Arithmetic

What is Arithmetic?

Arithmetics is among the oldest and elementary branches of mathematics, originating from the Greek word arithmos, meaning number. It involves the study of numbers, especially the properties of traditional operations on them such as addition, subtraction, division and multiplication.

Arithmetic

These are the basic operations, although the subject also involves advanced operations like computation of percentages, logarithmic functions, exponentiation and square roots.

Who discovered 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 Progressions

A sequence like 1, 5, 9, 13, 17, or 12, 7, 2, -3, -8 that follow a constant difference is known as arithmetic progressions. You can name the first term as a1, the common difference as d and the total number of terms as n.

Therefore, an explicit formula can be written as

an = a1 + (n-1)d

Example 1: 3, 7, 11 has a1 = 3, d = 4 and n = 5. Hence, the explicit formula is

an = 3 + (n-1).4

= 4n – 1

Example 2: 3, -2, -7 has a1 = 3, d = -5 and n = 4. Hence, the explicit formula is

an = 3 + (n-1)(-5)

= 8-5n

Arithmetic operations

The basic operations under arithmetics are addition, 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.

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 opposite of any digit to the digit itself gives the additive identity. For instance, the opposite of 5 is -5, therefore 5 + (-5) = 0.

Subtraction (−)

Subtraction can be labelled as inverse of addition. It computes the difference between two values, i.e, the minuend minus the subtrahend. If the minuend is greater than subtrahend, the difference is positive. If the minuend is less than subtrahend, the result is negative, and 0 if the numbers are equal.

Multiplication (×, · or *)

Multiplication also combines two values like addition and subtraction into a single value or the 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 x b. In software languages wherein only characters are used that are found in keyboards, it is often expressed as, a*b (* is called asterisk).

Division (÷ or /)

Division is the inverse of multiplication. 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 divisor for any well-defined positive number, else it is smaller than 1.


Practise This Question

Distance between Mumbai and Pune is (2y2+y+2). There are two routes to travel from Mumbai to Pune as shown in the figure.

What is the difference between the distances of the two routes?