NCERT Class 8 Mathematics Textbook for Blind Students made Screen Readable By Professor T K Bansal.

1.2.1 Closure Property

(1) Whole numbers

Let us revisit the closure property for all the operations on whole numbers in brief.

(a) Addition of Whole Numbers

Let us try the addition of some of the whole numbers:

0 + 5 = 5, a whole number
4 + 7 = question? ... . Is it a whole number?

In general, a + b is a whole number for any two whole numbers a and b.

Whole numbers are closed under addition.

(b) Difference of Whole Numbers

Let us try the difference of some of whole numbers:

5 − 7 = −2, which is not a whole number.

Whole numbers are not closed under subtraction.

(c) Multiplication of Whole Numbers

The multiplication of some of the whole numbers is as follows:

0 × 3 = 0, a whole number
3 × 7 = question? ... . Is it a whole number?

In general, if a and b are any two whole numbers, their product a × b is a whole number.

Whole numbers are closed under multiplication.

(d) Division of Whole Numbers
Check the following division:
5 ÷ 8 = 5/8, which is not a whole number.

Whole numbers are not closed under division.

You may check for the closure property under all the four operations for more whole numbers.

(2) Integers

Let us now recall the operations under which integers are closed.

(a) Addition of integers

Let us try the addition of some of the integers:
− 6 + 5 = − 1, an integer
Is − 7 + (− 5) an integer?
Is 8 + 5, an integer?
In general, a + b is an integer for any two integers a and b.

Integers are closed under addition.

(b) Subtraction of integers

Let us try the difference of some of integers:
7 − 5 = 2, an integer
Is 5 − 7, an integer?
− 6 − 8 = − 14, an integer
− 6 − (− 8) = 2, an integer
Is 8 − (−6), an integer?

In general, for any two integers a and b, a − b is an integer.

Check if b − a is also an integer.

Integers are closed under subtraction.

(c) Multiplication of integers

The multiplication of some of the integers is as follows:
5 × 8 = 40, an integer.
Is − 5 × 8, an integer?
− 5 × (− 8) = 40, an integer

In general, for any two integers a and b, a × b is also an integer.

Integers are closed under multiplication.

(d) Division of integers

Check the following division:
5 ÷ 8 = 5/8, which is not an integer.

Integers are not closed under division.

You have seen that whole numbers are closed under addition and multiplication but not under subtraction and division. However, integers are closed under addition, subtraction and multiplication but not under division.

(3) Rational numbers

Recall that a number which can be written in the form of p/q, where p and q are integers and q ≠ 0, is called a rational number. For example,
\[\frac{−2}{3},\ \frac{6}{7},\ \frac{9}{−5}, \]
are all rational numbers. Since the numbers 0, −2, 4 can be written in the form p/q, they are also rational numbers.
(Check it yourself!)

(a) Addition of Rational Numbers

You know how to add two rational numbers. Let us add a few pairs.
\[\frac{3}{8}\ +\ \frac{(−5)}{7}\ =\ \frac{(21\ +\ (−40))}{56}\]
\[=\ \frac{−19}{56}\]
is a rational number.

\[\frac{−3}{8}\ +\ \frac{(−4)}{5}\ =\ \frac{(−15 + (−32))}{40}\ =\ ..\ ..\ \]
Is it a rational number?

\[\frac{4}{7}\ +\ \frac{6}{11}\ =\ ..\ ..,\]
Is it a rational number?

We find that sum of two rational numbers is always a rational number.
Check it for a few more pairs of rational numbers.

We say that rational numbers are closed under addition. That is, for any two rational numbers a and b, a + b is also a rational number.

(b) Difference of Rational Numbers

Will the difference of two rational numbers again be a rational number?

