Irrational Numbers: Definition, Properties, and Operations

Question 1

What is an example of an irrational number?

Accepted Answer

√2

Answer

1/2

Answer

-3

Answer

0.123456789101112...

Question 2

Which of the following numbers is irrational?

Accepted Answer

√2

Answer

1/2

Answer

3.14

Answer

0.25

Question 3

Simplify √(4 + 9).

Accepted Answer

2√13

Answer

√4 + √9

Answer

√13

Answer

4√3

Question 4

Which operation is not valid for irrational numbers?

Accepted Answer

Division by a zero.

Answer

Addition

Answer

Subtraction

Answer

Multiplication

Question 5

An irrational number's decimal expansion is:

Accepted Answer

Non-terminating and non-repeating

Answer

Terminating

Answer

Repeating

Answer

Alternates between terminating and non-terminating.

Question 6

Which property is NOT applicable to irrational numbers?

Accepted Answer

Irrational numbers are greater than any rational number.

Answer

Irrational numbers cannot be exactly represented as a fraction of two integers.

Answer

Irrational numbers do not terminate or repeat when written as a decimal.

Answer

Irrational numbers can be represented by decimals that go on forever.

Question 7

What is the square root of 49?

Accepted Answer

7

Answer

14

Answer

21

Answer

√7

Question 8

Which of the following is the Pythagorean theorem?

Accepted Answer

a^2 + b^2 = c^2

Answer

a^2 - b^2 = c^2

Answer

c^2 - a^2 = b^2

Answer

a^2 + b^2 - c^2 = 0

Question 9

Identify the property of irrational numbers that is NOT true:

Accepted Answer

They are rational numbers.

Answer

They extend infinitely without repeating.

Answer

They cannot be written as a ratio of two integers.

Answer

Their decimal expansions are non-terminating and non-repeating.

Question 10

Find the rational approximation of √7:

Accepted Answer

2.6

Answer

2.4

Answer

2.8

Answer

3.0

Question 11

Which of the following is NOT a property of irrational numbers?

Accepted Answer

They are always greater than rational numbers.

Answer

They cannot be expressed as a fraction of two integers.

Answer

They are non-terminating and non-repeating decimals.

Answer

They can be approximated by rational numbers.

Question 12

Which of the following numbers is irrational?

Accepted Answer

√2

Answer

1/2

Answer

5

Answer

3.14

Question 13

What is the square root of 25?

Accepted Answer

5

Answer

√2

Answer

10

Answer

15

Question 14

Which of the following is a rational approximation of √7?

Accepted Answer

2.64

Answer

3.14

Answer

1.41

Answer

4.23

Question 15

What is the sum of √2 and √3?

Accepted Answer

√6

Answer

√5

Answer

5

Answer

6

Question 16

What is the quotient of √12 and √3?

Accepted Answer

√3

Answer

√4

Answer

4

Answer

6

Question 17

What is the square of √6?

Accepted Answer

12

Answer

6

Answer

3

Answer

18

Question 18

What is the cube of √2?

Accepted Answer

8

Answer

2

Answer

4

Answer

16

Question 19

Which of the following is an example of an irrational number used in real life? The speed of light is approximately 299,792,458 meters per second, which is an irrational number.

Accepted Answer

The speed of light

Answer

The population of the United States

Answer

The number of days in a year

Answer

The length of a football field

Question 20

When calculating the volume of a pyramid, why might irrational numbers be involved?

Accepted Answer

The base length and/or height of the pyramid could be irrational numbers.

Answer

Irrational numbers are needed to find the slant height of the pyramid.

Answer

The surface area of the pyramid requires irrational numbers in its calculation.

Answer

Irrational numbers are not encountered when determining the volume of a pyramid.

Question 21

When multiplying two irrational numbers, which step should you take to simplify the product?

Accepted Answer

Multiply the numbers inside the radical symbols and simplify if possible.

Answer

Add the numbers inside the radical symbols and simplify if possible.

Answer

Subtract the numbers inside the radical symbols and simplify if possible.

Answer

Divide the numbers inside the radical symbols and simplify if possible.

Question 22

Which of the following is equivalent to the square root of 4?

Accepted Answer

2

Answer

4

Answer

1/2

Answer

2/3

Question 23

Which of the following is the decimal representation of the irrational number √5?

Accepted Answer

The decimal representation of √5 is non-terminating and non-repeating.

Answer

2.236

Answer

2.2360679

Answer

2.2360679125

Question 24

Which of the following is the best approximation of the value of π?

Accepted Answer

3.14

Answer

3

Answer

3.1

Answer

3.141592

Question 25

Which of the following is the correct representation of the irrational number √3 on the number line?

