Arithmetic

Arithmetic is one of the oldest and elementary branches of mathematics, originating from the Greek word ‘Arithmos’, which means ‘number’. Arithmetic involves the study of numbers, especially the properties of traditional operations, such as:

• Addition
• Subtraction
• Multiplication
• Division

The arithmetic operators based on these operations are ‘+’, ‘-’, ‘×’ and ‘÷’. Let us learn here all the important topics of arithmetic with examples.

What is Arithmetic?

Arithmetic is the fundamental of mathematics that includes the operations of numbers. These operations are addition, subtraction, multiplication and division. Arithmetic is one of the important branches of mathematics, that lays the foundation of the subject ‘Maths’, for students.

History of Arithmetic

The Fundamental principle of number theory was provided by Carl Friedrich Gauss in 1801, according to which, any integer which is greater than 1 can be described as the product of prime numbers in only one way. Arithmetic is another name given to number theory. The four elementary operations in arithmetic are addition, subtraction, multiplication and division. All these operations are discussed in brief here.

Arithmetic operations

The basic operations under arithmetic are addition and subtraction, division and multiplication, although the subject involves many other modified operations.

Addition (+)

Addition is among the basic operations in arithmetic. In simple forms, addition combines two or more values into a single term, for example: 2 + 5 = 7, 6 + 2 = 8, where ‘+’ is the addition operator.

The procedure of adding more than two values is called summation and involves methods to add n number of values.

The identity element of addition is 0, which means that adding 0 to any value gives the same result. The inverse element of addition is the opposite of any value, which means that adding the opposite of any digit to the digit itself gives the additive identity. For instance, the opposite of 5 is -5, therefore 5 + (-5) = 0.

Examples of addition:

• 8 + 10 = 18
• 12 + 5 = 17

Subtraction (−)

Subtraction can be labelled as the inverse of addition. It computes the difference between two values, i.e., the minuend minus the subtrahend. The operator of subtraction is (-).

If the minuend is greater than the subtrahend, the difference is positive. If the minuend is less than the subtrahend, the result is negative, and 0 if the numbers are equal.

For example,

• 4  – 3 = 1
• 3 – 4 = -1

Multiplication (×)

Multiplication also combines two values like addition and subtraction, into a single value or product. The two original values are known as the multiplicand and the multiplier, or simply both as factors.

The product of a and b is expressed as a·b or a × b, where ‘×’ is the multiplication operator. In software languages wherein only characters are used that are found in keyboards, it is often expressed as, a*b (* is called asterisk).

For example,

• 4 × 5 = 20
• 2 × 3 = 6

Division (÷)

The division is the inverse of multiplication. The operator used for the division method is ‘÷’ or ‘/’ sometimes. It computes the quotient of two numbers, the dividend that is divided by the divisor.

The quotient is more than 1 if the dividend is greater than the divisor for any well-defined positive number else, it is smaller than 1.

For example,

• 10 ÷ 2 = 5
• 9 ÷ 3 = 3

What is Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers, where the difference between one term and the next is a constant. For example, 1, 4, 7, 10, 13, 16, 19, 22, 25, … is an arithmetic sequence with common difference equal to 3. It is also termed arithmetic progression and is commonly represented as:

a, a + d, a + 2d, a + 3d, a + 4d, ………. ,a + (n – 1) d

Where, 

a = first term

d = common difference between the terms

n = number of terms

Arithmetic Solved Problems

Question 1: The sum of the two numbers is 50, and their difference is 30. Find the numbers.

Solution: Let the numbers be x and y. Now, as per the given situation,

x + y = 50……………………(i)

and x – y = 30………………(ii)

We can write, x = 50-y, from eq.(i),

Therefore, putting the value of x in eq(ii), we get,

50 – y – y = 30

50 -2y = 30

2y = 50-30= 20

y = 20/2 = 10

and x = 50 – y = 50-10 =40

Therefore, the two numbers are 40 and 10.

Question 2: Solve 25 + 5 (27 ÷ 3) – 9.

Solution: 25 + 5(27 ÷ 3) – 9

⇒  25 + 5(9) – 9

⇒ 25 + 45 – 9

⇒ 70 – 9 = 61

&

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