We have,
\[\frac{−5}{7}\ −\ \frac{2}{3}\ =\ \frac{(−5\ ×\ 3\ −\ 2\ ×\ 7)}{21}\ =\ \frac{−29}{21}\]
which is also a rational number.

\[\frac{5}{8}\ − \frac{4}{}3}\ =\ \frac{(25\ −\ 32)}{40}\ =\ ..\ ..\ \]
Is it also a rational number?

\[\frac{3}{7}\ −\ \frac{−8}{5}\ =\ ..\ ..,\]
Is it also a rational number?

Try this for some more pairs of rational numbers.

We find that rational numbers are closed under subtraction. That is, for any two rational numbers a and b, a − b is also a rational number.

(c) Let us now see the product of two rational numbers.

\[\frac{−2}{3}\ ×\ \frac{4}{5}\ =\ \frac{−8}{15};\]

\[\frac{3}{7}\ ×\ 2}{5}\ =\ \frac{6}{35}\]

It can be seen that both these products are rational numbers.

\[\frac{−4}{5}\ ×\ \frac{−6}{11}\ =\ ..\ ..,\]
Is it also a rational number?

Take some more pairs of rational numbers and check whether their product is again a rational number.

We say that rational numbers are closed under multiplication. That is, for any two rational numbers a and b, a × b is also a rational number.

(d) We note that
\[\frac{−5}{3}\ ÷\ \frac{2}{5}\ =\ \frac{−25}{6}\]
is a rational number.

\[frac{2}{7}\ ÷\ \frac{5}{3}\ =\ ...\ \]
Is it also a rational number?

\[\frac{−3}{8}\ ÷\ \frac{−2}{9}\ =\ ..\ ..,\]
Is it too a rational number?

Can we say that rational numbers are closed under division?

However, We find that for any rational number a, a ÷ 0 is not defined.

So rational numbers are not closed under division. However, if we exclude zero then the collection of, all other rational numbers is closed under division.

TRY Filling in the blanks in the following table.

Numbers	Closed under addition	Closed under subtraction	Closed under multiplication	Closed under division
Rational numbers	Yes	Yes	.. ..	No
Integers	.. ..	Yes	.. ..	No
Whole numbers	.. ..	.. ..	Yes	.. ..
Natural numbers	.. ..	No	.. ..	.. ..

1.2.2 Commutative Property

(1) Whole numbers

Recall the commutativity of different operations for whole numbers with the help of the following examples.

(a) Addition of Whole numbers
We know that
0 + 7 = 7 + 0 = 7;
2 + 3 = 3 + 2 = 5.
For any two whole numbers a and b, a + b = b + a

That is, Addition of whole numbers is commutative.

(b) Subtraction of Whole numbers

We know that,
7 − 5 = 2, and 5 − 7 = − 2.

That is, A − b ≠ b − a,

Hence subtraction of whole numbers is not commutative.

(c) Multiplication of Whole numbers

We know that,
5 × 7 = 7 × 5 = 35.
8 × 0 = 0 × 8 = 0,
i.e., Multiplication of whole numbers is commutative.

(d) Division of Whole numbers

\[\frac{5}{7}\ ≠\ \frac{7}{5},\]
and hence, Division of whole numbers is not commutative.

Check whether the commutativity of the operations hold for natural numbers also.

(2) Integers

(a) Addition of Integers

5 + (−7) = (−7) + 5 = −2
Addition of integers is commutative

(b) Subtraction of Integers

Is 5 − (−3) = − 3 − 5?
Subtraction of integers is not commutative.

(c) Multiplication of Integers

5 × (−7) = (−7) × 5 = −35
Multiplication of integers is commutative.

(d) Division of Integers

Is 5 ÷ (−7) = (−7) ÷ 5?
Division of integers is not commutative.

