example Question #1 : Number Sets

If $A= \left \{ 10, 20, 30, 40, 50 \right \}$ , $B=\left \{ 10, 20, 40, 60 \right \}$ , and $C=\left \{ 20, 30, 40, 60 \right \}$ , then find the following set:

$A\cup B$

Possible Answers:

$\left \{ 10, 20, 30, 40, 50 \right \}$

$\left \{ 10, 20, 40 \right \}$

$\left \{ 10, 20, 40, 50, 60 \right \}$

$\left \{ 10, 20, 30, 40, 50, 60 \right \}$

$\left \{ 10, 20, 40, 60 \right \}$

Correct answer:

$\left \{ 10, 20, 30, 40, 50, 60 \right \}$

Explanation:

The union is the set that contains all the numbers from $A= \left \{ 10, 20, 30, 40, 50 \right \}$ and $B=\left \{ 10, 20, 40, 60 \right \}$ . Therefore the union is $\left \{ 10, 20, 30, 40, 50, 60 \right \}$ .

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Example Question #2 : Number Sets

If $A= \left \{ 10, 20, 30, 40, 50 \right \}$ , $B=\left \{ 10, 20, 40, 60 \right \}$ , and $C=\left \{ 20, 30, 40, 60\right \}$ , find the following set:

$A\cap B\cap C$

Possible Answers:

$\left \{ 40 \right \}$

$\left \{ 10, 20, 30, 40, 50, 60 \right \}$

$\left \{ 20, 40, 60 \right \}$

$\left \{ 20, 40 \right \}$

$\left \{ 20, 30, 40 \right \}$

Correct answer:

$\left \{ 20, 40 \right \}$

Explanation:

The intersection is the set that contains only the numbers found in all three sets. Therefore the intersection is $\left \{ 20, 40 \right \}$ .

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Example Question #1 : Number Sets

If $A= \left \{ 10, 20, 30, 40, 50 \right \}$ , $B=\left \{ 10, 20, 40, 60 \right \}$ , and $C=\left \{ 20, 30, 40, 60 \right \}$ , find the following set:

$B\cap C$

Possible Answers:

$\left \{ 10, 20, 40, 60 \right \}$

$\left \{ 40, 60 \right \}$

$\left \{ 20, 40, 60 \right \}$

$\left \{ 20, 40 \right \}$

$\left \{10, 20, 40 \right \}$

Correct answer:

$\left \{ 20, 40, 60 \right \}$

Explanation:

The intersection is the set that contains the numbers that appear in both $B=\left \{ 10, 20, 40, 60 \right \}$ and $C=\left \{ 20, 30, 40, 60 \right \}$ . Therefore the intersection is $\left \{ 20, 40, 60 \right \}$ .

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Example Question #4 : Number Sets

If $A=\left \{ 10, 20, 30, 40, 50\right \}$ , $B=\left \{ 10, 20, 40, 60 \right \}$ , and $C=\left \{ 20, 30, 40, 60 \right \}$ , find the following set:

$A\cap C$

Possible Answers:

$\left \{ 20, 30, 40 \right \}$

$\left \{ 30, 40 \right \}$

$\left \{ 20, 40 \right \}$

$\left \{ 20, 30, 40, 60 \right \}$

$\left \{ 10, 20, 30, 40, 50, 60 \right \}$

Correct answer:

$\left \{ 20, 30, 40 \right \}$

Explanation:

The intersection is the set that contains the numbers found in both sets. Therefore the intersection is $\left \{ 20, 30, 40 \right \}$ .

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Algebra II : Number Theory

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #71 : Number Theory

Example Question #5121 : Algebra Ii

Example Question #5122 : Algebra Ii

Example Question #5123 : Algebra Ii

Example Question #75 : Number Theory

Example Question #1 : Irrational Numbers

Example Question #1 : Number Sets

Example Question #2 : Number Sets

Example Question #1 : Number Sets

Example Question #4 : Number Sets

All Algebra II Resources

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