# What are integers

In this post we will discuss concept of integers, its types and its properties.

We have taken the help of number line so that you can understand the concept easily.

After that we will solve some questions related to this topic

## What is Integer?

All the positive, negative numbers and 0 comes under the definition of integer.

Positive Numbers are : 1, 2, 3, 4, 5 . . . . . . . . and so on.
Negative Numbers are: -1, -2, -3, -4 . . . . . and so on

All these numbers come under the category of integers.

Numbers that are not integers:
(a) Fractions are not integers
(b) Numbers with decimals ( like 2.3, 5.4, 6,1) are not integers
(c) Complex numbers are not integers

Integers on the number line

### What are positive integers

Numbers that are right to number 0 are positive integers.

These numbers are also called “counting numbers” as when you start counting you count from 1, 2, 3, 4, ……. and so on.

Whether 0 is included in positive integer?
0 is a unique digit which is neither positive and negative. So we don’t include zero as a part of positive integer.

Positive integer in number line

Positive Integer in Daily life

All the numbers you see around are the examples of positive integer.

Examples:
(a) Speed Limit Signs on Highways
(b) Temperature of Room
(c) Number of Pencils in box

### What are Negative Integers

In a number line, numbers which are left to 0 are negative integers.

These numbers are less than zero & has negative sign (” – “) in front of them

Examples of Negative integers are: -1, -2, -3, -4 . . . . and so on.

These numbers are pronounced with minus.
For example minus 1, minus 2, minus 3 . . . . etc.

Explanation of Negative Integers

Below is the image of Room Thermometer.

During winter, some cold countries room temperature goes below 0 degrees. When you read the thermometer, it will show the temperature in negative numbers.

Hence negative number has significance in daily life so remember the concept fully.

Negative Integers in Daily Life

Concept of negative integers can be found in

(a) Banking and Financial Transaction
When the money is debited from an account, the concept of negative integer is used

(b) Temperatures
There are some countries where temperature runs in negative.
Extreme cold countries have temperature like – 10 degree Celsius

(c) Medical Industry
During eye check up, the doctor check if the eye sight is weak or not.
The power of eyesight is given by both positive and negative integers

Negative Integers in Number Line

### Is number 0 an integer?

Yes, Number 0 is an integer.

It is neither positive nor negative.

Number 0 just lies on the boundary of positive and negative integer

### Even Integers

Integer which is multiple of 2 is known as Even Integers.

Examples of even Integers
⟹ -8, -6, -4, -2, 0, 2, 4, 6, 8, 10 . . . .
Note all the numbers are divisible by 2

Even Integers in Number Line

Important Points

(A) Multiplication of even Integer is also even integer
Example:
2 x 4 = 8
8 x 6 = 48
Both 8 and 48 are even integers

(B) Even Integers can be expressed mathematically as ⟹ 2k

### Odd Integer

Any integer which is not divisible by 2 is odd integer

Example
⟹ 1, 3, 5, -7, 7, 9, 11
Note all the numbers are not divisible by 2

Odd Integer in Number line

Important Points

(A) Multiplication of Odd Integer is also Odd integer
Example:
3 x 5 = 15
1 x 7 = 7
Both 15 and 7 are odd integers

(B) Addition & Subtraction of Odd Integer is Even
Example
1 + 3 = 4
5 + 7 = 12
Both 4 & 12 are even integers

(C) Odd Integers are mathematically expressed as ⟹ 2k + 1

### Consecutive Integer

Integers that are next to each other are consecutive integers.

Example
1 & 2 are consecutive integers
10 & 11 are consecutive integers

Consecutive Even Integers
Even Integers that are next to each other are consecutive even integer

Example
2 & 4 are consecutive even integer
14 & 16 are consecutive even integer

Consecutive Odd Integer
Odd Integers that are next to each other are Consecutive Odd Integers

3 & 5 are odd consecutive integers
9 & 11 are also odd consecutive integers

## Properties of Integer

Below is the collection of important properties of integer.