(3) Rational numbers

(a) Addition of rational numbers

We know how to add two rational numbers. Let us add a few pairs here.
\[\frac{−2}{3}\ +\ \frac{5}{7}\ =\ \frac{1}{21},\]
And
5/7 + (−2/3) = 1/21.
So, −2/3 + 5/7 = 5/7 + (−2/3)
Also, −6/5 + (−8/3) = …..
and −8/3 + (−6/5) = …
Is −6/5 + (−8/3) = (−8/3) + (−6/5)?
Is −3/8 + 1/7 = 1/7 + (−3/8)?
You find that two rational numbers can be added in any order. We say that addition is commutative for rational numbers. That is, for any two rational numbers a and b, a + b = b + a.

(b) Subtraction of rational numbers
Is 2/3 − 5/4 = 5/4 − 2/3?
Is ½ − 3/5 = 3/5 − ½?
You will find that subtraction is not commutative for rational numbers.

Note that subtraction is not commutative for integers and integers are also rational numbers. So, subtraction will not be commutative for rational numbers too.

(c) Multiplication of rational numbers
We have −7/3 × 6 /5
= −42/15
= 6/5 × (−7/3)
Is −8/9 × (−4/7) = −4/7 × (−8/9)?
Check for some more such products.

You will find that multiplication is commutative for rational numbers. In general, a × b = b × a for any two rational numbers a and b.

(d) Division of rational numbers
Is −5/4 ÷ 3/7 = 3/7 ÷ (−5/4)?
You will find that expressions on both sides are not equal.
So division is not commutative for rational numbers.

Complete the following table:

Numbers	Commutative for addition	Commutative for subtraction	Commutative for multiplication	Commutative for division
Rational numbers	Yes	...	...	...
Integers	...	No	...	...
Whole numbers	...	...	Yes	...
Natural numbers	...	...	...	No

1.2.3 Associative Property

(1) Whole numbers

Recall the associativity of the four operations for whole numbers through examples:

(a) Addition of Whole numbers

2 + (3 + 5) = (2 + 3) + 5 = 10,
Addition of whole numbers is associative

(b) Subtraction of Whole numbers

Is 3 − (5 − 3) = (3 − 5) − 3?
Subtraction of whole numbers is not associative

(c) Multiplication of Whole numbers

Is 7 × (2 × 5) = (7 × 2) × 5?
Is 4 × (6 × 0) = (4 × 6) × 0?
For any three whole numbers a, b and c
A × (b × c) = (a × b) × c
Multiplication of whole numbers is associative

(d) Division of Whole numbers

Check, 100 ÷ (25 ÷ 5) = 20, not = (100 ÷ 25) ÷ 5 = 4/5.
Division is not associative

Check for yourself the associativity of different operations for natural numbers.

(2) Integers

(a) Addition of Integers

Is (−2) + [3 + (− 4)]
= [(−2) + 3)] + (− 4)?
Is (− 6) + [(− 4) + (−5)]
= [(− 6) + (− 4)] + (−5)?
For any three integers a, b and c
a + (b + c) = (a + b) + c
Addition of integers is associative

(b) Subtraction of Integers

Is 5 − (7 − 3) = (5 − 7) − 3?
Subtraction of integers is not associative

(c) Multiplication of Integers

Is 5 × [(−7) × (− 8) = [5 × (−7)] × (− 8)?
Is (− 4) × [(− 8) × (−5)]
= [(− 4) × (− 8)] × (−5)?
For any three integers a, b and c
a × (b × c) = (a × b) × c
Multiplication of integers is associative

(d) Division of Integers

Is [(−10) ÷ 2] ÷ (−5?)
= (−10) ÷ [2 ÷ (− 5)]?
Division of integers is not associative

(3) Rational numbers

(a) Addition

We have −2/3 + [3/5 + (−5/6)]
= −21/3 + (−7/30)
= −27/30
= −9/10

[−2/3 + 3/5] + (−5/6)
= −1/15 + (−5/6)
= −27/30
= −9/10

So, −2/3 + [3/5 + (−5/6)] = [−2/3 + 3/5] + (−5/6)

Find −1/2 + [3/7 + (−4/3)] and [−1/2 + 3/7] + (−4/3).
Are the two sums equal?

