# Real Numbers

In the number system, real numbers are essentially a mixture of rational and irrational numbers. Generally, all the arithmetic operations can be performed on these numbers and can also be represented in the number line. Simultaneously, un-real numbers are imaginary numbers that cannot be represented in the number line. It is widely used to represent a complex number. In this article, we are going to cover real numbers, types, and their properties.

## What is the real number?

Real numbers are a mixture of rational and irrational numbers. They can be either positive or negative numbers and denoted by the symbol R. It contains all-natural numbers, decimals, and fractions. A real number can be a number that can be expressed by a point on the number line. Some examples of real numbers are 3.5, 0.003, 2/3, π, etc.

## Types of real numbers

We know that the set of real numbers is made up of both rational and irrational numbers. It simply indicates that if we pick up some number R, it's either rational or irrational. The various types of real numbers are as follows:

### Rational Numbers

Any number specified in the form of a p/q or ratio fraction is called a rational number. The numerator is defined as p, and the denominator is represented as q, where q is not equal to zero. A rational number may be an integer, whole, and natural number.

For example: 0.67, 17, 214, etc.

### Irrational Number

Irrational numbers are defined as the real number that cannot be represented as a simple fraction. In other words, it cannot be represented as a ratio, such as p/q, where p and q are integers, q≠0. It is the opposite of rational numbers.

For example, √5, √11, √21, etc.

### Natural Numbers

An integer greater than 0 is a natural number. Natural numbers start at 1 and move to infinity, for example, 1, 2, 3, 4, 5......... For beginners, we have to use natural numbers if we are timing anything in seconds. We don't use the decimal point in natural numbers, but we may use the commas when we have written large numbers, e.g., 1,500 and 156,597,720. A minus sign (-) can never be used with natural numbers because they should not be negative.

### Whole Numbers

The whole number is the number system that contains all positive integers from 0 to infinity. The whole number is an integer of 0 or greater number.

For example: 1, 2, 3, 4, 5…..

## Properties of Real Numbers

There are four primary assets that include commutative property, associative property, distributive property, and property of identification.

### Commutative Property

In mathematical computation, commutative property or commutative law states that the order of terms does not matter when performing an operation. This property is only applicable to the adding and multiplying method, such as a + b = b + a, and a b = b a. But it does not refer to the form of subtraction and division since a - b ≠ b - a and a/b ≠ b/a.

This property states that if we add the two integer numbers, the result will always be the same number even the number's position is changed. Let's take the two integer numbers that are A and B as follows;

A + B = B + A

For Example:

1. 2 + 3 = 3 + 2 = 5
2. 4 + 5 = 5 + 4 = 9
3. 7 + 8 = 8 + 7 = 15
4. 18 + 8 = 8 + 18 = 26

### Commutative property of multiplication

This property states that if we multiply the two integer numbers, we will get the same result, even if the integer number's position is interchanged. Let's take the two integers numbers that are A and B as follows;

A * B = B * A

For Example:

1. 4 * 2 = 2 * 4 = 8
2. 5 * 6 = 6 * 5 = 30
3. 12 * 3 = 3 * 12 = 36
4. 9 * 7 = 7 * 9 = 63

### Associative Property

The associative property is a property of certain binary operations. In propositional logic, associativity is a true substitution rule for logical proof expressions. Depending on the associative property, we may add or multiply regardless of how the numbers are grouped. In simple terms, it refers to the grouping of numbers. It is mostly used to regroup things, and every form of computation for time depends on the regrouping of things.

The addition property of associative follows associative property. The associative property of addition states that:

(x + y) + z = x + (y + z)

For example: Let's take an example to explain the associative property of addition. We are taking the value of x, y, z is 5, 6, 8.

Now,

(5 + 6) + 8 = 5 + (6 + 8)

(11) + 8 = 5 + (14)

19 = 19

L.H.S = R.H.S

### Associative property for multiplication

The rule for the associative property of multiplication is as follows:

(xy) z = x (yz)

For example: Let's take an example to explain the associative property of multiplication. We are taking the value of x, y, z is 3, 4, 8.

Now,

(3 * 4) * 8 = 3 * (4 * 8)

(12) * 8 = 3 * (32)

96 = 96

L.H.S = R.H.S

### Distributive Property

It is an algebraic property used to multiply a value with two or more other values in the parenthesis. When a factor is multiplied by the addition or sum of the two numbers, it is necessary to multiply each of the two numbers by a factor. After that, perform the addition operation. Symbolically, this property can be described as:

A (B + C) = AB + AC

For example: Let's take an example to explain the distributive property. We are taking the value of x, y, z is 2, 5, 7.

Now,

2 * (5 + 7)

2 * 5 + 2 * 7

10 + 14

24

### The distributive property for addition

When a value is multiplied by a number, the distributive property of multiplication over addition is applied.

For Example: Let's take an example to explain the distributive property for addition. We are taking the value of x, y, and z as 7, 9, and 3.

7 * (9 + 3)

7 * 9 + 7 * 3

63 + 21

84

### The distributive property for subtraction

For Example: Let's take an example to explain the distributive property for subtraction. Now, we are taking the values 4, 8, 5 for A, B, and C.

Now,

= 4 * (8 - 5)

= 4 * 3

= 12

### The distributive property of division

Using this property, we may divide the larger numbers by breaking these numbers into smaller factors.

For example: Let's take an example to explain the distributive property for the division. We are taking the value is 48/4.

We may write this number 48 as (12 + 36)

Here, we may apply the distributive property for division and breaking this number into small factors, and we get:

(12 / 4) + (36 / 4)

= 3 + 9

= 12

### Identity Property

This property includes the additive identity and multiplicative identity. We can perform various operations based on real numbers, including addition, multiplication, subtraction, and division.

The additive identity of the numbers is the property of the numbers used to perform additional operations. This property states that if we add any number with 0, the result will be the same. "Zero" is called an identity element, also defined as an additive identity.

For Example: Let's take an example to explain the additive identity property.

15 + 0 = 15

Here, 0 is the additive identity, and the result is the same number.

### What is multiplicative identity?

Multiplicative identity of numbers is a number property that is introduced when multiplication operations are carried out. This property states that if we multiplied by any number with 0, the result would provide the number as a product. "1" is a number's multiplicative identity. It is true if 1 itself is the number being multiplied.

For Example: Let's take an example to explain the multiplicative identity property.

Suppose a is a real number, the multiplicative identity property represented as:

A * 1 = A

1 * A = A

Some other examples are as follows:

10 * 1 = 10

1 * 10 = 1

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