Prime Mathematics 8

Prime Mathematics Book - 8

Area Revision Test (Sets)

1. Represent the given sets as stated in the brackets.
(a) A = {0,1, 8, 27, 64, 125} [rule method]
(b) B = { Whole numbers less than 10}

2. Form the subsets of :

(a) A = {a} (b) B = {1, 2}

3. Write the proper subsets of:

(a) {a, b} (b) {x, y, z}

4. If A and B are two overlapping sets under the universal set ∪. Draw the Venn
diagram and shade

(a) A ∪ B (b) A ∩ B (c) A − B (d) A ∪ B

5. Write the following in set notations

(i) A U (ii) A U

B B

6. If A = {x : x is a factor of 6} and B= {x : x ∈ N, n ≤ 5} list A∪B.
7. If A and B are the sets under universal set ∪ where n(∪) = 100, n(A) = 48,

n(B) = 54 and n( A ∪ B ) = 20, find n0(A).

8. Out of 48 students of a class the number of students who like coffee and tea
are in the ratio 3 : 2, 5 like both the beverages and 3 do not like any of the
beverages.

(i) Illustrate the above information in a Venn diagram
(ii) Find the number of students who like tea.

146 5 Sets

Prime Mathematics Book - 8

Answers
Exercise 1.1(Show to your teacher)
Exercise 1.2 (Venn diagram and set operation)
4. (a) {1, 5, 7, 11} (b) {1, 3, 5, 6, 7, 9 11, 12} (c) {3, 9}
U
U U
AB
A BA B





(d) {0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11} (e) {0, 2, 4, 8, 10} (f) {0, 1, 2, 4, 5, 6, 7, 8, 10, 11, 12}
U
UU
AB A BA B



5. (a) {1, 2, 3, 4, 5, 6, 8} (b) {2} (c) {3, 5, 6}

U U U
A BA B
AB



C C C

(e) {1, 2, 4} (f) {1, 2, 3, 4, 8}

(d) {0, 7, 9} U A U A U
B B B
A



C C C

6. (a) {1, 2, 3, 4, 5, 6} (b) {1, 2, 3} (c) {6} (d) {4,5}
7. (a) {1, 2, 3, 5, 7, 9U} (b) {3, 5, 7} (c) {1, 9}
AB
A U
2 31 B
U

AB

5

79

4, 6, 8, 10 (e) {2, 4, 6, 8, 10} (f) {1, 2, 4, 6, 8, 9, 10}
U
(d) {4, 6, 8 ,10} U A
U B
AB
AB

8. (a) {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15} (b) {6} (c) {1, 2, 4, 5, 8, 10}
U
A U U
B AB
AB

1, 5 8, 10 1, 5 8, 10 1, 5 8, 10
2, 4 2, 4 2, 4

6 6 6
3 12 3 12 3 12

9, 15 9, 15 7, 11, 13, 14 9, 15 C
7, 11, 13, 14
C 7, 11, 13, 14 C

Answers 147

Prime Mathematics Book - 8

(d) {3, 6, 12} (e) {7, 11, 13, 14} (f) { 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15}

U U U

AB AB AB

CC C

Exercise 1.3 (Cardinal number)
1. (a) (i) 36 (ii) 14 (iii) 15 (iv) 9 (b) (i) 22 (ii) 75 (iii) 23 (iv) 30 (c) (i) 62 (ii) 80 (d) (i) 88% (ii) 21% (iii) 29% (iv) 38%
2. (a) (i) 73 (ii) 82 (iii) 115 (iv) 5 (v) 120 (b) (i) 58 (ii) 38 (iii) 37 (iv) 70 (c) (i) 0 (ii) 115 (iii) 70
(d) (i) 20 (ii) 34 (iii) 14 (iv) 40
U
3. 5 4. 30 5. 70. A 6. 30, A U 7. 15
B
B

20 20 30 30 30 10

8. M U, 5 9. 30, M 10
40
A U

40 10 45 C

30 30

x 10

10. (a) (i) 170 (ii) 30 (iii)60 (iv) 40 (v) 11. (i) 20% (ii) 30% (iii) U

F U H N
130-70
n(F) 70 110-70 M n(M)=110 70%-20% 20% 50%-20% n0(N)
n0(F) = 60 =40 n0(M) = 50% = 30% n0(H∪N)

n0(F∪M) n0(H) 0

30 n(H∩N)

n(F∩M)

12. (i) (ii) 36 (iii) 24 13. (i) 280 (ii) 720 (iii)

C U P U
600-280
36-15 T = 320 680-280 L
= 21 280 = 400
n0(C) 15 24-15 n0(T) n0(L)
14. 12, M =9
n0(P) 0 n0(P∪L)
3 n0(C∪T)
n(P∩L)
n(C∩T)

U 15. a) 44 b) 6

S

60 24 24

12

148 Answers

At the end of this unit the student will be able to Estimated periods - 23
Objectives:

● get introduction of preexisting number systems like grouping, multiple grouping, ciphered system.
● know about the positional system of numerals.
● convert binary to denary and denary to binary.
● convert quinary number to denary and denary to quinary.
● add and subtract binary number.
● add and subtract quinary number.
● know decimal and vulgar fractions.
● convert terminating decimals to decimal fractions and non - terminating but recurring decimals into vulgar

fractions.
● know the properties of rational and irrational numbers.
● know the properties of surds.

in Egyptian Hieroglyphic (simple grouping)
ccxxxiv in Roman (grouping, additive & subtractive)

LQ#?$ in chinese (multiple grouping)

τ λ δ in Greek (cipheral)
234 in decimal (positional)

Teaching Materials:

List of prime numbers, List of ancient numerals, conversion table of numerals, graphs, square papers,
charts showing three of numbers.

Activities:

It is better to
● list of ancient numerals, conversion tables of numerals.
● explain and discuss about the development of number system.
● explain the process of conversion original and alternative methods.
● let the students know terminating recurring and non terminating and non recurring decimals by actual

division and division also with of square roots.
● discuss about the different properties of rational & irrationals(surds).
● discuss different types of rationalizing of actual.
● Explain the cardinality relations.

Prime Mathematics Book - 8

Unit - 1 Number System Estimated period : 6

Number System

To express numbers in written form called numerals, symbols are adopted and
arranged in order. In the history of development of mathematics, we find the following
major number systems.

1. Grouping system:
In this system symbols are put in groups in their respective positions. Under this

system, there were simple grouping system and multiplicative grouping system.

A. Simple grouping system:
Under this system comes Egyptian Hieroglyphic and Roman numeral system.
Egyptian Hieroglyphic numerals used to be based on scale of 10 (base 10).

They adopted symbols for 1 and powers of 10 as:

1 = (a stroke) 104 = (a pointing finger)
10 = (a heel bone) 105 = (a tadpole)
102 = (a coil of rope) 106 = (a man in astonishment)

103 = (a lotus flower)

Numbers were expressed using this symbols additivity each symbol being
repeated the required times.
Thus, 123456 = 1(105) + 2(104) + 3(103) + 4(102) + 5(10) + 6 would be

Written from right to left as

Roman numeral system was also simple grouping system (also base 10).
They adopted the symbols.

I, X, C, M for 1,10, 100 and 1000 and V, L, D, for 5, 50, 500 respectively.

Thus, 3284 = 3 x 1000 + 2 x 100 + 8 x 10 + 4 x 1

= 3 x 1000 + 2 x 100 + (50 + 30) + (5 - 1)

Would be write as: MMMCCLXXXIIII (Ancient additive way)

MMMCCLXXXIV (Modern subtractive way)

B. Multiplicative grouping system:
Under this system comes traditional Chinese numeral system. They adopted the symbols.

1 2 3 4 5 6 7 8 9 10 102 103

! L# $ % & Q( ) * = ?

e.t.c
Thus they wrote 4567
= 4 x 103 + 5 x 102 + 6 x 10 + 7

150 6 Real Number System

Prime Mathematics Book - 8

Vertically with group of two as: $
?
%
=
&
*
Q
2. Ciphered numeral system:

Under this system comes Greek alphabetic system. They adopted alphabets

for 1 to 9 and all the ciphered numbers (number with zero) as:

1 = α(alpha) 10 = ί(iota) 100 = ρ (rho)

2 = β(beta) 20 = κ(kappa) 200 = σ(sigma)

3 = γ(gamma) 30 = λ(lambda) 300 = τ(tau)

4 = δ(delta) 40 = µ(mu) 400 = υ(upsilon)

5 = ε(epsilon) 50 = ν(nu) 500 = φ(phi)

6 = ς(digamma) 60 = ξ(xi) 600 = χ(chi)

7 = ζ(zeta) 70 = ο(omicron) 700 = ψ(psi)

8 = η(eta) 80 = π(pi) 800 = ω(omega)

9 = θ(theta) 90 = ϕ(koppa) 900 = λ(sampi)

Thus, 345 was written as 300+40+5= τ µ ε

3. Positional numeral system:
Under this system comes today's number system (decimal, binary, octal, hexadecimal
e.t.c). In this system, after base b is selected, basic symbols, 0, 1, 2, 3, ..., b - 1
are adopted and there are b basic symbols called digits and any number N can be
written uniquely SaysstNem= )anhbans+aadno-1pbtne-1d+.t.h..e+ dai2gbit2s +0a, 11b,1 + a30., Decimal system(Hindu
Arabic Numeral 2, 4, 5, 6, 7, 8, 9. Lets
consider a number 5432 and see how the digits are arranged in their respective

positions. Positional Weight


Thousands Hundreds Tens Ones

5 4 3 2

2 x 100 = 2

3 x 101 = 30

4 x 102 = 400

5 x 103 = 5000
5432

Hindu Numerals, now known to us as Hindu Arabic Numerals along with zero was
developed some time between 250 A.D. to 800 A.D. from Hindu Brahmi numerals which
was ciphered system by Hindu mathematicians. It has dominated all the numerals
previously developed.

Number System 151

Prime Mathematics Book - 8

The distinguishing characteristics of positional number system is its base and use of
symbol 0 (zero) for voidness. The base of a number system is defined as the number of
different digits (symbols) used in the system.

Depending upon the base, we have the following number systems.

i. Decimal number system ii. Binary number system
iii. Quinary number system iv. Octal number system
v. Hexadecimal number system

After development of computer, it became necessary to use other system like binary,
octal and hexadecimal systems.

1.1 Binary number system:
Binary number system is a base 2 positional system. It uses two digits 0 and 1 called

bits. Each one of the digits in a binary number system has a place value or weights
expressed as powers of 2 as positional weights in decimal system are expressed as
powers of 10.

Consider a binary number 11001.