Take some more rational numbers, add them as above and see if the two sums are equal.
We find that addition is associative for rational numbers. That is, for any three rational numbers a, b and c,
a + (b + c) = (a + b) + c.

(b) Subtraction

You already know that subtraction is not associative for integers, then what about rational numbers.

Is −2/3 − [−4/5 − ½]
= [2/3 − (−4/5)]
= − ½?
Check for yourself.
Subtraction is not associative for rational numbers.

(c) Multiplication

Let us check the associativity for multiplication.
−7/3 × (5/4 ×2/9)
= −7/3 × 10/36
= −70/108
= −35/54
(−7/3 × 5/4) × 2/9 = …

We find that
−7/3 × (5/4 × 2/9) = (−7/3 × 5/4) × 2/9

Is 2/3 × (−6/7 × 4/5) = (2/3 × −6/7) × 4/5?

Take some more rational numbers and check for yourself.

We observe that multiplication is associative for rational numbers. That is for any three rational numbers a, b and c,
a × (b × c) = (a × b) × c.

(d) Division of rational numbers

Recall that division is not associative for integers, then what about rational numbers?

Let us see if ½ ÷ [−1/3 ÷ 2/5]
= [1/2 ÷ (−1/3)] ÷ 2/5

We have, L H S = ½ ÷ (−1/3 ÷ 2/5)
= ½ ÷ (−1/3 × 5/2) (reciprocal of 2/5 is 5/2)
= ½ ÷ (−5/6) = ...

RHS = [½ ÷ (−1/3)] ÷ 2/5
= (1/2 × −3/1) ÷ 2/5
= −3/2 ÷ 2/5 = …..

Is LHS = RHS? Check for yourself.
You will find that division is not associative for rational numbers.

Complete the following table:

Numbers	Associative for addition	Associative for subtraction	Associative for multiplication	Associative for division
Rational numbers	Yes	...	...	No
Integers	...	...	Yes	...
Whole numbers	Yes	...	...	...
Natural numbers	...	No	...	...

Example 1

Find 3/7 + (−6/11) + (−8/21) + (5/22)

Solution:

3/7 + (−6/7) + (8/21) + (5/22)
= 198/462 + (−252/462) + (−176/462) + (105/462)
(Note that 462 is the LCM of 7, 11, 21 and 22)
= (198 − 252 − 176 + 105)/462
= −125/462

We can also solve it as.
3/7 = (−6/11) + (−8/21) + 5/22
= [3/7 + (−81/21)] + [−6/11 + 5/22] (by using commutativity and associativity)
= [(9 + (−8))/21] + [(−12 + 5)/22]
(LCM of 7 and 21 is 21; LCM of 11 and 22 is 22)
= 1/21 + (−7/22)
= (22 − 147)/462
= −125/462

Do you think the properties of commutativity and associativity made the calculations easier?

Example 2

Find −4/5 × 3/7 × 15/16 × (−14/9)

Solution:

We have −4/5 × 3/7 × 15/16 × (−14/9)
= (− (4 × 3)/ (5 × 7)) × ((15 × (−14))/ (16 × 9)
= −12/35 × (−35/24)
= (−12 × (−35))/ (35 × 24) = ½

We can also do it as.
−4/5 × 3/7 × 15/16 × (−14/9)
= (−4/5 × 15/16) × [3/7 × (−14/9)] (Using commutativity and associativity)
= −3/4 × (−2/3) = ½

1.2.4 The role of zero (0)

Look at the following.
2 + 0 = 0 + 2 = 2 (Addition of 0 to a whole number)
− 5 + 0 = 0 + (−5) = −5 (Addition of 0 to an integer)
−2/7 + 0 = 0 + (−2/7) = −2/7 (Addition of 0 to a rational number)

You have done such additions earlier also. Do a few more such additions.

