Integers

PROPERTIES OF ADDITION AND SUBTRACTION OF INTEGERS

We have learnt about whole numbers and integers in Class VI. We have also learnt about addition and subtraction of integers.

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. Let us see whether this property is true for integers or not. Following are some pairs of integers. Observe the following table and complete it

What do you observe? Is the sum of two integers always an integer? Did you find a pair of integers whose sum is not an integer? Since addition of integers gives integers, we say integers are closed under addition. In general, for any two integers a and b, a + b is an integer

Closure under Subtraction

What happens when we subtract an integer from another integer? Can we say that their difference is also an integer? Observe the following table and complete it:

What do you observe? Is there any pair of integers whose difference is not an integer? Can we say integers are closed under subtraction? Yes, we can see that integers are closed under subtraction.

Thus, if a and b are two integersthen a – b is also an intger. Do the whole numbers satisfy this property?

 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 Are the following equal?

(i) (– 8) + (– 9) and (– 9) + (– 8)

(ii) (– 23) + 32 and 32 + (– 23)

(iii) (– 45) + 0 and 0 + (– 45)

Try this with five other pairs of integers. Do you find any pair of integers for which the sums are different when the order is changed? Certainly not. We say that addition is commutative for integer

In general, for any two integers a and b, we can say

a + b = b + a

We know that subtraction is not commutative for whole numbers.

Is it commutative for integers? Consider the integers 5 and (–3). Is 5 – (–3) the same as (–3) –5? No, because 5 – ( –3)

=5 + 3 = 8, and (–3) – 5 = – 3 – 5 = – 8.

Take atleast five different pairs of integers and check this. We conclude that subtraction is not commutative for integers.

Associative Property

Observe the following examples: Consider the integers –3, –2 and –5. Look at (–5) + [(–3) + (–2)] and [(–5) + (–3)] + (–2). In the first sum (–3) and (–2) are grouped together and in the second (–5) and (–3) are grouped together. We will check whether we get different results.

In both the cases, we get –10.

i.e., (–5) + [(–3) + (–2)] = [(–5) + (–2)] + (–3)

Similarly consider –3 , 1 and –7.

 ( –3) + [1 + (–7)] = –3 + __________ = __________

[(–3) + 1] + (–7) = –2 + __________ = __________

 Is (–3) + [1 + (–7)] same as [(–3) + 1] + (–7)?

 Take five more such examples. You will not find any example for which the sums are different. Addition is associative for integers.

In general for any integers a, b and c,we can say

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

 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?

Observe the following and fill in the blanks:

(i) (– 8) + 0 = – 8

(ii) 0 + (– 8) = – 8

(iii) (–23) + 0 = _____

 (iv) 0 + (–37) = –37

(v) 0 + (–59) = _____

(vi) 0 + _____ = – 43

(vii) – 61 + _____ = – 61

 (viii) _____ + 0 = _____

The above examples show that zero is an additive identity for integers. You can verify it by adding zero to any other five integers. In general, for any integer a

a + 0 = a = 0 + a

EXAMPLE: Write down a pair of integers whose

 (a) sum is –3                     (b) difference is –5

(c) difference is 2               (d) sum is 0

SOLUTION

(a) (–1) + (–2) = –3 or (–5) + 2 = –3

(b) (–9) – (– 4) = –5 or (–2) – 3 = –5

 (c) (–7) – (–9) = 2 or 1 – (–1) = 2

 (d) (–10) + 10 = 0 or 5 + (–5) = 0

Can you write more pairs in these examples?

Integers

 MULTIPLICATION OF INTEGERS

We can add and subtract integers. Let us now learn how to multiply integers.

 Multiplication of a Positive and a Negative Integer

We know that multiplication of whole numbers is repeated addition. For example, 5 + 5 + 5 = 3 × 5 = 15

Can you represent addition of integers in the same way? We have from the following number line,

(–5) + (–5) + (–5) = –15

But we can also write (–5) + (–5) + (–5) = 3 × (–5)

Therefore, 3 × (–5) = –15

Similarly (– 4) + (– 4) + (– 4) + (– 4) + (– 4) = 5 × (– 4) = –20

And (–3) + (–3) + (–3) + (–3) = __________ = __________

 Also, (–7) + (–7) + (–7) = __________ = __________

Let us see how to find the product of a positive integer and a negative integer without using number line

Let us find 3 × (–5) in a different way. First find 3 × 5 and then put minus sign (–) before the product obtained. You get –15.That is we find – (3 × 5) to get –15.

Similarly, 5 × (– 4) = – (5×4) = – 20.