(1) Commutative Property of Integer

If a & b are two integers then

a + b = b + a
a x b = b x a

It means that if we add/multiply the two integers and interchange their position in the calculation, the end result will be the same

Example
Let a = 2 & b = 5

a + b = 2 + 5 = 7
b + a = 5 + 2 = 7

Hence a + b = b + a

(2) Associative Property of Integer

If a, b and c are integers then

a + (b + c) = (a + b) + c
a x (b x c) = (a x b) x c

Example
Let a = 2, b = 4 and c = 6

a x (b x c) = 2 x (4 x 6) = 48
(a x b) x c = (2 x 4) x 6 = 48

(3) Distributive Property of Integer

If a, b and c are integer then

a x (b + c) = a x b + a x c

Example
let a = 2, b = 5, c = 7

LHS = a x (b + c)
= 2 x ( 5 + 7 )
= 2 x (12)
= 24

RHS = (a x b) + ( a x c)
= (2 x 5) + (2 x 7)
=10 + 14
= 24

Hence, LHS = RHS
a x (b + c) = (a x b) + ( a x c)

(04) Identity Property of Integer

If a is integer then

a + 0 = a
a x 1 = a

If we add the same integer with opposite sign the result will be zero

a + ( – a) = 0

(06) Multiplicative Inverse Property

If we multiply integer with its reciprocal, the result will be 1

a x (1/a) = 1

These were some common properties of integers.

Try to remember each of the properties as they are asked in school examination

## Common FAQ on Integers

(01) Is whole number same as integers?

No, Whole Numbers are part of Integers

Whole Numbers include 0 and positive integers.
Hence the negative integers does not come under whole number

Whole Numbers are : 0, 1, 2, 3, 4 . . .
Where Integers are : . . . .-3, -2, -1, 0 , 1, 2, 3, . . .

(02) Can integers be negative?

Yes, Integers includes negative numbers.

Example: -1, -4, -10 etc are all part of integers

(03) Are fractions and decimal numbers part of integers?

No, Fractions and decimals are not integers

(04) What are types of integer?

Three types of integers are there

(a) Zero
(b) Positive integers
(c) Negative integers

## Integer Worksheets

Given below are collection of questions related to integer.

All questions are to the standard of Grade 5.

Each question is provided with solution and explanation

(01) Using number line show the below integers

(a) 5
(b) -7
(c) 12
(d) -12
(e) 2
(f) 15
(g) -10

(02) Write all the integers between the given numbers on number line

(a) 0 and 5
(b) 10 and 14
(c) – 3 and 3
(d) -7 and -3
(e) -1 and 4
(f) 5 and 11

(a) 0 and 5

(b) 10 and 14

(c) – 3 and 3

(d) -7 and -3

(e) -1 and 4

(f) 5 and 11

(03) Compare the integers and put > (greater than), < (less than) or = (equal to) sign

(a) -17 ………… 17

(b) 14 …………. -4

(c) 10 ………. 7

(d) 9 ……….. 4

(e) -9 ……….. -7

(f) 0 ………. -3

(g) 5 ………….8

(h) 2 ………….-2

(a) -17 < 17

(b) 14 > -4

(c) 10 >. 7

(d) 9 > 4

(e) -9 < -7

(f) 0 > -3

(g) 5 < 8

(h) 2 > -2

(04) Check if the statement related to integers is True or False

(a) 0 is the smallest integer

(b) 5 is integer number

(c) 2.5 is an integer number

(d) Minus 4 is greater than -3
(i.e -4 > -3 )

(e) 5 is on right of 2 on the number line

(f) Minus 15 is greater than minus 17
(i.e -15 > -17)

(a) False
There are negative integers which are smaller than 0

(b) True

(c) False
2.5 is a decimal, it is not an integer

(d) False
-3 is greater than -4

(e) True
5 is greater than 2;
hence 5 is on right of 2 on number line

(f) True