What do you observe?
You will find that when you add 0 to a whole number, the sum is again that whole number. This happens for integers and rational numbers also.

In general, a + 0 = 0 + a = a, where a is a whole number
b + 0 = 0 + b = b, where b is an integer
c + 0 = 0 + c = c, where c is a rational number

Zero is called the identity for the addition of rational numbers. It is the additive identity for integers and whole numbers as well.

1.2.5 The role of 1

We have,
5 × 1 = 5 = 1 × 5 (Multiplication of 1 with a whole number)
−2/7 × 1 = 1 × (−2/7) = −2/7
3/8 × 1 = 1 × 3/8 = 3/8
What do you find?
You will find that when you multiply any rational number with 1, you get back the same rational number as the product.
Check this for a few more rational numbers.
You will find that,
a × 1 = 1 × a = a, for any rational number a.
We say that 1 is the multiplicative identity for rational numbers.

Is 1 the multiplicative identity for integers? For whole numbers?

THINK, DISCUSS AND WRITE

If a property holds for rational numbers, will it also hold for integers? For whole numbers? Which will? Which will not?

1.2.6 Negative of a number

While studying integers you have come across negatives of integers. What is the negative of 1?
It is − 1 because 1 + (− 1) = (−1) + 1 = 0
So, what will be the negative of (−1)? It will be 1.

Also, 2 + (−2) = (−2) + 2 = 0,
so we say 2 is the negative or additive inverse of −2 and vice versa. In general, for an integer a, we have, a + (− a) = (− a) + a = 0;
so, a is the negative of − a and − a is the negative of a.

For the rational number 2/3, we have,
2/3 + (−2/3) = (2 + (−2))/ 3 = 0
Also, (−2/3) + 2/3 = 0 (how?)
Similarly, −8/9 + .. .. = .. .. + (−8/9) = 0
.. .. + (−11/7) = (−11/7) + .. ... = 0

In general, for a rational number a/b, we have,
A/b + (−a/b) = (−a/b) = (−a/b) + a/b = 0
we say that −a/b is the additive inverse of a/b and a/b is the additive inverse of (−a/b).

1.2.7 Reciprocal

By which rational number would you multiply 8/21,, to get the product 1?
Obviously by 21/8,
since 8/21 × 21/8 = 1.

Similarly, −5/7 must be multiplied by 7/− 5 so as to get the product 1.

We say that 21/8 is the reciprocal of 8/21 and 7/− 5 is the reciprocal of −5/7.

Can you say what is the reciprocal of 0 (zero)?
Is there a rational number which when multiplied by 0 gives 1? Thus, zero has no reciprocal.

We say that a rational number c/d is called the reciprocal or multiplicative inverse of another non-zero rational number a/b if a/b × c/d = 1.

1.2.8 Distributive Property of multiplication over addition for rational numbers

To understand this, consider the rational numbers −3/4, 2/3 and −5/6
−3/4 × {2/3 + (−5/6)}
= −3/4 × {((4) + (−5))/6}
= −3/4 × (−1/6)
= 3/24 = 1/8

Also −3/4 × 2/3
= (−3 × 2)/ (4 × 3)
= −6/12 = −1/2
And −3/4 × −5/6 = 5/8

Therefore (−3/4 × 2/3) = (−3/4 × −5/6)
= −1/2 + 5/8 = 1/8

Thus, −3/4 × {2/3 + −5/6}
= (−3/4 × 2/3) + (−3/4 × −5/6)

Distributivity of Multiplication over Addition and Subtraction.
For all rational numbers a, b and c,
A (b + c) = A B + a c
A (b − c) = a b − a c

TRY THESE

Find using distributivity.
(1) {7/5 × (−3/5)} + {7/5 × 5/12}
(2) {9/16 × 4/12} + {9/16 × −3/9}

Example 3

Write the additive inverse of the following:
(1) −7/19
(2) 21/112

Solution:

(1) 7/19 is the additive inverse of −7/19 because
−7/19 + 7/19 = (−7 + 7)/19 = 0/19 = 0

(2) The additive inverse of 21/112 is −21/112 (Check!)