Accepted Answer

A point between 2 and 3

Answer

A point between 1 and 2

Answer

A point between 3 and 4

Answer

A point between 4 and 5

Question 26

Which of the following is NOT a property of irrational numbers? Irrational numbers are numbers that are not rational. Rational numbers can be expressed as a fraction of two integers.

Accepted Answer

They are rational numbers.

Answer

They cannot be written as a fraction of two integers.

Answer

They are decimals that do not terminate or repeat.

Answer

They can be approximated by rational numbers.

Question 27

Which of the following is the decimal approximation of π to two decimal places? When approximating irrational numbers, we can use a calculator or a table of values.

Accepted Answer

3.14

Answer

3.141

Answer

3.1

Answer

3.1415

Question 28

Which of the following is the least common multiple of √2 and √3? To find the least common multiple of two radicals, we can multiply the two radicals together.

Accepted Answer

2√6

Answer

√2

Answer

√6

Answer

6

Question 29

Which of the following is NOT a property of irrational numbers?

Accepted Answer

They are always larger than 1.

Answer

They cannot be expressed as a simple fraction of two integers.

Answer

They have non-terminating and non-repeating decimal representations.

Answer

They can be approximated by rational numbers.

Question 30

Which of the following is equivalent to √5?

Accepted Answer

2.236...

Answer

2.5

Answer

2.75

Answer

3

Question 31

Which of the following is the square root of 8?

Accepted Answer

4

Answer

2

Answer

6

Answer

8

Question 32

Select the correct relationship between rational and irrational numbers.

Accepted Answer

Rational and irrational numbers are distinct number sets.

Answer

All rational numbers are irrational.

Answer

All irrational numbers are rational.

Answer

Irrational numbers are a subset of rational numbers.

Question 33

How is the diagonal of a square related to irrational numbers?

Accepted Answer

If a square's side has an irrational length, its diagonal will also be irrational.

Answer

The diagonal of a square is always a whole number.

Answer

The diagonal of a square is always shorter than its sides.

Question 34

Which of the following rational numbers is the smallest?

Accepted Answer

0.125

Answer

0.25

Answer

0.375

Answer

0.5

Question 35

What is the result of 4√5 - √5?

Accepted Answer

3√5

Answer

5√5

Answer

7√5

Answer

9√5

Question 36

Which of the following is an irrational number?

Accepted Answer

2.14

Answer

1/2

Answer

5

Answer

100

Question 37

Classify irrational numbers. Which of the following statements about irrational numbers is false?

Accepted Answer

They are always greater than 1.

Answer

They are numbers that cannot be expressed as a fraction of two integers.

Answer

They do not terminate or repeat.

Answer

They can be represented on a number line.

Question 38

Identify the set of irrational numbers. Which of the following notations represents the set of irrational numbers?

Accepted Answer

R

Answer

Q

Answer

N

Answer

Z

Question 39

Understand the relationship between rational and irrational numbers. Which of the following is a rational approximation of π?

Accepted Answer

22/7

Answer

3.14

Answer

1.62

Answer

0.78

Question 40

Which of the following is NOT a property of irrational numbers?

Accepted Answer

They are always greater than 1.

Answer

They cannot be expressed as a fraction of two integers.

Answer

They cannot be written as a terminating or repeating decimal.

Answer

They are not rational numbers.

Question 41

Which of the following is the cube root of 27?

Accepted Answer

3

Answer

9

Answer

27

Answer

3√27

Question 42

Which of the following is the fourth root of 81?

Accepted Answer

3

Answer

9

Answer

27

Answer

81

Question 43

Which of the following is the rational approximation of π?

Accepted Answer

3.14

Answer

3

Answer

3.1

Answer

3.1415

Question 44

Which of the following is an irrational number?

Accepted Answer

π

Answer

0.123

Answer

1/2

Answer

√9

Question 45

A characteristic of an irrational number is that it:

Accepted Answer

Continues infinitely without repeating.

Answer

Can be represented as a fraction of two integers.

Answer

Can be approximated using rational numbers.

Answer

Cannot be greater than 1.

Question 46

Which approximation is most precise for the irrational number π?

Accepted Answer

3.1415

Answer

3.1

Answer

3.14

Answer

3.141

Question 47

Which of the following is NOT a characteristic of irrational numbers?

Accepted Answer

They can be expressed as a fraction of two integers.

Answer

They cannot be exactly represented as a decimal.

Answer

They are non-terminating and non-repeating decimals.

Question 48

Which of the following is the closest approximation of √3?

Accepted Answer

1.732

Answer

1.5

Answer

1.8

Answer

2

Question 49

What is the result of √2 + √3?

Accepted Answer

An irrational number

Answer

A rational number

Answer

1

Answer

0

Question 50

What is the decimal representation of 1/√2?

Accepted Answer

0.7071... (non-terminating, non-repeating)

Answer

1

Answer

0.5

Answer

0.75