In Maths, integers are the numbers which represent a value or a quantity. Integer word is derived from a Latin word, which means ‘whole’. These integers can be positive or negative or zero. Basically, integers are the combination of natural numbers, whole numbers, even and odd numbers, etc. But we do not mention fractions as a part of integers. Integers can easily be represented in a number line. The examples of integers are 0, 2, 5, 9. -11. -13, -15, etc. Hence, we can say the types of integers are:
- Positive integers
- Negative integers
- Zero
The positive integers include all the counting numbers (1,2,3,4,…) and it lies at the right side of the number line. The negative integers are located on the left side of the number line. And zero is located in between positive and negative integers.
When we represent integers in a sequence with an equal interval between them, they are said to be consecutive integers. That means here the integer follows the sequence where each subsequent integer is one more than the previous integer. Suppose, x is an integer, then x+1, x+2 is the consecutive integers. It is also divided into two parts: Odd and Even. In case of odd consecutive integers, the sequence starts with odd integers such as 1 or 3 or 5 and so on. But in the case of even, the sequence starts with even integers such as 2 or 4 or 6 and so on.
Properties
Just like the numbers we have learned in primary school, integers also follows some rules. They also hold the three major properties: Associative, Commutative and Distributive. When two integers are added, subtracted or multiplied, the result is also an integer. Suppose if A, B and C are three integers, then:
- Commutative Property: A+B=B+A and A.B = B.A
- Associative Property: A+(B+C) = (A+B)+A and A.(B.C)=(A.B).C
- Distributive Property: A.(B+C) = A.B+A.C
As a Set
Integers can also be represented in the form of a set. If Z denotes set of all the integers then:
Set Z = {…-3,-2,-1,0,1,2,3…}.
As a function
If a fraction is required to be expressed in the form of integer, then we use the concept of greatest integer function. This function round-off the real number to the nearest integer which is less than the number. For example, the nearest integer of 3.33 is 3.
Applications
Integers are not simply numbers on a paper; they have many real-life applications. Use of positive and negative numbers in the real world is diverse. They are essentially used to express two contradicting conditions.
For example, when the temperature in the thermometer is above zero, positive integers are used to indicate it, whereas negative integers denote the temperature below zero-point. These integers support one to compare and weigh two things like how big or small things are and hence can be expressed numerically.
Some real-life conditions where integers come into action are player’s records in golf, football and cricket tournaments, the rating of songs, in banks debits and credits are designated as positive and negative values respectively.
