# Integers

Do you know that the lowest temperature that hit Antarctica was -89â—¦C ?Â  Have you been in a building with floors marked as -3 , -2 etc ?Â  Now , have you noticed that all these numbers come with a negative sign ?

It is an interesting factor to notice that we need both positive and negative numbers in everyday life to express various measurements in different situations.Â  So , are these negative numbers also called natural numbers ? Let’s find out.

We have already learnt that natural numbers are 1,2,3,4…etc. We don’t specify negative numbers in this list . So , what are those numbers which include negative digits called ? They are called integers .

What are integers ?

The collection of negative numbers , zero and positive numbers ( i e , natural numbers )are called integers. Integers are …..-3,-2,-1,0,1,2,3….

• All the numbers that are not fractions or decimals are integers. For example : 0.385, 2/3,âˆš2 Â , etcÂ  are not integers , since each of these can be expressed as fractions. Whereas , -6,-99, 0, 76 , 54,etc are integers .
• There are two types of integers : Positive Integers + Â eg: 2,3,76,89,666 etc

and

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  Negative Integers –Â Â Â Â Â  Â Â Â Â Â Â eg : -2, -3, -99, -765 etc

0 is neither positive nor negative

How do integers look on our number line ?

We can see the number line with all the positive and negative integers . Here , if you stand on ‘0’ of the number line and look towards the positive side , you will find that the numbers are increasing as you move rightwards.

But , if you look towards the negative side , you will find that the value of the numbers are decreasing as you move leftwards . Yes ! Are you surprised ?Â  Think about it ..

In real life too , you must have heard your teachers tell you not to be a negative person , and to be lessen negativity in your life . The more the negativity , the lesser will be the value of the person . Similarly , with an example , you can understand that -10 is a smaller number than -4.Similarly , for all other cases.

• 0 is larger than all negative numbers .

Now that we know what integers are , let us learn some operations on integers :

OPERATIONS ON INTEGERS

• When two positive integers are added , the sum is also a positive integer . eg : 3 + 6 = 9 ;

100 + 20 =Â  120

• When two negative integers are added , the sum is also a negative integer .eg: -3+-6 =-9
• When a positive integer is added to a negative number , subtraction happens , i e ,

for eg : -3 + 5 =2 , and -5 + 3 = -2 . If you closely observe this example you can understand that

• when a smaller negative number is added to a larger positive number , the two numbers get subtracted and the difference will hold the sign of the larger number , i e , a positive sign . In the above example , -3+5=2 ; here , 5>3 , and hence the differenceÂ  that is 2 m will be +2 .
• when a larger negative number is added to a smaller positive number , the two numbers get subtracted and the difference will hold the sign of the larger number , i e , a negative sign . In the above example , -5+3= -2 ; here , 5>3 , and hence the differenceÂ  that is 2 will be -2 .
1. Subtraction of integers
• When two negative numbers are subtracted , the difference will either have a negative or a positive sign . This is explained below :

eg : -6 – (-3) = -3 .Â  In this example , we are subtracting -3 ; which is a larger negative number from -6 , which is a smaller negative number . Since we are trying to subtract a larger numberÂ  from a smaller number , the difference will naturally have a negative value.

eg: -3 -(-6) = 3 . In this example , we are subtracting -6 from -3 , i e , a smaller number is being subtracted from a larger number , which is normal. Hence , the difference is a positive value.

1. Multiplication of integers
• When two integers with the same sign are multiplied , the product always has a positive sign.

For eg : 2*3 = 6 ; and , -2*-3 = 6

• When two integers with different signs are multiplied , the product is always has a negative sign.

For eg: -2*3 =-6 , and ,Â Â Â  7* (-8) =-56

1. Division of integers
• When two integers with the same sign are divided , the quotient always has a positive sign.

For eg : 6/2 = 3 ; and , -6/-2 = 3

• When two integers with different signs are divided , the quotient is always has a negative sign.

For eg: -6/2 = -3 , and ,Â Â Â  56/ (-7) = -8

Now , you have learnt what integers are and their operations and properties.

1. The square root of a negative integer is called as an imaginary number
2. 1729 is the smallest integer that can be expresses as the sum of cubes of two other numbers , in two different ways, i e , 1729 = 9^3 + 10^3 = 1^3 + 12^3

Now get model solutions for chapter Integers in detail and step-by-step procedure to all questions in an NCERT textbooks only at BYJU’S.

#### Practise This Question

After listening to -1, you told him everything that 1 told you about 0. On this -1 asked- "Then, where is 0?” You thought and thought about this but couldn't get the answer. Then something struck your head. You asked -1: "Is '0' larger or smaller than you?”-1 replied: "Of course it is smaller than me, I am the king!!!”  Is this statement given by -1 true or false?

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