Arithmetic Operations

A mathematical study of the characteristics and manipulation of numbers is known as arithmetic. It includes basic operations like addition, subtraction, multiplication, and division which are essential to many other fields of mathematics.

In our daily lives, we use the addition, subtraction, multiplication, and division operations of basic mathematics to divide resources equally among a group of people or to calculate annual expenses. These operations are essential to many facets of our life.

Table of Contents:

• Four Basic Arithmetic Operators
• Mathematical Operations
• Basic Arithmetic Properties
• Arithmetic Operations with Whole Numbers

Let’s delve deeper into the specifics of each of these arithmetic operations.

Four Basic Arithmetic Operators

Four Basic Arithmetic Operators involve manipulating numbers to get a result.

In mathematics, the four fundamental arithmetic operations applicable to all real numbers are:

• Addition (represented by the symbol ‘+’)
• Subtraction (represented by the symbol ‘-‘)
• Multiplication (represented by the symbol ‘×’)
• Division (represented by the symbol ‘÷’)

The four basic arithmetic operations will be thoroughly covered in this part, along with their corresponding rules and examples.

Addition Definition:

A mathematical procedure called addition involves adding two or more values to get their sum. The ‘+’ symbol stands for addition. Because it is a commutative operation, the result is unaffected by the order in which the numbers are added. Any sort of number, including fractions, decimals, real numbers, and complex numbers, can be added.

Example: 6.25 + 9.75 = 16.

Rules of Addition:

• The following are the addition rules for integers:
• The addition of two positive integers results in a positive integer.
• The addition of two negative integers yields a negative integer.

When adding positive and negative integers, subtract the smaller integer from the larger one, and use the sign of the larger integer number.

Subtraction:

The arithmetic operation of subtraction includes calculating the difference between two numbers. ‘-‘ is the sign used to denote subtraction. The second term is deducted from the first term rather than added to it, similar to how addition works.

The opposite of addition is the operation of subtraction. It is done by adding the second term’s inverse, which is denoted by a negative sign. To identify the difference between two numbers or to calculate the amount left over after a certain quantity has been subtracted, subtraction is frequently utilised.

Example: 13-2

Subtraction Rules:

When performing subtraction with integers, there are certain rules to follow:

• If both integers have the same positive sign, the result will be a positive integer.
• If both integers have the same negative sign, the result will be a negative integer.
• If the signs of the integers are different, subtract the absolute values of the integers, and the sign of the result will be the same as the integer with the largest absolute value.

Multiplication

One way to think of the arithmetic operation of multiplication is as the repeated addition of the same integer. The letters “” or “*” stand in for it. The result of multiplying two or more numbers together is a single value, known as the product. The multiplicand 

and the multiplier are the two main components of multiplication. The multiplier is the amount by which the multiplicand is multiplied, whereas the multiplicand is the number being multiplied. The outcome of multiplying the multiplicand and multiplier together is the product.

Example: 13×2 = 26

Multiplication Rules

When performing multiplication with integers, there are certain rules to follow:

• When multiplying two positive integers, the result is a positive integer.
• When multiplying two negative integers, the result is a positive integer.
• When multiplying a positive integer and a negative integer, the result is a negative integer.

Division Definition:

The mathematical operation of division is represented by the symbol “.” It performs multiplication in the other direction. The dividend and the divisor are the two main components of the division operation. The divisor is the number by which the dividend is being divided, whereas the dividend is the number that is being divided.

 A single term value, known as the quotient, is produced when the dividend is divided by the divisor. The quotient will be larger than 1 if the dividend is greater than the divisor, but less than 1 if the dividend is less than the divisor.

Example: 18/9 =2

Division Rules:

When performing division with integers, there are certain rules to follow:

• When dividing two positive integers, the result is a positive integer.
• When dividing two negative integers, the result is a positive integer.
• When dividing an integer with a different sign, the result is a negative integer.

Mathematical Operations:

The four fundamental arithmetic operations—which we have already examined in the previous sections—are the basic mathematical operations.

The actions of addition and subtraction are the opposites of one another. In other words, if adding two numbers results in a third number, taking away one of the numbers from the third number will give you the fourth number.

For example:

14 + 9 = 23

Now, if we subtract 9 from 23, we get:

23 – 9 = 14

Thus, we get the original number.

Similarly, multiplication and division are also inverse operations.

If 6 x 8 = 48

Then,

48 / 8 = 6

Thus, we can see that these mathematical operations are interrelated. Additionally, these operations are the most basic form of mathematical calculations, and they can be easily comprehended by everyone.

Basic Arithmetic Properties

The basic arithmetic properties for real numbers are:

• Associative property
• Commutative property
• Distributive property
• Commutative Property

Commutative Property

The commutative property applies only to two fundamental arithmetic operations, namely addition and multiplication.

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

A+B = B+A	Example: 2 + 3 = 3 + 2
A x B = B x A	Example: 2 x 3 = 3 x 2

Associative Property

Similar to the commutative property, the associative property is also valid for the operations of addition and multiplication.

A+(B+C) = (A+B)+C	Example: 2 + (3+4) = (2+3)+4
Ax(BxC) = (AxB)xC	Example: 2 x (3 x 4) = (2 x 3) x 4

Distributive Property

The distributive property states that if A, B, and C are any three real numbers, then:

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

Arithmetic Operations with Whole Numbers

We can simply do the four fundamental arithmetic operations using integers. Positive, negative, and 0-digit whole numbers are all included in the category of integers. There are no fractional or decimal portions in them.

Two or more integers added together always result in a sum. For example, when we add the digits 3, 7, and 9, we get 3, 7, and 9, which equals 13. 13, therefore, exceeds all three addends in this case. Any number added to 0 always yields the same result, while any integer added to 1 yields that integer’s succeeding number.

When working with whole numbers, subtraction entails subtracting a smaller amount from a bigger amount to produce a difference that is less than the starting number. Any number is the same when you remove 0 from it, and any number is its predecessor when you subtract 1.

Multiplication tables can be used when multiplying two or more whole numbers. Except when multiplication by 1 or 0, the product is always greater than the sum of the two numbers. Any number multiplied by 0 will always result in 0, and any number multiplied by 1 will provide the same result.

A whole number may or may not be produced when two whole numbers are divided. The dividend is a multiple of the divisor if the quotient is a whole number. Otherwise, the quotient will be a decimal number.

Question 1:

Add 15 and 36 and then subtract 12 from the sum.

Solution:

On adding 15 and 36, we get:

Sum = 15 + 36 = 51

Now, subtracting 12 from the sum, we get:

51 – 12 = 39

Question 2: 

Solve: 15 + 15 + 15 + 15.

Solution:

Given, 15 + 15 + 15 + 15

It is clear that 15 is added to itself four times, thus, we can write;

4 times of 15 = 4 x 15 = 60

If we add them directly, the answer remains the same.

Question 3: 

Suppose we have to find the value of (8 x 3) ÷ 6 + 90 ÷ 10 – 8.

Solution:

Given,

(8 x 3) ÷ 6 + 90 ÷ 10 – 8

⇒ (24 ÷ 6) + (90 ÷ 10) – 8 [Using the BODMAS rule, we first do multiplication, then division, addition and subtraction]

⇒ 4 + 9 – 8

⇒ 5

Therefore, the value of the given expression is 5.