Positional weight


24 23 22 21 20

1 1 0 0 1

1 x 20 = 1 Decimal equivalent
0 x 21 = 0
0 x 22 = 0

1 x 23 = 8

1 x 24 =+ 16
25

Decimal to Binary Conversion:
→ Express the decimal number as the sum of the possible binary positional weights (or

place values) in expanded from.

→ Since positional values are obvious in positional number system, remove the
positional weights (place values). Remaining face values in order gives binary
equivalent.

For example, to convert 5010 to binary.

Binary positional weights (Place Values) 26 25 24 23 22 21 20
32 16 8 4 2 1
Decimal equivalent 64 32 16 0 0 2 0
1 x 25 1 x 24 0 x 23 0 x 22 1 x 21 0 x 20
Given denary number 50 1 1 0 0 1 0

In terms of binary place values

Binary equivalent

∴ 5010 = 1100102

152 6 Real Number System

Prime Mathematics Book - 8

Alternative Method:

The decimal number is divided successively by 2.

→ The quotient and remainder are noted down in each step.

→ The quotient of each step is divided by 2 in the next step.

→ The process is repeated until the quotient becomes zero.

The first remainder is the least significant bit (L.S.B.) and the last remainder is

the most significant bit (M.S.B.). Thus digits (bits) in the remainder arranged from

M.S.B. to L.S.B. given the binary equivalent.

For example to convert 5010 to binary

Quotient Remainder or Simply

5 0 ÷ 2 = 25 0 L.S.B. 2 50 Remainder
25 ÷ 2 = 12 1 0
12 ÷ 2 = 6 0 2 25 1
6 ÷ 2 = 3 0 2 12 0
3 ÷ 2 = 1 1 26 0
1 ÷ 2 = 0 1 M.S.B. 23 1
21 1
∴ 5010 = 1100102
0

Binary to decimal conversion:
To convert a binary to decimal, various bits (digits) are multiplied by their respective
positional weights (place values) and their sum gives the decimal equivalent.

GFoivreenxabminpalrey,ntuomcboenrv ert 1 100 102 to decimal. 1 0 0 1 0
1
Bit position 6th 5th 4th 3rd 2nd 1st
Positional weights (Place values) 25 24 23 22 21 20
Decimal equivalent 1 x 25 1 x 24 0 x 23 0 x 22 1 x 21 0 x 20
= 32 + 16 + 0 + 0 + 2 + 0 = 50
∴ 1100102 = 5010

Conversion from binary with decimal point to denary:
→ Various bits (digits) are multiplied by their respective positional weights (place

values) and their sum gives the denary equivalent.

→ Place values beyond decimal points are negative powers of 2.

For example, to 2c2o+nv0exrt2110+11.1x01220 to denary 0 x 2-2 + x 2-3
1 0 1.1012 =1x + 1 x 2-1 + 0 x 0.25 1 1 x 0.125
=1x 4 + 0 x 2 + 1 x 1 + 1 x 0.5 + +
= 4 + 0 + 1 + 0.5 + 0 + 0.125 = 5.62510

Number System 153

Prime Mathematics Book - 8

Conversion from denary with decimal point to binary we proceed in two steps.

I. For whole number (integral) part, we use usual successive division by 2 methods.
II. For the decimal part

→ Multiply the decimal parts successively by 2
→ Integral (whole number) parts and decimal parts in step are ruled.
→ Decimal part of each step is multiplied by 2 on the next step.

Process terminates or in some case doesn't terminate. If so take certain decimal places.
Whole number of first step is M.S.B. and last one is L.S.B.
For example, to convert 5.62510 to binary

Converting Whole number part to binary Converting decimal part to binary

2 5R Whole number

2 2 1 L.S.B. 0.625 x 2 = 1.25 M.S.B.
2 10
0.25 x 2 = 0.5 0
0 1 M.S.B.
0.5 x 2= 1.0 1 L.S.B.

∴ 5.62510= 101.1012
Similarly, to convert 12.34510 to binary

Converting Whole number part to binary Converting decimal part to binary

2 12 R Whole number
0 L.S.B.
26 0 0.345 x 2 = 0.690 0 M.S.B.
23 1
21 1 M.S.B. 0.69 x 2 = 1.38 1

0 0.38 x 2 = 0.76 0

0.76 x 2 = 1.52 1

∴ 12 = 1100 0.52 x 2 = 1.04 1 L.S.B.

∴ 0.345 =.01011....

∴ 12.34510= 1100.010112

Note: Conversion with decimal points is not in the course. You can practice for
your extra knowledge.

13.2 Binary Operations:
Here we will learn about addition and subtraction of binary numbers.

A. Addition of binary numbers:
→ Process of addition is same as in the case of addition of denary numbers.
→ Arrange vertically with digits of same places along same column.
→ Start adding the digits in the lowest place. If the sum is 2 or more than 2 (denary)
convert into binary. Write down the 1st digit and rest, add to the next higher
place digits as carry over.

→ Continue the process in all the column.

154 6 Real Number System

For example, to add 1112 and 1012 Prime Mathematics Book - 8

- In the place of 20, 1+1=2=102 write
23 22 21 20 Places down 0 and take carry over 1

1 1 1 Carry over - In 21 place, 1 +1=2=102 write 0 and
1 1 12 take 1 carry to next.

+ 1 0 12 - In 22 place, 1 +1+1=3=112 write down
1 and 1 as carry over.
1 1 0 02

∴ 1112+1012 = 11002

Similarly, to add 1012, 1102 and 1112

places
24 23 22 21 20 working row for carry over

1 0 1 1 20 place,1+1=2=102 write 0, carry 1
1 0 12 21 place, 1 +1+1=3=112 write 1, carry 1
1 1 02 22 place, 1 +1+1+1=4=1002 write 0, carry 10
1 1 12

1 0 0 1 02

∴ 1012+1102+1112 = 100102

Alternate method: 21 20 places 20 place,1+1=2 → 2 2
24 23 22 carry over 10
write 0,carry 1
1 2 1 1
1 0 12 21 place, 1 +1+1=3 → 2 3
1 1 02 1
1 1 12 write 1,carry 1 1

1 0 0 1 02 22 place, 1 +1+1+1=4→ 2 4
2 0
write 0,carry 2

23 place, 2 +0=2 → 2 2
1 0
write 0,carry 1

∴ 1012+1102+1112 = 100102

Number System 155

Prime Mathematics Book - 8

B. Subtraction of binary numbers:
→ Arrange vertically so that digits of same place lie on same column
→ Rules 02 - 02 = 02
12 - 12 = 02
12 - 02 = 12
02 - 12 = 12 with borrow
→ 1 borrow from next higher position = 2
E.g. to subtract 1011012 from 10101012

26 25 24 23 22 21 20 positions

2 0 2 borrow

1 0 1 0 1 0 1 In 20 position, 1 - 1 = 0

− 1 0 1 1 0 1 In 21 position, 0 - 0 = 0
1 0 1 0 0 0 In 22 position, 1 - 1 = 0
In 23 position borrow, (2 + 0) - 1 = 1

In 24 position, 0 - 0 = 0

Example 1: Add 11012 and 1112 21 20 In 25 position, 2 + 0 - 1 = 1
Solution: positions

24 23 22

1 1 1 1 carry over

1 1 0 12 In position of 20 →1+1 = 0 carry 1
In position of 21 → 1 +0+1 = 0 carry 1
+ 1 1 12 In position of 22 → 1 +1+1 = 1 carry 1

1 0 1 0 02

∴ The sum is 101002 In position of 23 → 1 +1 = 0 carry 1
In position of 24 → 1 = 1

Example 2: Complete the addition 1112+1102+1012

Solution: 23 22 21 20 positions
carry over
2 1 1 1

1 1 12
1 1 02
+ 1 0 12

1 0 0 1 02
∴ 1112+1102+1012 = 100102

156 6 Real Number System

Example 3: Subtract 1112 from 11102 Prime Mathematics Book - 8

Solution: 23 22 21 20 positions
borrows
2 2 2

1 1 1 02
 1 1 12

1 1 12
∴ 11102 - 1112 = 1112

Example 4: Simplify: 11012+10112-1112 1 1 2
2 2
Solution: 11012+10112-1112
1 1 1 1



1 1 0 12
+ 1 0 1 12

1 1 0 0 02 Again

= 110002 - 1112 1 1 0 0 02
= 100012  1 1 12

1 0 0 0 12

Exercise 1.1

1. Convert the following denary numbers into binary.
(a) 9 (b) 12 (c) 17 (d) 25 (e) 50
(f) 65 (g) 70 (h) 91 (i) 100 (j) 111

2. Convert the following binary numbers into denary.

(a) 112 (b) 1102 (c) 1112 (d) 1002 (e) 1012
(f) 111012 (g) 10101012 (h) 10002 (i) 11112 (j) 110110112

3. Find the sums: (c) 11012+1112+1002

(a) 112+10012 (b) 11012+10102
(d) 100102+1112+100012 (e) 1002+112+112

4. Complete the following subtractions:

a) 1112 - 1012 b) 101102 - 10112 c) 11102 - 1112
d) 1100002 - 11112 e) 10000102 - 1011112 f) 1010002 - 11112

Number System 157

Prime Mathematics Book - 8

1.2 Quinary number system:

Quinary number system is a positional system with based - 5 constituting 5 digits 0, 1,
2, 3, 4.

Consider a quinary number 12345



Positional weight 50


53 52 51

(M.S.D .) 1 2 3 4 (L.S.D.) Decimal equivalent

4 x 50 = 4

3 x 51 = 15

2 x 52 = 50

1 x 53 = 125

19410

Decimal to quinary conversion:

→ Express the decimal number as the sum of the possible quinary positional weights
(Place values) in expanded from.

→ Since positional values are obvious in positional number system, remove the
positional weights.

→ The remaining face values in order gives quinary equivalent.

For example, to convert 439210 to quinary

Quinary positional weights 55 54 53 52 51 50
Decimal equivalent 3125 625 125 25 5 1
(In term of quinary (rem.1267) (rem.17) (rem.17) (rem.17) (rem.2)
weights) 439210 1x3125 2x625 0x125 0x25 3x5 2x1
In terms of quinary place 1x55 2x54 0x53 0x52 3x51 2x50
values
Quinary equivalent 1 2 0 0 3 2

∴ 439210 = 1200325

Alternative Method:
→ Divide the decimal number by 5 successively.

→ Note down the quotient and remainder in each step.

158 6 Real Number System

Prime Mathematics Book - 8

→ Repeat the process until the quotient becomes zero.

→ The first remainder is the least significant digits (L.S.D.) and the last remainder is
the most significant digit (M.S.D.).

