# Real Numbers: Definition,Properties and Examples

Real Numbers include both rational and irrational numbers like fractions, integers, and more. The real numbers can be easily represented on a number line with zero in the middle, positive numbers on the right side, and negative numbers on the left side. However, not all the numbers are part of the rational numbers. There are some numbers like √-1, 2 + 3i, and -i, which are not included in the real numbers.A good knowledge of the real numbers is important for various competitive exams like JEENEETSSCBanking, and many state-level exams. To better understand this basic idea of the real numbers , let’s go into more detail.

## Definition of Real Numbers

Real numbers include all the numbers you can think of that are used in our everyday life. They include whole numbers like 1, 2, 3, and so on, as well as numbers with decimals like 3.14 (which you might know as pi). Even numbers like -5 (negative five) are real numbers. It includes positive and negative integers, fractions, and irrational numbers. However, there are some numbers as well that are not included in the real numbers like rational nor irrational are non-real numbers, like, √-1, 2 + 3i, and -i. These types of numbers are included in complex numbers.

Also Read: Basics of Prime Numbers

## Set of Real Numbers

The set of real numbers, denoted by R, includes all rational and irrational numbers. This includes a wide collection of numbers, including natural numbers (N), whole numbers (W), integers (Z), and rational and irrational numbers. The table below represents the numbers that come under the set of real numbers.

Also Read: Properties of Isosceles Triangle

## Types of Real Numbers

All numbers are either rational or irrational. There’s no number that is both or neither. This means every single number you can think of belongs into one of these two groups.

Rational numbers: These are numbers that can be written as a simple fraction. This includes whole numbers (like 1, 2, 3), fractions (like 1/2, 3/4), and decimals that either end (like 0.25) or repeat (like 0.333…).

Irrational numbers: These are numbers that can’t be written as a simple fraction. They have decimal parts that go on forever without repeating. Famous examples are numbers like pi (3.14159…) and the square root of 2 (1.41421…).

## Symbols of Real Numbers

There are certain symbols that we use to represent real numbers and the terms related to them. You can have a look at these symbols below.

• N – Natural numbers
• W – Whole numbers
• Z – Integers
• Q – Rational numbers

## Properties of Real Numbers

The properties of the real numbers can be found in the given section of the article. Have a look at the properties.

• Commutative Property: For addition and multiplication, the order of operands does not affect the result. For addition: a + b = b + a and For multiplication: a * b = b * a
• Associative Property: The grouping of operands does not affect the result for addition and multiplication. For addition: (a + b) + c = a + (b + c) and For multiplication: (a * b) * c = a * (b * c)
• Distributive Property: Multiplication distributes over addition. a * (b + c) = a * b + a * c
• Identity Property: Additive identity: There exists a number 0 such that a + 0 = a for any real number a. Multiplicative identity: There exists a number 1 such that a * 1 = a for any real number a.
• Inverse Property: Additive inverse: For every real number a, there exists an additive inverse -a such that a + (-a) = 0. Multiplicative inverse: For every non-zero real number a, there exists a multiplicative inverse 1/a such that a * (1/a) = 1.

## Difference Between Real Numbers and Integers

Real numbers are like all the numbers you can imagine on a number line, including whole numbers, fractions, and decimals that go on forever without repeating. Integers are a smaller group within real numbers. They are whole numbers, but they also include negative numbers. So, every integer is a real number, but not every real number is an integer. For example, 3 is both an integer and a real number, but 1/2 is a real number but not an integer.

## Representation of Real Numbers on Number Line

A number line is like a ruler for numbers. It’s a straight line with a middle point called zero. Numbers are placed at equal distances on either side of zero. Numbers to the right of zero are positive, and numbers to the left are negative.

Look at the number line. You can see numbers like -5/2, 0, 3/2, and 2 marked on it.

To show numbers on a number line:

1. Draw a straight line with arrows at both ends. Put a zero in the middle.
2. Make marks at equal distances on both sides of zero.
3. Label the marks with numbers.

## FAQs

What are real numbers?

Real numbers are any numbers that can be found on a number line. This includes whole numbers, fractions, decimals, and even numbers.

What is the difference between rational and irrational numbers?

Rational numbers can be written as a simple fraction, like 1/2 or 3/4. Irrational numbers cannot be written as a fraction.

Are all integers real numbers?

Yes, all integers are real numbers. Integers are whole numbers and their opposites (like -3, -2, -1, 0, 1, 2, 3), and they are part of the bigger group of real numbers.

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