Find in a similar way, 4 × (– 8) = _____ = _____, 3 × (– 7) = _____ = _____

6 × (– 5) = _____ = _____, 2 × (– 9) = _____ = _____

Using this method we thus have, 10 × (– 43) = _____ – (10 × 43) = – 430

Till now we multiplied integers as (positive integer) × (negative integer).

Let us now multiply them as (negative integer) × (positive integer).

We first find –3 × 5. To find this, observe the following pattern:

We have, 3 × 5 = 15

2 × 5 = 10 = 15 – 5

1 × 5 = 5 = 10 – 5

 0 × 5 = 0 = 5 – 5

So, –1 × 5 = 0 – 5 = –5

–2 × 5 = –5 – 5 = –10

 –3 × 5 = –10 – 5 = –15

We already have 3 × (–5) = –15

 So we get (–3) × 5 = –15 = 3 × (–5)

Using such patterns, we also get (–5) × 4 = –20 = 5 × (– 4)

Using patterns, find (– 4) × 8, (–3) × 7, (– 6) × 5 and (– 2) × 9

Check whether, (– 4) × 8 = 4 × (– 8), (– 3) × 7 = 3 × (–7), (– 6) × 5 = 6 × (– 5)

and (– 2) × 9 = 2 × (– 9)

 Using this we get, (–33) × 5 = 33 × (–5) = –165 We thus find that while 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 × (– b) = (– a) × b = – (a × b)

Multiplication of two Negative Integers

Can you find the product (–3) × (–2)?

Observe the following:

–3 × 4 = – 12

–3 × 3 = –9 = –12 – (–3)

 –3 × 2 = – 6 = –9 – (–3)

–3 × 1 = –3 = – 6 – (–3)

–3 × 0 = 0 = –3 – (–3)

–3 × –1 = 0 – (–3) = 0 + 3 = 3

–3 × –2 = 3 – (–3) = 3 + 3 = 6

Do you see any pattern? Observe how the products change. Based on this observation, complete the following: –3 × –3 = _____ –3 × – 4 = _____

Now observe these products and fill in the blanks:

– 4 × 4 = –16

– 4 × 3 = –12 = –16 + 4

– 4 × 2 = _____ = –12 + 4

– 4 × 1 = _____

– 4 × 0 = _____

– 4 × (–1) = _____

– 4 × (–2) = _____

 – 4 × (–3) = _____

From these patterns we observe that,

 (–3) × (–1) = 3 = 3 × 1

 (–3) × (–2) = 6 = 3 × 2

(–3) × (–3) = 9 = 3 × 3 and

(– 4) × (–1) = 4 = 4 × 1

 So, (– 4) × (–2) = 4 × 2 = _____

(– 4) × (–3) = _____ = _____

So observing these products we can say that the product of two negative integers is a positive integer. We multiply the two negative integers as whole numbers and put the positive sign before the product. Thus, we have (–10) × (–12) = + 120 = 120

Similarly (–15) × (– 6) = + 90 = 90

In general, for any two positive integers a and b,

(– a) × (– b) = a × b

Game 1

(i) Take a board marked from –104 to 104 as shown in the figure.

(ii) Take a bag containing two blue and two red dice. Number of dots on the blue dice indicate positive integers and number of dots on the red dice indicate negative integers.

(iii) Every player will place his/her counter at zero.

(iv) Each player will take out two dice at a time from the bag and throw them.

(v) After every throw, the player has to multiply the numbers marked on the dice

(vi) If the product is a positive integer then the player will move his counter towards 104; if the product is a negative integer then the player will move his counter towards –104.

(vii) The player who reaches either -104 or 104 first is the winner.

Integers

PROPERTIES OF MULTIPLICATION OF INTEGERS

Closure under Multiplication

1. Observe the following table and complete it:

What do you observe? Can you find a pair of integers whose product is not an integer? No. This gives us an idea that the product of two integers is again an integer. So we can say that integers are closed under multiplication. In general,

a × b is an integer, for all integers a and b

 Find the product of five more pairs of integers and verify the above statement

Commutativity of Multiplication

We know that multiplication is commutative for whole numbers. Can we say, multiplication is also commutative for integers? Observe the following table and complete it:

What are your observations? The above examples suggest multiplication is commutative for integers. Write five more such examples and verify.

In general, for any two integers a and b,

a × b = b × a

Multiplication by Zero We know that any whole number when multiplied by zero gives zero. Observe the following products of negative integers and zero. These are obtained from the patterns done earlier.

 (–3) × 0 = 0 0 × (– 4) = 0 – 5 × 0 = _____ 0 × (– 6) = _____

This shows that the product of a negative integer and zero is zero.

 In general, for any integer a,

a × 0 = 0 × a = 0

Multiplicative Identity

We know that 1 is the multiplicative identity for whole numbers. Check that 1 is the multiplicative identity for integers as well. Observe the following products of integers with 1.