Thus, the digits in the remainder in the order from M.S.D. to the L.S.D. gives the
quinary equivalent.

e.g. to convert 439210 to quinary. or Simply

Quotient Remainder

4392 ÷ 5 = 878 2 L.S.D. 5 4392 Remainder

878 ÷ 5 = 175 3 5 878 2

175 ÷ 5 = 35 0 5 175 3
35 ÷ 5 = 7 0
7 ÷ 5 = 1 2 5 35 0

1 ÷ 5 = 0 1 M.S.D. 57 0
51 2
1
0
∴ 439210 = 1200325

Quinary to decimal conversion:

To convert a quinary number to decimal

→ Multiply the various digits by their respective positional weights.

→ Sum of the term gives the decimal equivalent.

For example, to convert 1200325 to decimal.

Quinary number 1 2 0 0 3 2

Position of digits

(from right to left) 6th 5th 4th 3rd 2nd 1st

Positional weights 55 54 53 52 51 50

Decimal equivalent 1 x 55 2 x 54 0 x 53 0 x 52 3 x 51 2 x 50

= 3125 + 1250 + 0 + 0 + 15 + 2

= 439210
∴1200325 = 439210

Quinary Operations:

In this class we will learn about addition and subtraction of quinary numbers.

1. Addition of quinary numbers:
The process of quinary addition is also same as in the case of addition of denary
numbers.
→ Arrange vertically with digits of the same places along same column.

Number System 159

Prime Mathematics Book - 8

→ Start adding the digits in the lowest place. If the sum is 5 or more than 5
(denary) convert into quinary. Write down the 1st digit and rest, add to the
next higher place digit as carry over.

→ Continue the process in next columns.

For example, to add 10005, 14215 and 3035
Solution:

54 53 52 51 50 position

1 carry over

1 0 0 05 In the position of 50, 0 + 1 + 3 = 4 = 45
In the position of 51,0 + 2 + 0 = 2 = 25
1 4 2 15 In the position of 52, 0 + 4 + 3 = 7 = 125
write 2, carry 1
+ 3 0 35 In the position of 53, 1 + 1 + 1 = 3 = 35

3 2 2 45

∴ 10005+ 14215 + 3035 = 32245

Subtraction of Quinary numbers:
To subtract quinary numbers.

→ Arrange vertically with digits of same place along the same column.
→ 1 borrow from next higher position=5

For example, to subtract 14235 from 23415
Solutions:

53 52 51 50 position
result equivalent
8 3 6
1 5 5 borrow
2 1 3 4 1 1 5
 1 4 2 35 50 Place, borrow 1=5+1= 6,6-3= 35
51 Place, rem. 3-2 = 1 = 15
4 1 35 52 Place, borrow 1= 5+3 = 8,8-4= 45
53 Place, rem. 1-1 = 05
∴ 23415 - 14235 = 4135

Example 1: Convert 9810 into base 5 53 52 51 50
Solution: To convert 9810 to quinary 125 25 5 1

Quinary positional
Decimal equivalent

160 6 Real Number System

Prime Mathematics Book - 8

(In term of quinary weights) 9810 3 x 25 (rem.23) 4 x 5 (rem.3) 3 x 1
In terms of quinary place values
Quinary equivalent 3 x 52 4 x 51 3 x 50
3
43

∴ 9810 = 3435

Alternate method:

5 98 Remainder
5 19 3
4
53 3

0

∴ 9810 = 3435

Example 2: Convert 2345 into denary.

Solution: Quinary number 2 3 4

Positional weights 52 51 50

Decimal equivalent 2 x 52 3 x 51 4 x 50

= 2 x 52 + 3 x 51 + 4 x 50

= 50 + 15 + 4

=6910

∴2345 = 6910

Example 3: Complete the following additions.

(a) 1243314455 (b) 223044421332555
+ +

Solution: (a) 53 52 51 50 position
carry over(denary)
1 1 1

2 3 1 45
+ 1 4 3 45
35
4 3 0

∴ The sum is 43035

Number System 161

Prime Mathematics Book - 8

(b) 54 53 52 51 50 position
carry over
1 1 1 1
position
2 0 4 3255 equivalent(denary)
3 4 1 borrow

+ 2 4 2 35

1 3 4 3 35
∴ The sum is 134335

Example 4: Subtract 12345 from 30135
Solution:

53 52 51 50

2 4 8

5 5 5

3 1 0 1 1 1 3 5
- 1 2 3 45

1 2 2 45
∴ The difference is 12245

Example 5: Simplify: 430125 - 120345 + 432115
Solution:

1 1

4 3 0 1 25
+4 3 2 1 15 5
35
1 4 1 2 2 35 45
45
= 430125-120345+432115 Again
= 1412235-120345 3 5 1 5
= 1241345
1 4 1 2 2
- 1 2 0 3

1 2 4 1 3

Example 6: Convert 1011012 to quinary.
Solution:

Firstly converting 1011012 to denary

1011012 = 25 × 1 + 24 × 0 + 23 × 1 + 22 × 1 + 21 × 0 + 20 × 1

= 32+8+4+1

= 4510

162 6 Real Number System

Prime Mathematics Book - 8

Secondly converting 4510 to quinary 5 45

5 9 0

5 1 4 = 1405

0 1

∴ 1011015=1405

Exercise 1.2

1. Convert the following denary numbers into quinary.

(a) 586 (b) 1234 (c) 678 (d) 684 (e) 318 (f) 2480
(g) 50 (h) 987

2. Convert the following quinary numbers into denary.

(a) 245 (b) 235 (c) 145 (d) 2435 (e) 4315 (f) 3215
(g) 1025 (h) 120345 (i) 10005 (j) 102345

3. Complete the following additions.

( a) 2345 (b) 204 5 (c) 333 5 (d) 12033455 (e) 12345 (f) 11105
+ 135 + 345 + 2225 + + 23405 + 40405

(g) 43215 (h)+430002255 (i) 1+22333444555 (j) +21044442555 (k) +144123234201555 (l) +14043142432455 5
+ 4325

4. Complete the following subtractions.

( a)1 34000055 (b)2331455 (c)204000255 (e) 2312114055 (f) 1010204055 (g)2040210055 (h) 3122134455
 

5. Convert the following quinary numbers into binary numbers.

(a) 435 (b) 1245 (c) 4125 (d) 205

6. Convert the following binary numbers into quinary numbers.

(a) 1102 (b) 110112 (c) 1010112 (d) 11011012

Number System 163

Prime Mathematics Book - 8

Unit - 2 Integers Estimated period : 4

Review:

Sum of two natural numbers is also a natural number. Similarly product of two
natural numbers is also a natural number. Thus natural numbers are closed under
addition and multiplication. As 2-2 = 0 which is not a natural number. Thus including
0(zero) in natural numbers, came whole numbers (w) = {0, 1, 2, 3, .....}. Also whole
numbers are not closed under subtraction . As 3-4 = -1 is a negative number which is
not a whole number, thus including negative whole numbers came integers. The set of
all the numbers both positive and negative whole numbers including zero(0) is called
integers. The set of integers is denoted by I or Z.

∴ Z = { ......, -3, -2, -1, 0, 1, 2, 3, .......}
"Z" came form German word "Zahlen" meaning "to count".
Z+ = {1, 2, 3, +,....} are positive integers
Z- = {..., -4, -3, -2, -1} are negative integers

→ Zero(0) is neither a positive nor a negative number. It is a neutral number.

→ In a number line positive integers are put to the right and negative integers to
the left of zero(0).

→ In a number line every number is greater than any integer on its left side.


-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
Thus 2> 1> 0> -1> -2 and so on

→ +1, -1; +2, -2; +3, -3; e.t.c. are called opposite integers or additive inverse
number to each other.

→ Integers are ordered. If a and b are two integers, then either a > b or a < b or a = b

→ If a>b then a is to the right of b, if b>a then b lies to the right of a and if a = b
then a and b represent the same point on the number line.

These laws are called the laws of Trichotomy.

2.1 Operation with integers
A. Addition of integers

(a) If the integers to be added are of same sign, their numerical values are added and
the sign remain same.

For example, (+3) + (+2) = + (3 + 2) = +5, (-3) + (-2) = -(3 + 2) = -5

(b) If the integers to be added are of opposite sign, find the difference of their
numerical values and take the sign of the integer of greater numerical value.

For example, (+5) + (-3) = + (5 - 3) = + 2 (-3) + (-7) = - (7 - 3) = - 4

164 6 Real Number System

Prime Mathematics Book - 8

Properties of addition of integers

(a) Addition of integers is closed: If a and b are two integers then a+b is also an integers.

For example, (+2) + (+3) = + 5 which is also an integers

(-5) + (+2) = -3 which is also an integers

(b) Addition of integers is commutative: If a and b are two integers, then a + b = b + a

For example, (+8) + (+7) = +15 and (+7) + (+8) = +15

∴ (+8) + (+7) = (+7) + (+8)

(c) Addition of integers is associative: If a, b, and c are integers, then (a + b) + c = a + (b + c)

For example, {(+1) + (+2)} + (+3) = (+3) + (+3) = (+6)

(+1) + {(+2) + (+3)} = (+1) + (+5) = (+6)

∴ {(+1) + (+2)} + (+3) = (+1) + {(+2) + (+3)}

(d) Existence of additive inverse: If a is an integer, then -a is its additive inverse. Every
integer has its inverse in the set of integer. Sum of the integer and its additive
inverse is zero.

For example, (+5) + (-5) = (-5) + (+5) = 0

(e) Existence of additive identity: If a and b are integers such that a + b = a then b = 0

For example, (+1) + 0 = (+1)

(-2) + 0 = (-2)

Note: Subtraction of integers is same as addition of the additive inverse of the

integer to be subtracted.
e.g. (+5) - (+2) = (+5) + (-2) = +(5 - 2) = (+3)
(+7) - (-4) = (+7) + (+4) = +(7 + 4) = (+11)

B. Multiplication of integers:

We know in expression 2a, a is unit and the 2a ← unit
multiplier 2 shows how many times the unit a Coefficient(multiplier)
is added.

