Integers

The notation of a number is connected with the act of counting. So, the natural numbers 1, 2, 3, 4, 5, …… are also called counting numbers. We use ten symbols or digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 to denote large numbers by using position of the digits in the given number.

So far, we have been dealing with natural numbers and whole numbers. While studying the properties of whole numbers, we found that the closure property does not hold good for the subtraction of natural numbers as well as whole numbers. This is because of the following:

25 - 20 = 5 is a whole number as well as a natural number.

25 - 25 = 0 is a whole number but not a natural number.

25 - 30 = - 5 which is neither a whole number nor a natural number

Hence, negative numbers are needed and are added in our number system.


The numbers discovered first were natural numbers i.e., 1, 2, 3, 4, 5, 6, ......

If 0 is included to the collection of natural numbers, we get a collection of whole numbers. i.e., 0, 1, 2, 3, 4, 5, 6, ......



There are negative numbers too, i.e., the system of whole numbers together with negative numbers are called integers i.e., -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, ......

The system of integers consists of the following:

(i) natural numbers i.e. 1, 2, 3, 4, 5, 6,......

(ii) zero i.e., 0

(iii) whole numbers i.e., 0, 1, 2, 3, 4, 5, 6, ......

(iv) negative numbers i.e., -1, -2, -3, -4, -5, -6, ......


What are integers?

The negative numbers, zero and the natural numbers together are called integers.

A collection of numbers which is written as …….. -4, -3, -2, -1, 0, 1, 2, 3, 4……… .

These numbers are called integers.



According to set theory

Integers, {-6, -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5, +6, ......}

For example:

(i) -8, -2, 0, 2, 8

(ii) -7, -1, 0, 5, 12

(iii) -15, -9, 11, 21

(iv) -10, -3, 0, 9, 14

(v) -6, -4, 0, 7, 19


New numbers denoted by -1, -2, -3, -4, -5, -6, ...... called minus one, minus two, minus three, minus four...... respectively are introduced corresponding to numbers 1. 2, 3, 4, 5, 6, ...... such that

1 + (-1) = 0 

2 + (-2) = 0

3 + (-3) = 0

4 + (-4) = 0

5 + (-5) = 0

6 + (-6) = 0 and so on.

1 and -1 are opposite numbers.

2 and -2 are opposite numbers.

3 and -3 are opposite numbers.

4 and -4 are opposite numbers.

5 and -5 are opposite numbers.

6 and -6 are opposite numbers and so on.

The combination of the new collection of timbers with whole timber 0, 1, 2, 3, 4, 5, 6, ...... are called integers.


We usually represent numbers by making evenly spaced points one unit apart on a straight horizontal line which is called a number line. The line extends indefinitely to the left and to the right. A point O on the line, called the origin corresponds to the number 0. So, point A at a distance of 9 units to the right of zero represents the number 9 on the number line shown below.

Positive Numbers

What lies on the left of 0? Let us think about some of the familiar things to understand about the positive number and the negative number.

We very often hear that the temperature during winter went down to -2°. We know that the temperature in -2° is very cold, it is colder than 0°c. It means that -2 is smaller than 0. Let us see in a thermometer where -2 is located. It is marked below 0.


To measure the depth below the ground level we use negative numbers. Here, 0 indicated ground level. On a number line a number that is right to the 0 is positive and a number left side to the 0 is negative. So, we have a new set of numbers -1, -2, -3, -4, -5, ….. These are known as the negative numbers. 

Integers

On the number line there are positive numbers, negative numbers and 0. These are called integers. 0 is neither positive nor negative.

Integers on Number Line

The sign before the number indicates its direction to the right or left of 0. For example +5 indicates that the number lies to the right of 0 and -5 indicates that the number lies to the left of 0. Thus, integers are also called directed numbers. If a number has no sign it means that it is a positive integer.


Negative Integers:

The negative numbers ………. -5, -4, -3, -2, -1 are called negative integers.

Thus, examples of negative integers are ……… -5, -4, -3, -2, -1.


Note:

We use the symbol ‘-’ to denote negative integers and the same symbol is used to indicate subtraction. But the context will always make it clear whether we mean negative integer or subtraction.


Positive Integers:

The natural numbers 1, 2, 3, 4, 5, ……… are called positive integers.

Thus, examples of positive integers are 1, 2, 3, 4, 5, ………. .


Note:

Positive integers are also written as +1, +2, +3, +4, +5, ………… however, the plus sign (+) is usually omitted and understood.


The number 0 is simply an integer. It is neither positive nor negative.

i.e., 'Zero' is an integer which is neither positive not negative.


In real life, integers are used to represent opposite situations.

For example:


Positive Integers

Negative Integers

(i)

Profit

Loss

(ii)

Deposits in banks

Withdrawals from banks

(iii)

Temperatures above 0°C

Temperatures below D°C

(iv)

Above sea level

Below sea level



Frequently asked questions

1. Are all natural numbers integers?

Answer:

Yes, all natural numbers are integers.

The natural numbers are 1, 2, 3, 4, 5, 6, …… 

The negative numbers, zero and the natural numbers together are called integers., such as -4, -3, -2, -1, 0, 1, 2, 3, and so on. Since all whole numbers are natural numbers, all natural numbers are also integers.


2. Are all whole numbers integers?

Answer:

Yes, all whole numbers are integers.

The negative numbers, zero and the natural numbers together are called integers., such as .... -4, -3, -2, -1, 0, 1, 2, 3, ... and so on. So, all whole numbers are integers.


3. Which integers are not whole numbers?

Answer:

All negative integers are not whole numbers.


4. Write the smallest positive integer?

Answer:

The smallest positive integer is 1.

Note: 0 is the smallest whole number, but 0 is neither positive nor negative.

You might like these

● Integers

Representation of Integers on a Number Line.

Addition of Integers on a Number Line.

Rules to Add Integers.

Rules to Subtract Integers.






5th Grade Numbers Page 

5th Grade Math Problems 

From Integers to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. Subtracting Integers | Subtraction of Integers |Fundamental Operations

    Jun 13, 24 02:51 AM

    Subtracting integers is the second operations on integers, among the four fundamental operations on integers. Change the sign of the integer to be subtracted and then add.

    Read More

  2. Properties of Subtracting Integers | Subtraction of Integers |Examples

    Jun 13, 24 02:28 AM

    The properties of subtracting integers are explained here along with the examples. 1. The difference (subtraction) of any two integers is always an integer. Examples: (a) (+7) – (+4) = 7 - 4 = 3

    Read More

  3. Math Only Math | Learn Math Step-by-Step | Worksheet | Videos | Games

    Jun 13, 24 12:11 AM

    Presenting math-only-math to kids, students and children. Mathematical ideas have been explained in the simplest possible way. Here you will have plenty of math help and lots of fun while learning.

    Read More

  4. Addition of Integers | Adding Integers on a Number Line | Examples

    Jun 12, 24 01:11 PM

    Addition of Integers
    We will learn addition of integers using number line. We know that counting forward means addition. When we add positive integers, we move to the right on the number line. For example to add +2 and +4…

    Read More

  5. Worksheet on Adding Integers | Integers Worksheets | Answers |Addition

    Jun 11, 24 07:15 PM

    Worksheet on Adding Integers
    Practice the questions given in the worksheet on adding integers. We know that the sum of any two integers is always an integer. I. Add the following integers:

    Read More