(–3) × 1 = –3             1 × 5 = 5

 (– 4) × 1 = _____      1 × 8 = _____

 1 × (–5) = _____       3 × 1 = _____

 1 × (– 6) = _____      7 × 1 = _____

This shows that 1 is the multiplicative identity for integers also. In general, for any integer a we have,

 a × 1 = 1 × a = a

What happens when we multiply any integer with –1?

Complete the following:

(–3) × (–1) = 3

3 × (–1) = –3

(– 6) × (–1) = _____

 (–1) × 13 = _____

(–1) × (–25) = _____

18 × (–1) = _____

What do you observe? Can we say –1 is a multiplicative identity of integers? No.

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 × (–1) = (–1) × a = – a

Associativity for Multiplication

Consider –3, –2 and 5.

 Look at [(–3) × (–2)] × 5 and (–3) × [(–2) × 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) × (–2)] × 5 = 6 × 5 = 30 and (–3) × [(–2) × 5] = (–3) × (–10) = 30

So, we get the same answer in both the cases. Thus, [(–3) × (–2)] × 5 = (–3) × [(–2) × 5]

 Look at this and complete the products:

[(7) × (– 6)] × 4 = __________ × 4 = __________

7 × [(– 6) × 4] = 7 × __________ = __________

Is [7 × (– 6)] × 4 = 7 × [(– 6) × 4]?

Does the grouping of integers affect the product of integers? No. In general, for any three integers a, b and c

(a × b) × c = a × (b × c)

Take any five values for a, b and c each and verify this property. Thus, like whole numbers, the product of three integers does not depend upon the grouping of integers and this is called the associative property for multiplication of integers.

Distributive Property

We know 16 × (10 + 2) = (16 × 10) + (16 × 2) [Distributivity of multiplication over addition]

Let us check if this is true for integers also.

 Observe the following:

 (a) (–2) × (3 + 5) = –2 × 8 = –16

and [(–2) × 3] + [(–2) × 5] = (– 6) + (–10) = –16

So, (–2) × (3 + 5) = [(–2) × 3] + [(–2) × 5]

(b) (– 4) × [(–2) + 7] = (– 4) × 5 = –20

and [(– 4) × (–2)] + [(– 4) × 7] = 8 + (–28) = –20

So, (– 4) × [(–2) + 7] = [(– 4) × (–2)] + [(– 4) × 7]

(c) (– 8) × [(–2) + (–1)] = (– 8) × (–3) = 24

and [(– 8) × (–2)] + [(– 8) × (–1)] = 16 + 8 = 24

So, (– 8) × [(–2) + (–1)] = [(– 8) × (–2)] + [(– 8) × (–1)]

Can we say that the distributivity of multiplication over addition is true for integers also? Yes. In general, for any integers a, b and c,

a × (b + c) = a × b + a × c

Take atleast five different values for each of a, b and c and verify the above Distributive property

Now consider the following:

Can we say 4 × (3 – 8) = 4 × 3 – 4 × 8?

 Let us check:

 4 × (3 – 8) = 4 × (–5) = –20

4 × 3 – 4 × 8 = 12 – 32 = –20

So, 4 × (3 – 8) = 4 × 3 – 4 × 8.

Look at the following:

 ( –5) × [( – 4) – ( – 6)] = ( –5) × 2 = –10

[( –5) × ( – 4)] – [( –5) × ( – 6)] = 20 – 30 = –10

So, ( –5) × [( – 4) – ( – 6)] = [( –5) × ( – 4)] – [ ( –5) × ( – 6)]

Check this for ( –9) × [ 10 – ( –3)] and [( –9) × 10 ] – [ ( –9) × ( –3)]

You will find that these are also equal. In general, for any three integers a, b and c,

a × (b – c) = a × b – a × c

Take atleast five different values for each of a, b and c and verify this property.

Integers

DIVISION OF INTEGERS

We know that division is the inverse operation of multiplication.

Let us see an example for whole numbers.

Since 3 × 5 = 15

So 15 ÷ 5 = 3 and 15 ÷ 3 = 5

 Similarly, 4 × 3 = 12 gives 12 ÷ 4 = 3 and 12 ÷ 3 = 4

 We can say for each multiplication statement of whole numbers there are two division statements.

Can you write multiplication statement and its corresponding divison statements for integers?

 l Observe the following and complete it

From the above we observe that :

 (–12) ÷ 2 = (– 6)

(–20) ÷ 5 = (– 4)

(–32) ÷ 4 = (– 8)

(– 45) ÷ 5 = (– 9)

We observe that 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.

l We also observe that: 72 ÷ (–8) = –9 and 50 ÷ (–10) = –5

72 ÷ (–9) = – 8 50 ÷ (–5) = –10

So we can say that when we divide a positive integer by a negative integer, we first divide them as whole numbers and then put a minus sign (–) before the quotient.