Consider the multiplication of two integers

(i) (+3) × (+2) Count the unit (+2) three times to the direction of(+2)

∴ (+3) × (+2) = (+6)

+2 +2 +2
unit
-2 -1 0 1 2 3 4 5 6 7

Integers 165

Prime Mathematics Book - 8

(ii) -3 × (+2)

Count the unit (+2) three times to the left of 0 (opposite direction of(+2))

∴ (-3) × (+2) = -6

+2
unit

(iii) (+3) × (-2) -7 -6 -5 -4 -3 -2 -1 0 1 2 3

Count a unit (-2) three times to the left of 0 (to the same side a direction of

(-2))
∴ (+3) × (-2) = -6

-2
unit

(iv) (-3) × (-2) -7 -6 -5 -4 -3 -2 -1 0 1 2 3

Count a unit (-2) three times from a 0 to the opposite direction of the unit (-2)

i.e. to the right of 0.
-2
∴ ( -3) × (-2) = +6 unit



→ Consider unit.
-2 -1 0 1 2 3 4 5 6 7

→ Coefficient (multiplier) shows how many time the unit is to be counted.

→ If the multiplier is (+) count the units from 0 to the same direction of the unit. If
the multiplier is (-), count the unit form 0 to the opposite direction of the unit.

Also observe the following patterns

Since the positive integers are also whole numbers and product of whole numbers is
also a whole number, the product of two positive integer is a positive integer.

e.g. (+3) × (+2) = (+6) ∴ (+) × (+) = (+)

Pattern I Pattern II Pattern III
(+3) × (+2) = (+6) (+2) × (+3) = (+6) (-2) × (+3) = (-6)
(+2) × (+2) = (+4) (+2) × (+2) = (+4) (-2) × (+2) = (-4)
(+1) × (+2) = (+2) (+2) × (+1) = (+2) (-2) × (+1) = (-2)
(+2) × 0 = 0 (-2) × 0 = 0
0 × (+2) = 0 (+2) × (-1) = (-2) (-2) × (-1) = (+2)
(-1) × (+2) = (-2) (+2) × (-2) = (-4) (-2) × (-2) = (+4)
(-2) × (+2) = (-4) (+2) × (-3) = (-6) (-2) × (-3) = (+6)
(-3) × (+2) = (-6) ∴ (+) × (-) = (-) ∴ ( -) × (-) = (+)
∴ (-) × (+) = (-)

Rule of signs:- (+) × (+) = (+) (-) × (+) = (-)
(+) × (-) = (-) (-) × (-) = (+)

Thus, the product of same signs is plus & the product of different signs is minus.

166 6 Real Number System

Prime Mathematics Book - 8

Note:
(i) Rule of signs hold for division also (+) ÷ (+) = (+); (+) ÷ (-) = (-); (-) ÷ (+) = (-); (-) ÷ (-) = (+)
(ii) The rule of signs hold in operation of any real numbers.

Properties of multiplication of integers:
(a) Closure property: Multiplication of two integers is closed i.e. if a and b are any two

integers, then a × b is also an integer.

For example: (+3) × (+2) = (+6) which is also an integer.

(+2) × (-3) = (-6) which is also an integer.

(-5) × (-6) = (+30) which is also an integer.

(b) Commutative property: Product of two integers remain unchanged if their positions
(roles) are interchanged i.e. if a and b are any two integers, then a × b = b × a.

For example: (+2) × (-3) = (-6)

(-3) × (+2) = (-6)

∴ (+2) × (-3) = (-3) × (+2)

(c) Associative property: Product of three or more than three integers remain
unchanged if the order in which they are grouped is changed i.e. If a, b and c are
any three integers then, (a × b) × c = a × (b × c)

For example: {(+2) × (+3)} × (-5) = (+6) × (-5) = (-30)

(+2) × {(+3) × (-5)} = (+2) × (-15) = (-30)

∴ {(+2) × (+3)} × (-5) = (+2) × {(+3) × (-5)}

(d) Distributive property: If the sum or difference of two integers is multiplied by an
integer, the result is equal to the sum or difference of the product of the integer
and the term of the sum or difference i.e. a, b and c are any three integers, then
a(b + c) = ab + ac and a (b - c) = ab - ac

For example: (+2) × {(+3) + (+4)} = (+2) × (+7) = (+14)

(+2) × (+3) + (+2) × (+4) = (+6) + (+8) = (+14)

∴ (+2) × {(+3) + (+4)} = (+2) × (+3) + (+2) × (+4)

(e) Existence of multiplicative identity: To any integer, there exist an integer such that
their product is the integer itself. If a × b = a, then b is an identity element.

For example: (+3) × (+1) = (+3) (-3) × (+1) = (-3)

∴ (+1) is an identity element

Note: Division of integers is not closed so division is not an operational part of integers.


Integers 167

Prime Mathematics Book - 8

C. Simplification of integers
We observed different operations with integers and their properties. Different
operations in mixed form is called simplification which should be done in certain
order. We use a well known rule "BODMAS", i.e. perfrom the operation in the order as

B = brackets (i.e. perfrom the operation within brackets)
O = of (perfrom operation with "of" which is multiplication (times))
D = division (÷)
M = multiplication (×)
A = addition (+)
S = subtraction (-)

Operations with brackets also in the order as:
= vinculum/semi-bracket
( ) = small brackets/round brackets/parenthesis
{ } = middle brackets/braces

[ ] = large brackets/square brackets


Note:

→ Addition (+) or subtraction, any one can be performed first.

→ Operation of number within bracket with number just out it is same as "of"
e.g. 12 ÷ 3 (6 - 4) = 12 ÷ 3 of 2 = 12 ÷ 6 = 2

→ The order of operations holds for operations of any real numbers.

→ While dividing an integer by another integer, the quotient may or may not
be an integer.


Example 1: Simplify: 20 - 27 ÷ 9 × 2 + 4

Solution:

Here, 20 - 27 ÷ 9 × 2 + 4

= 20 - 3 × 2 + 4 [performed operation of ÷]

= 20 - 6 + 4 [performed operation of ×]

= 20 + 4 - 6

= 24 - 6 [performed operation of +]

= 18 [performed lastly operation of -]

Example 2: Simplify: 13 + [22 - {14 + (5 - 3) × 8 ÷ 4}]

Solution: Here, 13 + [22 - {14 + (5 - 3) × 8 ÷ 4}]
= 13 + [22 - {14 + 2 × 8 ÷ 4}] [performed operation of ( )]
= 13 + [22 - {14 + 2 × 2}] [performed operation of ÷]

168 6 Real Number System

Prime Mathematics Book - 8

= 13 + [22 - {14 + 4}] [performed operation of ×]
= 13 + [22 - 18] [operation within { }]
[operation within [ ]]
= 13 + 4

= 17

Example 3: Simplify: 48 ÷ 8 - 2[3 + {8 - 3(3 + 4 - 2)}]

Solution: 48 ÷ 8 - 2[3 + {8 - 3(3 + 4 - 2)}] = 48 ÷ 8 - 2[3 - 7]
= 48 ÷ 8 - 2[3 + {8 - 3(3 + 2)}] = 48 ÷ 8 - 2[-4]
= 48 ÷ 8 - 2[3 + {8 - 3 of 5}] = 48 ÷ 8 + 8
= 48 ÷ 8 - 2[3 + {8 - 15}] =6+8
= 48 ÷ 8 - 2[3 + {-7}] = 14

Example 4: Write mathematically and simplify: Treble of difference of 9 and 4 is divided by 3.
Solution: The statement can be written mathematically as
= 3 of (9 - 4) ÷ 3 = 3 of 5 ÷ 3 = 15 ÷ 3 = 5

Exercise 2.1

1. Simplify: (b) 22 ÷ {19 - 2(1 + 3)} (c) 20 - {8 - (5 × 3 + 2)}
(a) 8 × 3 - 4 × 27 ÷ 9 + 2

(d) 7 - 24 ÷ [1 - 5 {7 + (7 - 15)}] (e) 12 - 5 × 3 + 12 ÷ 4 (f) 5[152 + 3{7 - 8(9 - 2)}]

(g) 11 × 11 ÷ [-11 ÷ {22 - (21 - 10)}] (h) -64 ÷ 16 + [12 × {6 ÷ (24 ÷ 10 - 2)}]

(i) 120 ÷ 4[400 ÷ 4{7 + (18 + 7 - 22)}] (j) 2 of (10 - 4) ÷ 3

(k)[{ { [ 5+1
5 20 x (25 - 13 - 3) ÷ (-5)

[ { { [ 1
(l) 202 - 9 42 + (56 - 8 + 9) + 108

2. Simplify: (b) 30 - 10 + 1 of (18 ÷ 2 + 3 × 2)
(a) 19 - 3(5 + 2 × 5 ÷ 10) 3

(c) 5 × 12 ÷ 2[-9 ÷ {15 + (3 - 15)}] (d) 84 ÷ 7[44 ÷ {10 - (4 - 10)}]

(e) 20 ÷ 2(1 + 6 ÷ 18 × 12)

3. Write the mathematical expression and simplify:
(a) 5 times the sum of 11 and 6 is subtracted from 100.
(b) Subtract 3 from one sixth of 4 times 15 and then multiply by 4.
(c) Divide 32 by the difference of 17 and 9 and subtract it from 10.
(d) Subtract 5 from the sum of 5 and 3 and then multiply the result by 4.
(e) Subtract thrice the sum of 2 and 4 from 25 and subtract the result from 27.
(f) Subtract the sum of quotient when difference if 24 and 4 is divided by 4 and 2 from 18.
(g) Divide 64 by the sum of 4 and difference of 13 and 9 and then subtract 8 from the result.
(h) Add the sum of 1 and one part of 8 part of 72 to the sum of 72 and one fourth of 72.

Integers 169

Prime Mathematics Book - 8 Scientific Notation Estimated period : 8

Unit - 3

Approximation and precision: • Rounding off
• Significant figure
Approximation is a mathematical device by which • Scientific notation
figures (numbers) are rounded off. There may be various
reasons for this. However, the purpose of rounding the figure
off is to simplify and make them easy to grasp.

n various cases like measurements in scientific
researches, measurements dealing with precious metals and
jewels are more minutely taken and more precise and accurate
results are expected. But in various spheres of inquiry where
figures/numbers are dealt with, only approximate results
are required for example the earth completes a complete
revolution around the sum in 365 days 5 hours 48 minutes and
46 seconds but we say one year is 365 days. Similarly, mean
distance between the sum and the earth is 149598500 km but
we simply say that the sum is 15,00,00,000 km (15 crore km).

Methods of approximation

A. Rounding off of numbers:

In this method when large number of digits occur in an integer, the number is rounded
off to the nearest ones, tens, hundreds, thousands etc. and if large number of
digits occur after decimal, the decimal number is rounded off to its nearest tenths,
hundredths, thousandths etc.

In case of integers, rounding off to given degree of precision, the digits after the
precision point are ignored keeping zeros in their places simply if the last digit (from
right to left) so ignored is less than 5 and adding one to the number if the last digit
so ignored is 5 or greater than 5.