Example 4

Verify that − (− x) is the same as x for
(1) x = 13/17
(2) x = −21/31

Solution:

(1) We have, x = 13/17
The additive inverse of x = 13/17 is − x = −13/17
since 13/17 + (−13/17) = 0.

The same equality 13/17 + (−13/17) = 0,
shows that the additive inverse of −13/17 is 13/17
or − (−13/17) = 13/17,
i.e., − (− x) = x.

(2) Additive inverse of x = −21/31 is −x = 21/31
since −21/31 + 21/31 = 0

The same equality −21/31 + 21/31 = 0,
shows that the additive inverse of 21/31 is 21/31,
i.e., − (− x) = x.

Example 5

Find 2/5 × −3/7 − ¼ − 3/7 × 3/5

Solution:

2/5 × −3/7 − 1/14 − 3/7 × 3/5
= 2/5 × −3/7 × − 3/7 × 3/5 − 1/14 (by commutativity)
= 2/5 × −3/7 + (−3/7) × 3/5 − ¼
= −3/7 (2/5 + 3/5) − 1/14 (by distributivity)
= −3/7 × 1 − 1/14
= (−6 − 1)/14 = −1/2

EXERCISE 1.1

Q1. Using appropriate properties find.
(1) −2/3 × 3/5 + 5/2 − 3/5 × 1/6
(2) 2/5 × (−3/7) − 1/6 × 3/2 + 1/14 × 2/5

A1.
(1) 2
(2) −11/28

Q2. Write the additive inverse of each of the following.
(1) 2/8
(2) −5/9
(3) −6/−5
(4) 2/−9
(5) 19/−6

A2.
(1) −2/8
(2) 5/9
(3) −6/5
(4) 2/9
(5) 19/6

Q3. Verify that − (− x) = x for.
(1) x = 11/15
(2) x = −13/17

Q4. Find the multiplicative inverse of the following.
(1) − 13
(2) −13/19
(3) 1/5
(4) −5/8 × −3/7
(5) −1 × −2/5
(6) −1

A4.
(1) −1/13
(2) −19/13
(3) 5
(4) 56/15
(5) 5/2
(6) −1

Q5. Name the property under multiplication used in each of the following.
(1) −4/5 × 1
= 1 × −4/5 = −4/5

(2) −13/17 × −2/7
= −2/7 × −13/17

(3) −19/29 × 29/−19 = 1

A5.
(1) 1 is the multiplicative identity
(2) Commutativity
(3) Multiplicative inverse

Q6. Multiply 6/13 by the reciprocal of −7/16

A6. −96/91

Q7. Tell what property allows you to compute 1/3 × (6 × 4/3) as (1/3 × 6) × 4/3.

A7. Associativity

Q8. Is 8/9 the multiplicative inverse of −1 (1/8)? Why or why not?

A8. No, because the product is not 1.

Q9. Is 0.3 the multiplicative inverse of 3 1/3? Why or why not?

A9. Yes, because 0.3 × 3 1/3 = 3/10 × 10/3 = 1

Q10. Write.
(1) The rational number that does not have a reciprocal.
(2) The rational numbers that are equal to their reciprocals.
(3) The rational number that is equal to its negative.

A10.
(1) 0
(2) 1 and (−1)
(3) 0

Q11. Fill in the blanks.
(1) Zero has ________ reciprocal.
(2) The numbers ________ and ________ are their own reciprocals
(3) The reciprocal of − 5 is ________.
(4) Reciprocal of 1/x, where x ≠ 0 is ________.
(5) The product of two rational numbers is always a _______.
(6) The reciprocal of a positive rational number is ________.

A11.
(1) No
(2) 1, −1
(3) −1/5
(4) x
(5) Rational number
(6) positive