In general, for any two positive integers a and b

a ÷ (–b) = (– a) ÷ b where b ≠ 0

l Lastly, we observe that

(–12) ÷ (– 6) = 2;(–20) ÷ (– 4) = 5; (–32) ÷ (– 8) = 4; (– 45) ÷ (–9) = 5

So, we can say that when we divide a negative integer by a negative integer, we first divide them as whole numbers and then put a positive sign (+). In general, for any two positive integers a and b

 (– a) ÷ (– b) = a ÷ b where b ≠ 0

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.

Justify it by taking five more examples of your own. l

*We know that division is not commutative for whole numbers. Let us check it for integers also.

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. You can verify it by taking five more pairs of integers.

*Like whole numbers, any integer divided by zero is meaningless and zero divided by an integer other than zero is equal to zero i.e., for any integer a, a ÷ 0 is not defined but 0 ÷ a = 0 for a ≠ 0.

* When we divide a whole number by 1 it gives the same whole number. Let us check whether it is true for negative integers also. Observe the following : (– 8) ÷ 1 = (– 8) (–11) ÷ 1 = –11 (–13) ÷ 1 = –13 (–25) ÷ 1 = ____ (–37) ÷ 1 = ___

(– 48) ÷ 1 = ____

 This shows that negative integer divided by 1 gives the same negative integer.

So, any integer divided by 1 gives the same integer. In general, for any integer a,

a ÷ 1 = a

* What happens when we divide any integer by (–1)? Complete the following table

(– 8) ÷ (–1) = 8         11 ÷ (–1) = –11

 13 ÷ (–1) = ____       (–25) ÷ (–1) = ____

(–37) ÷ (–1) = ____    – 48 ÷ (–1) = ____

What do you observe? We can say that if any integer is divided by (–1) it does not give the same integer.

*Can we say[(–16) ÷ 4] ÷ (–2) is the same as (–16) ÷ [4 ÷ (–2)]?

We know that [(–16) ÷ 4] ÷ (–2) = (– 4) ÷ (–2) = 2

and (–16) ÷ [4 ÷ (–2)] = (–16) ÷ (–2) = 8

So [(–16) ÷ 4] ÷ (–2) ≠ (–16) ÷ [4 ÷ (–2)]

Can you say that division is associative for integers? No. Verify it by taking five more examples of your own

EXAMPLE 

 In a test (+5) marks are given for every correct answer and (–2) marks are given for every incorrect answer. (i) Radhika answered all the questions and scored 30 marks though she got 10 correct answers. (ii) Jay also answered all the questions and scored (–12) marks though he got 4 correct answers. How many incorrect answers had they attempted?

SOLUTION

(i) Marks given for one correct answer = 5

So, marks given for 10 correct answers = 5 × 10 = 50

 Radhika’s score = 30

Marks obtained for incorrect answers = 30 – 50 = – 20

Marks given for one incorrect answer = (–2)

Therefore, number of incorrect answers = (–20) ÷ (–2) = 10

 (ii) Marks given for 4 correct answers = 5 × 4 = 20

 Jay’s score = –12

Marks obtained for incorrect answers = –12 – 20 = – 32

Marks given for one incorrect answer = (–2)

 Therefore number of incorrect answers = (–32) ÷ (–2) = 16

EXAMPLE  A shopkeeper earns a profit of ₹1 by selling one pen and incurs a loss of 40 paise per pencil while selling pencils of her old stock.

(i) In a particular month she incurs a loss of  ₹5. In this period, she sold 45 pens. How many pencils did she sell in this period?

(ii) In the next month she earns neither profit nor loss. If she sold 70 pens, how many pencils did she sell?

SOLUTION

(i) Profit earned by selling one pen = ₹1

Profit earned by selling 45 pens = ₹45, which we denote by + ₹45

Total loss given = ₹5, which we denote by – ₹ 5

Profit earned + Loss incurred = Total loss

Therefore, Loss incurred = Total Loss – Profit earned

= ₹ (– 5 – 45) = ₹ (–50) = –5000 paise

Loss incurred by selling one pencil = 40 paise which we write as – 40 paise So, number of pencils sold = (–5000) ÷ (– 40) = 125

(ii) In the next month there is neither profit nor loss.

So, Profit earned + Loss incurred = 0

i.e., Profit earned = – Loss incurred.

 Now, profit earned by selling 70 pens = ₹ 70

Hence, loss incurred by selling pencils = ₹ 70 which we indicate by – ₹70 or – 7,000 paise.

Total number of pencils sold = (–7000) ÷ (– 40) = 175 pencils.