B. Significant figures:
Measurements are often expressed by decimal numbers. let a measurement be
0.037cm then we assume that the measurement is made to the nearest 0.001cm and
there are 37 such 0.001cm (units) in the measurement.

We take, 0.037cm = 37 x 0.001cm

Where, 37 is the quantity and 0.001cm is the unit and say that 3 and 7 are significant
digits in the measurement. Similarly, if the measurement be 2.06 x 106 units, then we
assume that the measurement is made to the nearest 106 unit and there are 2.06 of
these 106 units in the measurement and say that 2,0 and 6 are significant digits in the
measurement.

170 6 Real Number System

Prime Mathematics Book - 8

Significant digits (figures) are the digits which indicate how many times the unit
of measurement is contained in the given measurement.
These are the number of digits used to express the given number (quantity) with
stated accuracy (or unit of precision).
The following rules help you to round off a given number (measure) to the stated
significant digits,
(i) All the non zero digits (1 to 9) are significant

(ii) All the digits in whole number representing a measurement are significant
Example: 2340 has 4 significant figures (s.f.)
23000 (has 5 s.f.)
Since 2340. Represent some measurement say 2340m. Where 1m is the unit and
2340 is the quantity. Such whole numbers are written with decimal at the end.

(iii) Captive zeros (zero/s sandwiched between two non-zero digits) are significant
irrespective of decimal point.

Example: 10.23 (has 4 s.f.), 23004 (has 5 s.f.)

(iv) In case of exact numbers, trailing zeros ( zero/s that come after) are non
significant.

Example: 25000 (has 2 s.f.)
Exact number (which comes from counting) are written without decimal at

the end.

(v) In case of measurement number (decimal numbers), zero/s that come before
non zero digit are non significant.
Example: 0.0123 (has 3 s.f.)
0.000126 ( has 3 s.f.)

(vi) Zero/s to the right of decimal and a non zero digit are significant (i.e. trailing
zeros after decimal are significant.)
Example: 2.0g (has 2 s.f.)
2.000g (has 4 s.f.)

(vii) Whenever the result is required to stated significant decimal places,
the integral part is not considered for significant figure and the required
significant figure are to be taken from decimal part only.

C. Scientific Notation

The number which can be expressed as ax 10n ? Where 1<a<10 and n∈Z is known as the

scientific notation of the number.

Observe these two quantities:
Weight of the earth= 5,977,000,000,000,000,000,000,000 kg
Weight of an electron = 0.000,000,000,000,000,000,000,000,000,0009019558 kg

It is inconvenient to express (read and write) and remember such quantities.
The quantity 5,977,000,000,000,000,000,000,000 kg can be convenient written or
5.977 x 1000,000,000,000,000,000,000,000 kg = 5.977 x 1024 kg

Scientific Notation 171

Prime Mathematics Book - 8

Similarly the quantity 0.000,000,000,000,000,000,000,000,000,0009019558 kg can be
conveniently writes as:

9.019558 ÷ 10000,000,000,000,000,000,000,000,000,000 kg

= 9.019558 ÷ 1031kg

= 9.019558 × 10−31kg

The way we write like this is scientific notation. Scientific notation (commonly
referred to as standard form or standard index form) is the way of writing the numbers
that are too big or too small to be conveniently written in decimal form as a × 10n where
‘a’, is a decimal with one digit at its integral part (1≤a<10) and n∈z. The quantity ‘a’ if
it consists many digits, can be rounded off to its suitable scientifically digits.

Thus, weight of an electron = 9.019558 × 10-31 kg = 9.02 × 10-31 kg

Following rules help you to write a given number in scientific notation.

(i) The number in scientific notation is written in decimal form as a × 109
where 1 ≤ a < 10 and n ∈ z

Example:

23500000 = 5.35 × 10000000 = 5.35 × 107 0.0000987 = 9.87 × 1 =9.87×10-5
100000

(ii) If the number is greater than 1, find the significant digit at the highest place value
and separate this digit with decimal point from other digits. Count the number of
digits after the decimal point which is the required power of 10.

Example: 35700000000 = 3.57 × 1010

[Significant digit at the highest place value being 3 and after the decimal point
then are 10 digits]

(iii) If the number is less than 1, find the significant digit at the highest place value
and separate if from other digits with decimal point after it. Count the number of
digits after which the decimal point is shifted, which gives the required power of
10 with negative sign.

Example: 0.00000000456 = 4.56 ×10-9

[Significant digit of the highest place value is 4 and decimal point is shifted to
the right after 9 digits.]

Example 7: Express the following numbers in scientific notation.
(a) 234000000 b) 471.368 c) 0.005678 d) 0.000000000693
Solutions:

(a) 234,000,000 = 2.34×108 [The digit at the highest place value is 2 decimal

172 6 Real Number System

(b) 471.368 Prime Mathematics Book - 8
(c) 0.005678
point is shifted 8 digits to left.]
= 4.71364×102 [The digit at the highest place value is 4 and

decimal point is shifted 2 digits left.]

= 5.678×10-3 [The digit at the highest place value is 5 and
decimal point is shifted after 3 digits to the right.]

(d) 0.000000000693 = 6.93×10-10[The digit at the highest place value is 6 and
decimal point is shifted 10 digits after it to the right.]

Operations involved with scientific notation.

(i) Addition/Subtraction :

→ Make the power of 10 same.

→ Add/subtract the quantities with the common power of 10.

→ Use the rule of addition/subtraction of significant figure.

Example 8: Simplify: 3.124 ×105 + 7.3 × 105

Solutions:
= (3.124 + 7.3) × 105
= 10.424 × 105
= 10.4 × 105 (up to 1 d. place)
= 1.04 × 10 × 105

= 1.04 × 106

Example 9: Simplify: 11.23 × 104 − 23.4 × 103

Solution: = 11.23 × 104 − 23.4 × 103 = 11.23 × 104 − 2.34 × 104 = (11.23 − 2.34) × 104

= 8.89 × 104

(ii) Multiplication / Division:

→ Multiply/divide quantities and unit parts.

→ Use rule of multiplication/division of significant figure in the quantity and/rule of
index in unit part.

→ If necessary, put the result in scientific natation.

Scientific Notation 173

Prime Mathematics Book - 8

Example 10: Simplify: (3.24 × 103) × (4.69 × 104)
Solution: = (3.24 × 103) × (4.69 × 104)
= (3.24 × 4.69) × (103 × 104)

= 15.1956 × 107
= 15.2 × 107 = 1.52 × 108

Example 11: Simplify: (19.72 × 1014) ÷ (3.16 × 106)
Solution: (19.72 × 1014) ÷ (3.16 × 106) = 6.24 × 1014-6 = 6.24 × 108

Example 12: Simplify: 2.5 × 105 + 6.7 × 106 − 2.3 × 104
Solution: = 2.5 × 105 + 6.7 × 106 − 2.3 × 104

= 25.0 × 104 + 670.0 × 104 − 2.3 × 104
= (25.0 + 670.0 − 2.3) × 104
= 697.3 × 104
= 6.973 × 106

Example 13: Simplify: (6.4 × 1024) × (2.7 × 1022)
(3.2 × 1015 )2

Solution: (6.4 × 1024 ) × (2.7 × 1022 )
(3.2 × 1015 )2
4 s.f.

= 6.4 × 2.7 × 1024 × 1022 17.28
(3.2)2 1030 10.24
= × 1016 = 1.6875 x 1016

= 17.28 × 10 46-30
10.24
4 s.f.

= 1.688 × 1016 (to 4s.f.)

Exercise 3

1. Express the following numbers in scientific notation.

(a) 24500000 (b) 600000000 (c) 2460000000000 (d) 36.124
(e) 1240.52 (f) 0.0000258 (g) 0.000000009 (h) 0.00078

174 6 Real Number System

Prime Mathematics Book - 8

2. Write the following number into normal form.

(a) 4.8 × 105 (b) 7.84 × 107 (c) 5.432 × 1012 (d) 1.245 × 10-4
(e) 7.24 × 10-6 (f) 3.0 × 10-10

3. Simplify the following. (b) 1.6 × 105 + 2.4 × 104
(a) 2.5 × 104 + 3.4 × 104 (d) 1.68 × 108 − 4.42 × 106
(f) 5.4 × 10−10 − 3.2 × 10−9 − 6.4 × 10−8
(c) 2.16 × 105 − 6.12 × 105

(e) 4.8 × 105 + 1.5 × 103 − 3.1 × 102

4. Simplify the following: (b) (1.8 × 104) × (1.51 × 106)
(a) (7.2 × 104) × (2.3 × 103)

(c) (5.6 × 10-4) ÷ (2.8 × 10-10) (d) (6.648 × 105) ÷ (3.2 × 10-2)

(e) (9.8 × 10-5) × (5.1 × 107) ÷ (1.2 × 10-2) (f) (7.9 × 10−2) ÷ {(1.23 × 10−4) × (2.46 × 106)}

(g) 6.67 × 10-11 × 2.4 × 1018 × 1.6 × 1020 (h) (7.5 × 10-8) × (4.8 × 1024) × (4.8 × 1024)

(3.2 × 1012)2 (2.5 × 1015) × (1.4 × 10-8)

5. Express the following in scientific notation.
a) Distance between the Earth and the Sun is about 15,00,00,000km.
b) The annual budget for a fiscal year 2071/72 was Rs. 6,79,70,00,00,000
c) Weight of electron is 0.0000008257gm.

Scientific Notation 175

Prime Mathematics Book - 8

Unit - 4 Rational & Irrational Numbers Estimated period : 5

Introduction

The set of counting numbers {1,2,3,…….} is called Natural numbers and is denoted by
N. Natural numbers are closed (closer property holds) under addition and multiplication
but may or may not be closed under subtraction and division.

1 + 2 = 3 (Sum of natural numbers is also a natural number)
2 × 3 = 6 (Product of natural numbers is also a natural number)

But 2 - 2 = 0 which is not a natural number. Then we need another set of numbers
which also includes zero (0). Thus whole numbers were introduced. The set of natural
numbers including zero (0) is called Whole numbers and are represented by W.

Also whole numbers are not closed under subtraction and division.

Example: 2 - 5 = - 3 which is not a whole number. So it was necessary to have next set of
numbers which includes negative numbers. Thus integers were introduced.

The set of positive and negative whole numbers including zero (0) is called Integers.
Set of integers is denoted by I or Z.

∴ Z = {…., -3 , -2 , -1 , 0 , 1 , 2 , 3 , …...}

Z comes from the German word 'zahlen' meaning 'to count'. Integers are closed

under addition, subtraction and multiplication.

