 # Arithmetic Operations

Arithmetic operations is a branch of mathematics, that involves the study of numbers, operation of numbers that are useful in all the other branches of mathematics. It basically comprises operations such as Addition, Subtraction, Multiplication and Division.

These basic mathematical operations (+, -, ×, and ÷) we use in our everyday life. Whether we need to calculate the annual budget or distribute something equally to a number of people, for every such aspect of our life, we use arithmetic operations.

Let us understand each of the basic arithmetic operations in detail.

## Basic Arithmetic Operations

The four basic arithmetic operations in Maths, for all real numbers, are:

• Addition (Finding the Sum; ‘+’)
• Subtraction (Finding the difference;  ‘-’)
• Multiplication (Finding the product; ‘×’ )
• Division (Finding the quotient; ‘÷’)

Let us discuss all these four basic arithmetic operations with rules and examples in detail.

The addition is a mathematical process of adding things together. The addition process is denoted by ‘+’ sign. It involves combining two or more numbers into a single term. In addition process, the order does not matter. It means that the addition process is commutative. It can involve any type of number whether it be a real or complex number, fraction, or decimals.

Example: 4.13 + 3.87 = 8

The addition of more than two numbers, values or terms is also known as a summation of terms and can involve n number of values.

The following are the addition rules for integers:

• Addition of two positive integers is a positive integer
• Addition of two negative integers is a negative integer
• While adding positive and negative integers, subtract the integers and use the sign of the largest integer number

## Subtraction Definition

The subtraction operation gives the difference between two numbers. Subtraction is denoted by ‘-‘ sign. It is almost similar to addition but is the conjugate of the second term. It is the inverse process of addition. The addition of the term with the negative term is known as subtraction. This process is mostly used to find how many are left when some things are taken away.

Example: 15 – 7

The term can also be re-written as 15 + (-7)

### Subtraction Rules

The following are the subtraction rules for integers:

• If both the signs of the integers are positive, the answer will be the positive integer
• If both the signs of the integers are negative, the answer will be the negative integer
• If the signs of the integers are different, subtract the values, and take the sign from the largest integer value.

## Multiplication Definition

Multiplication is known as repeated addition. It is denoted by ‘×’ or ‘*’. It also combines with two or more values to result in a single value. The multiplication process involves multiplicand, multiplier. The result of the multiplication of multiplicand and the multiplier is called the product

Example: 2 × 3 = 6

Here, “2” is the multiplier, “3” is the multiplicand, and the result “6” is called the product.

The product of two numbers says ‘a’ and ‘b’ results in a single value term ‘ab,‘ where a and b are the factors of the final value obtained.

### Multiplication Rules

The following are the multiplication rules for the integers.

• The product of two positive integers is a positive integer
• The product of two negative integers is a positive integer
• The product of positive and negative integer is a negative integer

## Division Definition

The division is usually denoted by ‘÷‘ and is the inverse of multiplication. It constitutes two terms dividend and divisor, where the dividend is divided by the divisor to give a single term value. When the dividend is greater than the divisor, the result obtained is greater than 1, or else it would be less than 1.

Example: 4 ÷ 2 = 2

Here, “4” is the dividend, “2” is the divisor, and the result “2” is called the quotient.

### Division Rules

The following are the division rules for integers:

• The division of two positive integers is a positive integer
• The division of two negative numbers is a positive integer
• The division of integers with different signs results in the negative integer.

## Mathematical Operations

The basic mathematical operations are the four arithmetic operations that we have already learned in the above sections.

Addition and subtraction are inverse operations of each other. It means if the addition of two numbers gives the third number, then subtraction of an added number from the third number will result in the original number.

Example:

4 + 7 = 11

Now, if we subtract 7 from 11, we get;

11 – 7 = 4

Thus, we got the original number.

Similarly, multiplication and division are also inverse operations.

If 4 x 5 = 20

Then,

20/5 = 4

Thus, we can see, these mathematical operations are related to each other. Also, these operations are the simplest form of mathematical calculations, which can be easily understood by everyone.

## Basic Arithmetic Properties

The basic arithmetic properties for real numbers are:

• Commutative property
• Associative property
• Distributive property

### Commutative Property

This property is applicable only for two arithmetic operations, i.e., addition and multiplication.

Suppose A and B are two numbers, then, according to commutative property;

 A+B = B+A Example: 1 + 2 = 2 + 1 A x B = B x A Example: 1 x 2 = 2 x 1

Thus, the order of numbers in addition and multiplication does not change the result.

### Associative Property

Like commutative property, the associative property is also applicable to addition and multiplication.

 A+(B+C) = (A+B)+C Example: 1 + (2+3) = (1+2)+3 Ax(BxC) = (AxB)xC Example: 1 x (2 x 3) = (1 x 2) x 3

Thus, if we change the grouping of numbers, the result does not change.

### Distributive Property

According to the distributive property, if A, B and C are any three real numbers, then,

 A x (B + C) = A x B + A x C

Example: 2 x (3 + 4) = (2 x 3) + (2 x 4)

2 x 7 = 6 + 8

14 = 14

Hence, proved.

## Solved Examples

Q.1: Add 23 and 40 and then subtract 20 from the sum.

Solution: On adding 23 and 40, we get;

Sum = 23 + 40 = 63

Now subtracting 20 from the sum, we get;

63 – 20 = 43

Q.2: Solve: 20 + 20 + 20 + 20 + 20.

Solution: Given, 20 + 20 + 20 + 20 + 20

It is clear that 20 is added to itself five times, thus, we can write;

5 times of 20 = 5 x 20 = 100

Q.3: Find the value of (6 x 4) ÷ 12 + 72 ÷ 8 – 9.

Solution: Given,

(6 x 4) ÷ 12 + 72 ÷ 8 – 9

⇒ (24 ÷ 12) + (72 ÷ 8) – 9  [BODMAS rule]

⇒ 2 + 9 – 9

⇒ 11 – 9

⇒ 2

## Frequently Asked Questions on Arithmetic Operations

Q1

### What are the four basic arithmetic operations?

The four basic arithmetic operations in Maths are:
Subtraction
Multiplication
Division

Q2

### Are integers closed under division operation?

No, the integers are not closed under the division operation. But the set of integers are closed for the arithmetic operations such as addition, subtraction and multiplication.

Q3

### What are the rules for adding integers?

The rules for adding integers are:
The addition of two positive integers results in positive-sum
The addition of two negative integers results in negative-sum
The addition of positive and negative integers takes the sign of the largest integer value and subtract the given integers

Q4

### What are the symbols of four basic operations in Mathematics?

The four basic operations with symbols are:
Subtraction → ‘ -’
Multiplication → ‘×’
Division → ‘÷’
Q5

### What does the four arithmetic operations represents?

Additions represents the sum of two values.
Subtraction is the difference between two values
Multiplication is the product of two numbers
Division is the method of dividing one number by another.
Quiz on Arithmetic Operations