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CK-12 FlexBooks explains Real numbers can be broken down into different types of numbers such as rational and irrational numbers. They can be visualized using number lines and operated on using set symbols and operators.

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Real NumBeRs



Study Guides

Big Picture

Real numbers are used to measure the quantity of things in life. Almost any number you can think of is most likely to

be a real number. Real numbers can be broken down into different types of numbers such as rational and irrational

numbers. They can be visualized using number lines and operated on using set symbols and operators. General

guidelines and rules are created to work with real numbers.

Key Terms

, as long as the denominator is not equal to 0.

Rational Number: Ratio of one integer to another:

Integer: A rational number where the denominator is equal to 1. Includes natural numbers, negative natural

numbers, and 0.

Natural Numbers: Counting numbers such as 1, 2, 3.

Whole Numbers: All natural numbers and 0.

Non-Integer: A rational number where the denominator is not equal to 1.

Proper Fraction: Numerator is less than denominator. Represents a number less than one.

Improper Fraction: Numerator is greater than denominator. Represents a number greater than 1.

Equivalent Fractions: Two fractions that represent the same amount.

or �.

Irrational Number: Number that cannot be expressed as a fraction, such as

Understanding Real Numbers


Here are some common symbols used in algebra:

1. Sum or product of two rational numbers is rational.

Example: 2 + 3 = 5







2. Sum of rational number and irrational number is irra-




× or ·


Example: 2 +

= 2 +

÷ or /


3. Product of nonzero rational number and irrational

square root, nth root


number is irrational.

| |

absolute value

Example: 3 ·

= 3



Example: 5 · � = 5 �


not equal


4. Difference between two whole rational numbers is not

approximately equal

always a positive number.

less than, less than or

Example: 5 - 4 = 1

<, ≤

equal to

Example: 5 - 9 = -4

greater than, greater

>, ≥

5. Quotient of a whole rational divisor and a whole

than or equal to

dividend is not always a whole number.

{ }

set symbol


an element of a set

( ), [ ]

group symbols


This guide was created by Nicole Crawford, Jane Li, and Jin Yu. To learn more

Page 1 of 3

about the student authors, visit



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Real NumBeRs




Exponents and nth Roots


nth Roots

Exponent is a short-hand notation for repeated multi-

The nth root is the inverse operation of raising a number


to the nth power. So the inverse operation of xn = y is



2 · 2

2 = 23. We say that 2 is raised to the power

of 3.

= 4 because 42 = 16

is called the radical sign


2 · 2 · 2 · 2

2 = 25. We say that 2 is raised to the

When n=2, we usually write

, not

, and we call

power of 5.

it the square root.

For x , we say that x is raised to the power of n.


When n=3, we call it the cube root.

•  x

and n are variables, symbols that are used to

represent a value.

If the nth root can’t be simplified (reduced) to a rational

If n=2, we can also say x squared. If x=3, we

number without the radical sign (

), the number is


say x cubed.


= 8, so it is a rational number.

cannot be reduced any further and is irrational.

We can get an approximate value for irrational square

roots by using the calculator. In decimal form, the

number will look like an unending string of numbers.


≈ 1.414 when rounded to three decimal


Fractions and Decimals

A rational number is just a ratio of one number to another written in fraction form as


Every whole number is a rational number where the denominator equals 1.

A denominator equal to 1 is sometimes called the invisible denominator because it is not usually written out:


Fraction bar: the line that separates the numerator and the denominator. The denominator b ≠ 0.

A proper fraction represents a number less than one because a < b, while an improper fraction represents a

number greater than one because a > b.

A negative fraction is usually written with the negative sign to the left of the fraction


could equal


Improper fractions can be rewritten as an integer plus a proper fraction (mixed number).


Whenever we can write two fractions equal (=) to each other, we have equivalent fractions.


, so


are equivalent fractions.

The fractions

are equivalent as long as c ≠ 0.

A fraction can be converted into a decimal - just divide the numerator

by the denominator.

Figure: Equivalent fractions


The ... means that the decimal goes on forever.

Not all decimals can be converted into fractions.

If the numbers after the decimal point (.) never repeats and never ends, the number is irrational.

Any number that can’t be written as a fraction is irrational.

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Real NumBeRs





Sets are used to define groups of elements. In math,

Difference: the set of elements that belong to A only

sets can be used to define different types of numbers,

Denoted as A \ B

such as even and odd numbers. Outside of math, sets

can also be used for other elements such as sets of keys

or sets of clothing. The different types of sets (as shown

below) are used to classify the objects in the sets.

We can list the elements (members) of a set inside the

symbols { }. If A = {1, 2, 3}, then the numbers 1, 2,

and 3 are elements of set A.

Numbers like 2.5, -3, and 7 are not elements of A.

If A is the group of whole numbers and B is the

We can also write that 1

A, meaning the number

group of natural numbers, A \ B is 0

1 is an element in set A.

The order here matters! B \ A means the set of

If there are no elements in the set, we call it a null

elements belonging to B only.

set or an empty set.

Complement Set: all elements in a set that is not A

Union: the set of all elements that belong to A or B

Denoted as Ac

Denoted as A


B \ A is equal to Ac. If A is the group of whole

The union of rational numbers and irrational

numbers and B is the group of natural numbers,

numbers is all real numbers.

Ac is null (there are no elements in set B that is not

Intersection: the set of elements that is true for both

also in A)

A and B

Disjoint Sets: when sets A and B have no common

Denoted as A



Rational and irrational numbers are disjoint sets.


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