Again, 4 ÷ 2 = 2 which is an integer but 2 ÷ 4 = 1 is not an integer.
2

So it is necessary to introduce next set of numbers which also includes numbers in

the form of ratios. Thus rational numbers were introduced. The set of numbers which
p
can be expressed in the form of the ratio q where p, q ∈ z and q≠0 is called Rational

numbers and is denoted by 'Q' where 'Q' stands for quotient and the word rational comes

from the word ratio.

Integers are ratios with denominator 1.

Note: Rational numbers include integers, fraction, terminating decimals and
non-terminating but recurring decimals.

176 6 Real Number System

Prime Mathematics Book - 8

4.1 Decimal numbers

1. Terminating decimals.

Observe the following division

i. 5 =2.5 ii. 3 = 0.12 iii. 5 = 0.3125
2 25 16

Note: P
q
→ A fraction is a terminating decimal only when the denominator q can be

expressed as q = 2m.5n where m, n = 0, 1, 2, ………….

→ Every terminating decimal can be converted into decimal fraction.


In each of the above examples the division is exact (without any remainder), Such
decimals are called terminating decimals.

Converting terminating decimal into decimal fraction:
Steps:
→ Write the decimal with denominator 1 and put decimal after it and as many zeros

as there are digits after decimal in the numerator.
→ Remove decimals from numerator and denominator.
→ Convert the fraction into lowest term.

Example 1: Convert the following decimals into fractions

(a) 0.2 (b) 0.12 (c) 0.3125 (d) 0.015625

Solutions:

(a) 0.2 (b) 0.12 (c) 0.3125 (d) 0.015625

0.2 = 0.12 = 0.3125 = 0.015625
= 1.0 1.00 1.0000 1.000000

=2 = 12 = 3125 = 15625
10 100 10000 1000000

= 1 = 3 = 5 = 1
5 25 16 64

Rational & Irrational Numbers 177

Prime Mathematics Book - 8

2. Non terminating but recurring decimals.

In some fractions like 1 , 1 , etc, we observe that division is not exact but digits in
3 7

the quotient repeat after a certain stage forcing the decimal expansion go on. In
other words we get a repeating blocks of digits in the quotient. We say that such
expression is non terminating but recurring or periodic.

Example: 1 = 0.333333…. 1 = 0.142857142837142857….
3 7

In the decimal expansion of 1 , 3 repeats in the quotient after decimal we write it
3

as 0.3. Similarly since the block of digits 142857 repeats in the quotient of decimal

expansion of 1 , we write it as 0.142857. Similarly, 2.3424242. is written as 2.342.
7

Converting non-terminating but recurring decimals into P form :
q

Example 2: Convert 0.3 into P form.
q

Solution: Let x = 0.3 (i) ∴ 10x = 3.3 (ii)

Subtracting, we get

9x = 3 or, x = 3
9

∴ 0.3 = 1
3

Example 3: Convert the following into P form. a) 0.27 b) 1.234
q

Solution: (a) let x = 0.27 (i) ∴ 100x = 27.27 (ii)

Subtracting we get , 99x = 27 or, x = 27 = 3 ∴ 0. 27 = 131
(b) 1.234 99 11

let x = 1.234 [Multiplying both side by 10]

10x = 12.34 (i) [Again multiplying both sides by 100]

and 1000x = 1234.34 (ii)

Subtracting (i) from (ii) we get

990x = 1222

or, x = 1222 = 1 116
990 495

∴ 1.234 = 1 116
495

178 6 Real Number System

Prime Mathematics Book - 8

Note:
P

- A fraction q is recurring decimal only when its denominator q has prime
factors other than 2 or 5.

- Every recurring decimals can be converted into vulgar fraction.

16.3 Properties of rational numbers
Let Q be the set of rational numbers and a and b be any member of Q, then

1) a, b ∈ Q ⇒ a + b ∈ Q i.e. rational numbers are closed under addition.

e.g. 1 + 2 = 3 + 4 = 7 which is also a rational number.
2 3 6 6

2) a, b ∈ Q ⇒ a - b ∈ Q i.e. rational numbers are closed under subtraction.

e.g. 1 - 2 = 5-4 = 1 which is also a rational number.
2 5 10 10

3) a, b ∈ Q ⇒ ab ∈ Q i.e. rational numbers are closed under multiplication.

e.g. 1 × 2 = 1 which is also a rational number.
2 5 5

4) a, b ∈ Q ⇒ b ≠ o ⇒ a ∈ Q i.e. rational numbers are closed under division.
b

e.g. 1 ÷ 2 = 1 × 5 = 5 which is also a rational number.
3 5 3 2 6

Thus, the rational numbers are closed under all the four fundamental operations
of arithmetic.

5) The set of rational numbers are ordered

i.e. a, b ∈ Q ⇒ either a > b or b > a or a = b.

6) If a and b are two different rational numbers, then a+b is a rational number
which lies between a and b. 2

a+b
i.e. if a, b ∈ Q and a < b, then a < 2 < b.
Continuing this process, we find that there are infinitely many rational numbers

between two different rational numbers (also called density property).

Rational & Irrational Numbers 179

Prime Mathematics Book - 8

11 1 + 2 7+6 13
3 7 21 42
If 3 and 7 are two different rational numbers then = = is a
rational numbers which lies between 1 2 2 2
and .
37

7) Identity element of addition and multiplication:

a) Existence of additive identity: If a be a rational number then there is one and
only on rational number 0 (zero) such that a + 0 = 0 + a = a.

Here, the rational number 0 is called the Identity element of addition.

b) Existence of multiplicative identity: If a be a rational number, then
there is one and only one rational number 1 such that a ×1 = 1 × a = a.

Here, the rational number 1 is called the identity element of multiplication.

8) Inverse element of addition and multiplication.

a) Existence of additive inverse: If a be a rational number, then there is one and
only one rational number -a such that a+(-a) = (-a) + a = 0.

Here -a is additive inverse of a and similarly a is the additive inverse of -a.

b) Existence of multiplicative inverse: If a be a rational number (a ≠ 0), then there
1 1 1
is one and only one rational number a such that a × a = a × a = 1

Here, a and 1 are multiplicative inverse of each other.
a

Example 4: Insert a rational number between 4 , 5 and arrange in ascending order.
5 6
Solution:
Given two of fractions are 4 and 5 where L.C.M of 5 and 6 is 30.
5 6

∴ 4 = 4 x 6 = 24 and 5 = 5 x 5 = 25
5 5 6 30 6 6 5 30

Since 24 < 25, 4 < 5 4 5
5 6 5 6
+
4 5 24 + 25 1 49
A rational number between 5 and 6 is 2 = 30 x 2 = 60

Thus the number in ascending order are 4 , 4690, 5 .
5 6

Since there exist infinitely many rational numbers between two rational numbers,
a rational number between them are not unique.

180 6 Real Number System

Prime Mathematics Book - 8

Exercise 4.1

1. State whether the following fractions give terminating or non terminating but
recurring decimals, give reason.

a) 1 b) 13 c) 17 d) 11 e) 1 2
4 40 44 12 64 f) 7

2. Convert the following into P form.
q

a) 0.625 b) 1.45 c) 0.0724 d) 2.125

3. Convert the following recurring decimals into P form.
q

a) 0.6 b) 0.16 c) 0.15 d) 0.742

4. Insert a rational number between the following and arrange in ascending order.

a) 2 and 3 b) 1 and 2 c) 2 and 2.5 d) 4 and 5
9 8 2 3

5. Insert two rational numbers between the following and arrange in ascending order.

a) 1 and -1 b) 3 and 5 c) 4 and 4.5 d) -1 and -1
2 4 3 2

6. Write the additive inverse of the following rational numbers.

4 b) -1171 c) -16 7
a) 7 25 d) 22

7. Write the multiplicative inverse of the following rational numbers.

1 b) 3 c) -11 d) -21
a) 2 4 15 31

8. Write the additive identity and multiplicative identity of the following rational

numbers.

a) 4 b) -3 c) 15 d) 7
7 5 19 9

-3
9. a) Subtract 5 from its additive inverse.

b) What is the sum of -3 and its additive inverse.
7

c) Divide 19 by its multiplicative inverse.
20

10. Show the relation among R.z and 3m. in a Venn-diagram.

11. Show the relation among Q, R, I in a Venn-diagram.

Rational & Irrational Numbers 181

Prime Mathematics Book - 8

4.2 Irrational numbers:

Under certain operations, we 1 2 1.4142...

obtain the decimal expansion of 1 1
24 100
numbers to be non terminating and 4 96
400
nonrecurring. For example, finding 281 281
1 11900
the square root of 2. 11296
2824 60400
We obtain, 2 =1.41421356237 4 56564
30950488016887242096....... and 3836
forcing above iteration go on, we 28282
2
never get termination or repetition
of similar blocks of digits. 28284


Similarly, ratio of circumference and

diameter of a circle.

Which we write c = π = 3014159265358979323846264338327950 .... also give non
d
terminating and non recurring decimal expansion. Such numbers can not be converted
P
into q form and are called irrational numbers.

The numbers which are not rational are called irrational numbers Here we will

discuss not about other irrationals but about the surds.

Surds
If nth root of a is b then we write a1/n=b, where a and b are real numbers (a, b∈R) and

n is a natural number, a1/n is commonly denoted by radical sign n a where a is called
radicand, n its index or order and the expression n a is called radical. We also find nth
root of a rational number is not always a rational number.
Example: 2 , 3 , 3 5 , 5 3 , etc are real but not rational numbers. Such numbers are
called surds. In other word7s, nth root of a number a i.e. n a is called a surd, if it is an
irrational number.

A. Some Properties of Surds (irrational numbers)

i) The sum or difference of a rational number and an irrational number.
Example: 3 + 2 , 5 -3, 8- 3 etc are irrational numbers.
ii) Product of a rational number and an irrational number is always an

irrational number. e.g. : 2 2 , 3 5 , etc which are also called mixed surds.
iii) The set of irrational number is not always closed under addition, subtraction,

multiplication and division.

182 6 Real Number System

Prime Mathematics Book - 8

For example:

2 + 3 is an irrational number but 3 +(- 3 ) = 0 is rational number.
5 + 3 is an irrational number but 5 - 5 = 0 is rational number.
3 . 2 = 6 is an irrational number but 3 ÷ 3 = 1 is rational number.
18 ÷ 2 = 6 is an irrational number but 18 ÷ 2 = 9 is rational number.

