Integers form a bigger collection of numbers which contains whole numbers and negative numbers.
Here we will study more about integers, their properties and operations.
A method that can do one again. Here we are learning with the integers in different way.
We learned and represented more integers on the Number line now hear we learn the Arrangement or representing of the integers in ascending and descending order.
The descending order of these integers is 14, 8, 3, ...
On a number line when we
Add a positive integer, we move to the right.
Add a negative integer, we move to the left.
Subtract a positive integer, we move to the left.
Subtract a negative integer, we move to the right.
State whether the following statements are correct or incorrect. Correct those which are wrong:
When two positive integers are added we get a positive integer.
When two negative integers are added we get a positive integer.
When a positive integer and a negative integer are added, we always get a negative integer.
The additive inverse of any integer a is − a and additive inverse of (− a) is a.
Additive inverse of an integer 8 is (− 8) and additive inverse of (− 8) is 8.
In general, for any two integers a and b, a + b is an integer.
Addition of integers gives integers, we say integers are closed under addition.
if a and b are two integers then a − b is also an intger.
In general, for any two integers a and b, we can say
a + b = b + a
In general for any integers a, b and c, we can say
a + (b + c) = (a + b) + c
When we add zero to any whole number, we get the same whole number. Zero is an additive identity for whole numbers.
a + 0 = a = 0 + a
We know that multiplication of whole numbers is repeated addition.
5 + 5 + 5 = 3 x 5 = 15
Multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a minus sign (−) before the product. We thus get a negative integer.
In general, for any two positive integers a and b we can say
a x (− b) = (− a) x b = −(a x b)
If the number of negative integers in a product is even, then the product is a positive integer; if the number of negative integers in a product is odd, then the product is a negative integer.
The product of two integers is again an integer. So we can say that integers are closed under multiplication.
In general, a x b is an integer, for all integers a and b.
In general, for any integer a, a x 0 = 0 x a = 0
0 is the additive identity whereas 1 is the multiplicative identity for integers. We get additive inverse of an integer a when we multiply (−1) to a, i.e. a x (−1) = (−1) x a = − a
The product of three integers does not depend upon the grouping of integers and this is called the associative property for multiplication of integers.
The commutativity, associativity and distributivity of integers help to make our calculations simpler.
Division is the inverse operation of multiplication.
Since 3 x 5 = 15
So 15 ÷ 5 = 3 and 15 ÷ 3 = 5
when we divide a negative integer by a positive integer, we divide them as whole numbers and then put a minus sign (−) before the quotient.
For any integer a, a ÷ 0 is not defined but 0 ÷ a = 0 for a ≠ 0.
Any integer divided by 1 gives the same integer.
a ÷ 1 = a
Closure under Addition
We have learnt that sum of two whole numbers is again a whole number. For example, 17 + 24 = 41 which is again a whole number. We know that, this property is known as the closure property for addition of the whole numbers.
Closure under Subtraction
What happens when we subtract an integer from another integer? Can we say that their difference is also an integer?
Commutative Property
We know that 3 + 5 = 5 + 3 = 8, that is, the whole numbers can be added in any order. In other words, addition is commutative for whole numbers.
Can we say the same for integers also?
We have 5 + (− 6) = −1 and (− 6) + 5 = −1
So, 5 + (− 6) = (− 6) + 5
Associative Property
Observe the following examples:
Consider the integers −3, −2 and −5.
Look at (−5) + [(−3) + (−2)] and [(−5) + (−3)] + (−2).
Additive Identity
When we add zero to any whole number, we get the same whole number. Zero is an additive identity for whole numbers. Is it an additive identity again for integers also?
Multiplication of Integers
We can add and subtract integers. Let us now learn how to multiply integers.
We know that multiplication of whole numbers is repeated addition. For example, 5 + 5 + 5 = 3 x 5 = 15
Multiplication of two Negative Integers
Can you find the product (−3) x (−2)?
Observe the following:
−3 x 4 = − 12
−3 x 3 = −9 = −12 − (−3)
−3 x 2 = − 6 = −9 − (−3)
−3 x 1 = −3 = − 6 − (−3)
−3 x 0 = 0 = −3 − (−3)
−3 x −1 = 0 − (−3) = 0 + 3 = 3
−3 x −2 = 3 − (−3) = 3 + 3 = 6
Product of three or more Negative Integers
We observed that the product of two negative integers is a positive integer.
What will be the product of three negative integers? Four negative integers?
Let us observe the following examples:
(− 4) x (−3) = 12
Associativity for Multiplication
Consider −3, −2 and 5.
Look at [(−3) x (−2)] x 5 and (−3) x [(−2) x 5].
In the first case (−3) and (−2) are grouped together and in the second (−2) and 5 are grouped together.
We see that [(−3) x (−2)] x 5 = 6 x 5 = 30 and (−3) x [(−2) x 5] = (−3) x (−10) = 30 So, we get the same answer in both the cases.
Thus, [(−3) x (−2)] x 5 = (−3) x [(−2) x 5]
Distributive Property
We know 6 x (10 + 2) = (16 x 10) + (16 x 2) [Distributivity of multiplication over addition]
Let us check if this is true for integers also.
Observe the following:
(−2) x (3 + 5) = −2 x 8 = −16 and [(−2) x 3] + [(−2) x 5] = (− 6) + (−10) = −16
So, (−2) x (3 + 5) = [(−2) x 3] + [(−2) x 5]
Properties of Division of Integers
Observe the following table and complete it:
What do you observe? We observe that integers are not closed under division.
You can see from the table that (− 8) ÷ (− 4) ≠ (−4) ÷ (− 8).
Is (− 9) ÷ 3 the same as 3 ÷ (− 9)?
Is (− 30) ÷ (−6) the same as (− 6) ÷ (− 30)?
Can we say that division is commutative for integers? No.