# Real Numbers

Real numbers are values that can be expressed as an infinite decimal expansion. Real numbers include Integers, Natural Numbers, and others we will talk about in the coming sections. Examples of real numbers are ¼, pi, 0.2, and 5.

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Real numbers can be represented classically as a long infinite line that covers negative and positive numbers.

## Number types and symbols

The numbers you use to count are known as whole numbers and are part of rational numbers. Rational numbers and whole numbers compose also the real numbers, but there are many more, and the list can be found below.

Venn diagram of numbers

## Types of real numbers

It is important to know that for any real Number picked, it is either a rational number or an irrational number which are the two main groups of real numbers.

### Rational numbers

Rational numbers are a type of real numbers that can be written as the Ratio of two integers. They are expressed in the form p / q, where p and q are integers and not equal to 0. Examples of rational numbers are$\frac{1}{2},\frac{10}{12},\frac{3}{10}$ . The set of rational numbers is always denoted by Q.

#### Types of rational numbers

There are different types of rational numbers and these are

• Integers, for example, -3, 5, and 4.

• Fractions in the form p / q where p and q are integers, for example, ½.

• Numbers that do not have infinite decimals, for example, ¼ of 0.25.

• Numbers that have infinite decimals, for example, ⅓ of 0.333….

### Irrational numbers

Irrational numbers are a type of real numbers that cannot be written as the Ratio of two integers. They are numbers that cannot be expressed in the form p / q, where p and q are integers.

As mentioned earlier, real numbers consist of two groups – the rational and irrational numbers, $\left(R-Q\right)$expresses that irrational numbers can be obtained by subtracting rational numbers group (Q) from real numbers group (R). That leaves us with the irrational numbers group denoted by Q '.

#### Examples of irrational numbers

• A common example of an irrational Number is 𝜋 (pi). Pi is expressed as 3.14159265….

The decimal value never stops and does not have a repetitive pattern. The fractional value closest to pi is 22/7, so most often we take pi to be 22/7.

• Another example of an irrational number is $\sqrt{2}$. the value of this is also 1.414213 ..., $\sqrt{2}$ is another number with an infinite decimal.

## Properties of real numbers

Just as it is with integers and natural numbers, the set of real numbers also has the closure property, commutative property, the associative property, and the distributive property.

• #### Closure property

The product and sum of two real numbers is always a real number. The closure property is stated as; for all a, b ∈ R, a + b ∈ R, and ab ∈ R.

If a = 13 and b = 23.

then 13 + 23 = 36

so, 13 × 23 = 299

Where 36 and 299 are both real numbers.

• #### Commutative property

The product and sum of two real numbers remain the same even after interchanging the order of the numbers. The commutative property is stated as; for all a, b ∈ R, a + b = b + a and a × b = b × a.

If a = 0.25 and b = 6

then 0.25 + 6 = 6 + 0.25

6.25 = 6.25

so 0.25 × 6 = 6 × 0.25

1.5 = 1.5

• #### Associative property

The product or sum of any three real numbers remains the same even when the grouping of numbers is changed.

The associative property is stated as; for all a, b, c ∈ R, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.

If a = 0.5, b = 2 and c = 0.

Then 0.5 + (2 + 0) = (0.5 + 2) + 0

2.5 = 2.5

So 0.5 × (2 × 0) = (0.5 × 2) × 0

0 = 0

• #### Distributive property

The distributive property of multiplication over addition is expressed as a × (b + c) = (a × b) + (a × c) and the distributive property of multiplication over subtraction is expressed as a × (b - c) = (a × b) - (a × c).

If a = 19, b = 8.11 and c = 2.

Then 19 × (8.11 + 2) = (19 × 8.11) + (19 × 2)

19 × 10.11 = 154.09 + 38

192.09 = 192.09

So 19 × (8.11 - 2) = (19 × 8.11) - (19 × 2)

19 × 6.11 = 154.09 - 38

116.09 = 116.09

## Real Numbers - Key takeaways

• Real numbers are values that can be expressed as an infinite decimal expansion.
• The two types of real numbers are rational and irrational numbers.
• R is the symbol Notation for real numbers.
• Whole numbers, natural numbers, rational numbers, and irrational numbers are all forms of real numbers.

#### Flashcards in Real Numbers 15

###### Learn with 15 Real Numbers flashcards in the free StudySmarter app

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What are real numbers?

Real numbers are values that can be expressed as an infinite decimal expansion.

What are real numbers with examples?

Every real number picked is either a rational number or an irrational number. They include 9, 1.15, -6, 0, 0.666 ...

What is the set of real numbers?

It is the set of every number including negatives and decimals that exist on a number line. The set of real numbers is noted by the symbol R.

Are irrational numbers real numbers?

Irrational numbers are a type of real numbers.

Are negative numbers real numbers?

Negative numbers are real numbers.

## Test your knowledge with multiple choice flashcards

How are rational numbers expressed?

Which of these is a rational number with infinite decimals?

What is an irrational number?

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