B. Some facts about the Surds.

→ 3 + 5 ≠ 3+5 , 5 - 2 ≠ 5-2 , 3 + 3 ≠ 6
but 3 + 3 = 2 3 it is just like x + x = 2x

5 3 -2 3 =(5 - 2) 3 = 3 3 it is just like 5x - 2x = (5 - 2)x
→ If a and b are positive rational numbers,

then a 2 = a, ab = a . b and a = a.
b b

For example,

3 2= 3

6 = 2×3 = 2 . 3 , 27 = 9×3 = 9 × 3 = 3 3 .

5 = 5 = 5 ; 12 12 4 =2
9 9 3 3= 3=

63 + 7 = 9×7 + 7 = 9 × 7 + 7 = 3 7 + 7 = 4 7

→ If a, b, c and d are national numbers and P is an irrational number and a+b P
= c + d P then a = c and b = d e.g. If a + b 2 = 3 + 5 2

→ a = 3 and b = 5
→ If a and b are irrational numbers, then a < b if a < b, a > b if a > b.
→ If a is a positive rational number and b is an irrational number. then, a< b if

a2 < b, a > b if a2 > b.

C. Rationalization

If the product of two irrational numbers (surds) is a rational number, then each
number is called the rationalizing factor of other.
Rational & Irrational Numbers 183

Prime Mathematics Book - 8

For example:

i) 3 × 3 = 3, x+y × x+y = x + y
∴ Rationalizing factor of whole square root is itself.

ii) (3+ 2 )(3- 2 ) = 32 - 2 2 = 9 - 2 = 7
( a + b ) ( a - b ) = a2 - b2 = a - b
∴ Rationalizing factor of an irrational in binomial form is its conjugate.
The process of multiplying a surd by its rationalizing factor is called rationalization.

Note:

→ Rationalization is not meant for converting an irrational or surd to rational.
Irrational number cannot be converted to rational.

→ In an irrational, in the form of fraction, either numerator or denominator is
rationalized making the value of the irrational remaining same.

22 5 25 (denominator is rationalized)
e.g. : 5 = 5 × 5 =5

3 = 3 × 3 = 3 (numerator is rationalized)
5 5 3 15

→ It is customary to rationalize denominator.

D. Comparison of surds:

Irrational numbers are ordered i.e. if a and b are two irrational numbers then either
a > b or a < b or a = b.
Since, n a = a1/n = ap/pn = pn ap, an irrational number (surd) can be written in any order.
To compare two surds m a and n b .
→ Change the order to same order.
→ Compare the radicands.
Thus m a = mn an and n b = mn bm then

If radicands an = bm, two surds are equal

If the radicands an > bm, m a > n b

If the radicands an < bm, m a < n b

184 6 Real Number System

Prime Mathematics Book - 8

Example 5: Compare 2 and 3 3

Solution: Here, L.C.M. of 2 and 3 is 6.
∴ 2 = 2 2 = 2×3 23 = 6 8
3 3 = 3×2 32 = 6 9
Since 8 < 9 (radicands)
∴68 <69
i.e. 2 < 3 3

Example 6: Simplify : a) 45 b) 18- 2

Solutions: a) 45 = 9×5 = 3 5
b) 18- 2 = 9×2 - 2 = 3 2 - 2 = 2 2

Example 7: Rationalize the denominator of : 7 -2
7 +2
7 -2
Solution: 7 +2

Here the denominator is 7 +2.

Its rationalizing factor is its conjugate i.e. 7 -2

∴ 7 -2 = 7 -2 × 7 -2 ( 7 -2)2
7 +2 7 +2 7 -2 = 72 -22

( 72 -2. 7 .2+22) = 7-4 7 +4 = 11-4 7
=
7-2 5 5

E. To insert irrational numbers between two rational numbers:

Example 8: Insert an irrational number between 2 and 3.
Solution: Given rational number 2 and 3
As 22 = 4 and 32 = 9 and 4 < 5 < 6 < 7 < 8 < 9
∴ 2 < 5 < 6 < 7 < 8 < 3
The required irrational number can be the square root of any natural numbers

between 4 and 9.
Hence, 5 lies between 2 and 3. To insert rational numbers between two

irrational numbers:

Rational & Irrational Numbers 185

Prime Mathematics Book - 8

Example 9: Insert a rational number between 2 and 3

Solution: Given irrationals 2 and 3
As ( 2 )2 = 2 and ( 3 )2 = 3
We can take any nay rational number between 2 and 3 which is a perfect square

of a rational numbers. One such number is 2.25 and Since 2 < 2.25 < 3
∴ 2 <1.5 < 3
Hence 1.5 is a rational number between 2 and 3 .

F. To insert irrational numbers between two irrational numbers:

Example 10: Insert two irrational numbers between 2 and 5
Solution: Given irrational numbers 2 and 5
As ( 2 )2 = 2 and( 5 )2 = 3
And 2 < 3 < 4 < 5
∴ 2 < 3 < 4 < 5

∴ 3 is an irrational number between 2 and 5 (VA = 2 is a rational)

Note: → Between two rationals there exists infinite number of irrationals.
→ Between two irrational numbers there exist infinite number of rationals.
→ Between two irrational numbers there exist infinite number of irrationals.

G. Real numbers
The set of numbers including rational and irrational numbers is called real numbers.
Set of Real numbers is denoted by R.

The given tree diagram gives you a clear vision of real number system.

Real numbers

Rational numbers Irrational numbers

Integer Fraction

Positive integers Decimal fraction (Terminating decimals)
Zero (0) Vulgar fraction (Non terminating but

Negative integers recurring decimal)

186 6 Real Number System

Prime Mathematics Book - 8

Exercise 4.2

1. State which of the following numbers are irrational.

a) 2+ 5 b) -2 3 - 2 c) 5 d) -2 . 5
5 50

e) (2- 3 )(2+ 3 ) f) (3+ 5 )2 g) (5 7 )2 h) (3- 6 )2

2. Compare the following. d) 25 and 3 125
a) 2 and 3 3 b) 3 5 and 4 6 c) 3 3 and 4 4

3. Simplify: c) 18 5 - 5 20
a) 5 3 + 3 27 b) 90 f) 4 12 × 2 3

d) 6 3 + 4 75 - 2 48 e) 2 24 + 54 - 2 6 - 96

g) 2 2 × 3 5 h) 32 ÷ 18 i) 45 ÷ 80

4. (a) Insert an irrational number between 3 and 5.

(b) Insert a rational number between 5 and 7
(c) Insert two irrational numbers between 2 and 7
(d) Insert four irrational numbers between 2 3 and 3 2

5. Rationalize the denominators of:

a) 5 b) 2 3 12
3 3 5 c) 3 +1 d) 3 - 2

1 4 g) 2 45 5 h) 6 + 3
e) 5 - 3 f) 3 5 -2 3 3- 6- 3

6. Simplify:

a) 4 + 6 b) 2 - 1 c) 3 - 1 53
2 d) -
3 55
34

e) 1 + 2+ 5 +3 3 +1 3 -1 g) 5 -3 - 5 +3
20 5 5 -3 f) 3 -1 + 3 +1 5 +3 5 -3

Rational & Irrational Numbers 187

Prime Mathematics Book - 8

7. Simplify:
a) 3 3 + 2 27

b) 5 5 + 3 5 + 2 5

c) 7 3 + 2 5 - 9 2 - 8 5 - 6 3 + 4 2

d) 4 27 - 5 75+ 7 108 - 16 12

8. Simplify:
a) 243 ÷ 12

b) ( 5 + 2 ) + ( 3 + 2 )

c) (3 6 - 2 3 ) (5 6 + 3 3 )

d) 3 + 12- 15
e) 3
5 14 45
3 + 12 + 5 27 - 243

188 6 Real Number System

Prime Mathematics Book - 8

Area Revision Test (Number System)

1. (a) convert 2410 to binary. (b) convert 1010112 to denary.
(c) convert 21710 to quinary. (d) convert 1435 to denary.

2. Complete the following operations. 3335 (d) 43425
+2225 -14345
(a) 101112 (b) 1110102 (c)
+100012 -101112

3. Simplify: (a) 1012 + 1112 - 1102 (b) 301425 + 13425 - 2345

4. Convert: (a) 101102 to quinary (b) 1245 to binary

5. Simplify: (a) 22 ÷ {19 − 2(1 + 3)} (b) 16 − 27 ÷ 9 × 2 + 2

(c) 5[152 + 3{7 − 8(9 − 2)}] (d) 120 ÷ 4[400 ÷ 4{7 + (18 + 7−22 )}]

6. Simplify: (a) (+2) × (−3) × (−8) (b) (−8) ÷ (−4) × (−6) (c) (−4) + (+10)

7. Show the following in number line. (a) (+3) × (−2) (b) (−4) × (+3)

8. Write mathematical expressions and simplify:
(a) Subtract 3 from one sixth of 4 times is and then multiply by 4.
(b) Divide 64 by the sum of 4 and difference of 13 and 9 and then subtract 6 from it.

9. (a) Round off 2345 to the nearest tens, hundreds and thousands.

(b) Round off 3.456 to the nearest ones, tenths and hundredths.

(c) Find the number of significant figure in: (i) 7.06 (ii) 0.00057

(d) Round off the following: (i) 3.875 (to 1 s.f.) (ii) 3.87 (to 2 s.f.)

10. (a) Simplify to necessary significant figure: 12.345 x 6.04

(b) Express in scientific notation: (i) 1230.45 (ii) 0.000000008

(c) Write in normal form: (i) 3.5 x 105 (ii) 3.0 x 10-8

11. (a) State whether the following fractions are terminating or non-terminating decimals.
(ii) 5
(i) 9 12
number
(b) Inser4t0a rational b-e1t1we. eAnlso12 and 2
(c) unite 3
Write additive inverse of its multiplicative inverse.
17
(d) Rationalize the denominator of 1
3-1
1 2
12. (a) Insert two rational numbers between 2 and 3

(b) Convert 0.24 into vulgar fraction (c) Compare 2 and 3 3

13. (a) Simplify: 3+1 + 3 -1 (b) Convert 0.16 into vulgar fraction.
3-1 3+1

Number System 189

Answers

Exercise 1.1 (Binary) (g) 10001102 (h) 10110112

1. (a) 10012 (b) 11002 (c) 100012 (d) 110012 (e) 1100102 (f) 10000012
(i) 11001002 (j) 1101112

2. (a) 3 (b) 6 (c) 7 (d) 4 (e) 5 (f) 29 (g) 85 (h) 8 (i) 15 (j) 219

3. (a) 11002 (b) 101112 (c) 110002 (d) 1010102 (e) 10102
4. (a) 102 (b) 10112 (c) 1112 (d) 1000012 (e) 100112 (f) 110012

Exercise 1.2 (Quinary Number System)

1. (a) 43215 (b) 144145 (c) 102035 (d) 102145 (e) 22335 (f) 344105 (g) 2005 (h) 124225
2. (a) 14 (b) 13 (c) 9 (d) 73 (e) 116 (f) 86 (g) 27 (h) 894 (i) 125 (j) 694

3. (a) 3125 (b) 243 (c) 11105 (d) 3425 (e) 41245 (f) 102005 (g) 103035 (h) 43045 (i) 20125 (j) 10005
(h) 14305
(k) 202435 (l) 201145
4. (a) 4005 (b) 2035 (c) 10435 (d) 2403 (e) 10415 (f) 43215 (g) 140405
5. (a) 1112 (b) 1102 (c) 1112 (d) 101102 (e) 1001112 (f) 1002
6. (a) 115 (b) 1025 (c) 1335 (d) 4145

Exercise 2.1 (Integers)

1. (a) 14 (b) 2 (c) 29 (d) 3 (e) 0 (f) 25 (g) −121 (h) 20 (i) 3 (j) 4 k) 9 l) 85
(d) 3 (e) 2 3. (a) 15 (b) 28 (c) 6 (d) 12
2. (a) 1 (b) 25 (c)−10 (h) 99

(e) 20 (f) 11 (g) 0

Exercise 3 (Scientific notation)

1. (a) 2.45 × 107 (b) 6.0 × 108 (c) 2.46 × 1012 (d) 3.642 × 101 (e) 1.24052 × 103

(f) 2.58 × 10−5 (g) 9.0 × 10−9 (h) 7.8 × 10−4

2. (a) 480000 (b) 78400000 (c) 5432000000000 (d) 0.0001245 (e) 0.00000724 (f) 0000000003

3. (a) 5.9 × 104 (b) 1.84 × 105 (c) −3.96 × 105 (d) 1.6358 × 108 (e) 4.8119 × 105 (f) −6.666 × 10−8

4. (a) 1.656 × 108 (b) 2.718 × 108 (c) 2.0 × 106 (d) 2.08 × 107 (e) 4.165 × 105 (f) 2.61 × 10−4

(g) 2.5 × 103 (h) 4.34 × 1034

5. a) 1.5 x 108 b) 6.797 x 1011 c) 8.25 x 10-7

Exercise - 4.1 (Decimal Numbers) (d) non-terminating but recurring

1. (a) terminating (b) terminating (c) non-terminating but recurring
(e) terminating (f) non-terminating but recurring

2. (a) 5 (b) 29 (c) 181 (d) 17 3. (a) 2 (b) 1 (c) 5 (d) 49
8 20 2500 8 3 6 33 66

4. (a) 2 , 43 , 3 (b) 1 , 7 , 2 11 (d) 4, 4.5, 5 [Answers may be varied]
9 144 8 2 12 3 (c) 2, 2 4, 2 2

190 Answers

5. (a) −1, 0, 1, 1 (b) 5 11 23 , 3 −1 −5 −9 −1
2 4, 8, 8 2 (c) 4, 4.25, 4.375, 4.5 (d) 2 , 12 , 12 , 3 [Answers may be varied]

6. (a) −4 (b) +11 (c) +16 (d) −7 7. (a) 2 4 (c) −15 (d) −31 8. 0(zero) in each case
7 17 25 22 (b) 11 21

3

9. (a) 6 (b) 0 (c) 361
5 400

Exercise 4.2 (Irrational Numbers)

1. Show to your teacher.

2. (a) 2 < 3 3 (b) 3 5 > 4 6 (c) 3 3 > 4 4 (d) 25 = 3 125

3. (a) 14 3 (b) 3 10 (c) 8 5 (d) 18 3 (e) 6 (f) 48 (g) 6 10 (h) 4 (i) 3
4. (a) 10 (b) 2.25 (c) 3 , 5 (d) 13 , 3 4
14 , 15 , 17

5. (a) 5 3 (b) 2 15 (c) 3 −1 (d) 2( 3 + 2 ) (e) 5+ 3 (f) 4(3 5 + 2 3 )
3 15 2 2 33

(g) 4 5 (2 3 + 5 ) (h) 3 + 2 2

7 25 −2 5 −7
5 2
6. (a) 4 + 3 2 (b) 6− 3 (c) (d) 17 3 (e) (f) 4. (g) 3 5
3 12

7. (a) 9 3 (b) 4 5 (c) 3 - 6 5 - 5 2 (d) 3 3

8. (a) 9 (b) 15 + 6 + 10 + 2 (c) 72 - 3 2 (d) -2 3 (e) 43 3
2 3

Answers 191

At the end of this unit the student will be able to Estimated periods - 5
Objectives:
● to know the relation between ratio, proportion and percentage.
● solve the problems of ratio and proportion.
● solve the problems on percentage.
● know about the discount and VAT.
● solve the problems of percentage including discount and VAT.

Teaching Materials:

Grid, graph, menu and chart of ratio and proportion, chart of menu with discount and VAT.

Activities:

It is better to
● discuss about the ratio and proportion.
● discuss about the solution of problems related to proportions.
● discuss about the percentage in a group.
● discuss to solve the problems of percentage with tax.

Unit - 1 Prime Mathematics Book - 8

Ratio, Proportion and Percentage Estimated period : 5

1.1 Ratios

Ratio is a way of comparing two quantition of same kind (unit) is taking ratio. If a and

b are two quantities having same units, then a , also denoted by a : b is the ratio of a
b
to b and b also denoted by b : a is the ratio of b to a.
a

In a ratio a : b, a and b are called the terms of the ratio. The symbol (:), read as "is to"
is used to denote ratio. Here, a is called antecedent and b is called consequent.

While taking ratio, the terms should be of same unit.
- Generally, a ratio is expressed in its lowest term.
- The terms of the ratio should be integers.

Example 1: Find the ratio of 8 months to 2 years.

Solution: We know, 2 years = 2 x 12 months = 24 months 81 = 1 or 1 : 3
8 months 24 3 3

Now, the ratio of 8 months to 2 years = 24 months =

Example 2: Express the following ratios in simple forms.

(a) 0.75 : 3 (b) 21 : 63
4 4
Solution:
0.75 75 3 1 1
(a) 0.75 : 3 = 3.00 = 300 =4 =1:4

214 12 4
634
9 9 1x
4
(b) 21 : 63 = = 4 = 4 = 1 = 1 : 3
4 4 27 273 3

4

Example 3: Lands possessed by Moti Lal and Panna Lal are in the ratio 5 : 3. If Moti Lal

has 15 ropanies, how much land does Panna Lal posses ?

Solution: Let the area of land possessed by Moti Lal = x and that by Panna Lal is = y

Here, x = 15 ropanies.

Given that x : y = 5 : 3

or, 15 ropanies = 5
y 3

Ratio, Proportion and Percentage 193

Prime Mathematics Book - 8

or, 15 ropanies x 3 = 5y

3 15 ropanies x 3

or, y = 51

∴ y = 9 ropanies

Therefore, Panna Lal possesses 9 ropanies of land.

Example 4: In a project Mr. Chaudhari, Mr. Kedia and Mrs. Sherchan invested in the ratio
2 : 3 : 4. If the total investment in the project is Rs.27,00,00,000, how much
did they invest repeatedly ?

Solution: Here, total investment = Rs.27,00,00,000

Investment in the ratio 2 : 3 : 4 be 2x, 3x and 4x

Then, 2x + 3x + 4x = Rs.27,00,00,000

or, 9x = Rs.27,00,00,000

or, x = 3
Rs.27,00,00,000

91

∴ x = Rs. 3,00,00,000

Mr. Chaudhari's share = 2x = 2 x Rs.3,00,00,000 = Rs.6,00,00,000

Mr. Kedia's share = 3x = 3 x Rs.3,00,00,000 = Rs.9,00,00,000

Mrs. Sherchan's share = 4x = 4 x Rs.3,00,00,000 = Rs.12,00,00,000

Example 5: What should be added to the term of the ratio 2 : 3 to make it 3 : 2 ?
Solution: Here, given ratio 2 : 3

Let the number to be added is x such that

or, 2+x = 3
3+x 2

or, 4 + 2x = 9 + 3x

∴ 9 + 3x = 4 + 2x

or, 3x - 2x = 4 - 9 = -5

Hence, -5 should be added to the terms of the ratio 2 : 3 to make it 3 : 2.

194 7 Ratio, Proportion and Percentage

Prime Mathematics Book - 8

Exercise 1.1

1. Write the following ratios in simple form.

(a) 24 kg to 84 kg (b) 3 m to 225 m (c) 750 gm to 4 kg (d) 250 ml to 6 l

(e) Rs.18 to 90 p (f) 0.75 to 8 (g) 5 3 to 3 2 (h) 2 3 to 12
4 7 5

2. a) What should be added to the terms of the ratio 1:2 to make it 2:1 ?

b) What should be subtracted from the terms of the ratio 2:5 to make it 5:2 ?

c) Two numbers are in the ratio 9:13. When 2 is subtracted from each the ratio becomes
2:3, find the numbers.

d) Two numbers are in the ratio 7:5. When 3 is added to each, the ratio becomes 4:3,
find the numbers.

3. a) In class VIII of a school, the number of boys to number of girls is 3 : 4. If the number
of boys is 24, find the number of girls in the class.

b) In a map drawn with scale of 1 : 20000, distance between two places is 8 cm. Find the
actual distance between the two places.

c) In a family expenditures on food and education are in the ratio 3 : 5. If expenditure
education is Rs. 7000, Find the expenditure on food.

d) Suhana and Krishtina spent in the ratio 7 : 10. If Suhana spent Rs. 17500, how much
did the Krishtina spend ?

4. a) Divide Rs.78 in the ratio 8:5.
b) Mr. Yadav divided Rs. 6,00,000 to his son and daughter in the ratio 7:5. How much will

each get ?
c) Angle of a triangle are in the ratio 1:2:3. Find the angles.
d) Anish, Neha and Dorje invested in the ratio 4:5:6 in a business. If they made profit of

Rs. 6,00,000 in a year, how much will the share of each ?

1.2 Proportion

Consider the data:

Class No. of boys No. of girls

VII 20 30
VIII 24 36

Ratio of the number of boys to the number of girls in class VII = 20 = 2 = 2:3
30 3

Ratio of the number of boys to the number of girls in class VIII = 24 = 2 = 2:3
36 3

Here we observed that the ratio the number of boys and girls in two classes are same
and we say the number of boy and girls in two class are in proportion and write as 20 :
30 :: 24 : 36 [read as "20 is to 30 as 24 is to 36"]

Ratio, Proportion and Percentage 195