# Number System - An Introduction

## Choose the correct option

#### Question 1

In a decimal number system, the base of a number is represented by

1. 2
2. 10
3. 16
4. All of them

10

Reason — The decimal number system uses 10 digits (from 0 to 9) hence its has a base of 10.

#### Question 2

The base of an octal number is represented by:

1. 2
2. 8
3. 7
4. None

8

Reason — The octal number system uses 8 digits (from 0 to 7) hence its has a base of 8.

#### Question 3

To convert an octal number to its binary equivalent, each octal digit is expressed as

1. 3 bits form
2. 4 bits form
3. 8 bits form
4. All of them

3 bits form

Reason — Since the digits from 0 to 7 need a maximum of 3 bits to be represented in binary form hence in Octal to Binary conversion each octal digit is expressed as 3 bits form.

#### Question 4

Sixteen raised to the power zero (16⁰) is equivalent to

1. 0
2. 1
3. 0 and 1
4. None

1

Reason — Any number raised to the power of 0 is 1.

#### Question 5

An octal number system uses the digits from

1. 0 to 8
2. 1 to 8
3. 0 to 7
4. All of them

0 to 7

Reason — The octal number system is a base 8 number system as it uses the digits from 0 to 7.

#### Question 6

The base of a hexa-decimal number is represented by

1. H16
2. 16
3. 15
4. None

16

Reason — The hexa-decimal number system uses 16 digits (from 0 to 15) hence its has a base of 16.

#### Question 7

In a hexa-decimal number system, 'B' represents the digit

1. 11
2. 12
3. 14
4. 13

11

Reason — In hexa-decimal number system, the digits 0 to 15 are represented by the letters A to F.

#### Question 8

To express a hexa-decimal number to its binary equivalent, each hexa-decimal digit is expressed as

1. 2 bits form
2. 3 bits form
3. 4 bits form
4. None

4 bits form

Reason — Since the digits from 0 to 15 need a maximum of 4 bits to be represented in binary form hence in Hexa-decimal to Binary conversion each hexa-decimal digit is expressed as 4 bits form.

#### Question 9

The binary equivalent of a hexa-decimal digit 12(C) is represented by

1. 1010
2. 1011
3. 1101
4. 1100

1100

Reason — The hexa-decimal digit 12(C) is represented as 1100.

#### Question 10

The hexa-decimal equivalent digit of 1011 (4 bits form) is

1. 14
2. 15
3. 11
4. 12

11

Reason — The hexa-decimal equivalent digit of 1011 (4 bits form) is 11.

## Fill in the blanks

#### Question 1

The binary system consists of two digits 0 and 1.

#### Question 2

A decimal number system uses the digits from 0 to 9.

#### Question 3

The base in the decimal number system is written as 10.

#### Question 4

A binary number system is written with 2 as the base.

#### Question 5

In a decimal to binary conversion, the first remainder is known as Least Significant Bit (LSB) and the last remainder is Most Significant Bit (MSB).

20 = 1

Octal
Digit
Binary
Equivalent
5
7
1
6
3
Digit
Binary
Equivalent
8
11
4
15
9

Octal
Digit
Binary
Equivalent
5101
7111
1001
6110
3011
Digit
Binary
Equivalent
81000
111011
40100
151111
91001

## Case-Study Based Questions

#### Question 1

Your teacher has assigned you a task to give a presentation on conversion of octal numbers into binary numbers and vice-versa. You are asked to create some aids to support your presentation. You have created two tables, Table 1 and Table 2, to demonstrate some examples.

Octal NumberBinary Equivalent
1001
5.......
3010
6110
Binary NumberOctal Equivalent
100.......
1117
0112
00111016

In the above tables, some entries have either been missed or incorrect. Answer the following questions based on the above case:

(a) What will be filled in the blank space of Table 1?

(b) Find and rectify the incorrect binary equivalent in Table 1.

(c) Fill the appropriate octal equivalent in the blank space of Table 2.

(d) Find and rectify the incorrect octal equivalent in Table 2.

(a) 101

(b) In Table 1, the binary equivalent of octal number 3 is incorrect. The correct value is 011.

(c) 4

(d) In Table 2, the octal equivalent of binary number 011 is incorrect. The correct value is 3.

## Convert the following to their binary equivalents

#### Question 1

(78)10

2QuotientRemainder
2780 (LSB)
2391
2191
291
240
220
211 (MSB)
0

Therefore, (78)10 = (1001110)2

#### Question 2

(99)10

2QuotientRemainder
2991 (LSB)
2491
2240
2120
260
231
211 (MSB)
0

Therefore, (99)10 = (1100011)2

#### Question 3

(141)10

2QuotientRemainder
21411 (LSB)
2700
2351
2171
280
240
220
211 (MSB)
0

Therefore, (141)10 = (10001101)2

#### Question 4

(123)10

2QuotientRemainder
21231 (LSB)
2611
2300
2151
271
231
211 (MSB)
0

Therefore, (123)10 = (1111011)2

## Convert the following to their decimal equivalents

#### Question 1

(10101)2

Binary
No
PowerValueResult
1 (LSB)2011x1=1
02120x2=0
12241x4=4
02380x8=0
1 (MSB)24161x16=16

Equivalent decimal number = 1 + 4 + 16 = 21

Therefore, (10101)2 = (21)10

#### Question 2

(10000)2

Binary
No
PowerValueResult
0 (LSB)2010x1=0
02120x2=0
02241x4=4
02380x8=0
1 (MSB)24161x16=16

Equivalent decimal number = 16

Therefore, (10000)2 = (16)10

#### Question 3

(11001)2

Binary
No
PowerValueResult
1 (LSB)2011x1=1
02120x2=0
02240x4=0
12381x8=8
1 (MSB)24161x16=16

Equivalent decimal number = 1 + 8 + 16 = 25

Therefore, (11001)2 = (25)10

#### Question 4

(101010)2

Binary
No
PowerValueResult
0 (LSB)2010x1=0
12121x2=2
02240x4=0
12381x8=8
024160x16=0
1 (MSB)25321x32=32

Equivalent decimal number = 2 + 8 + 32 = 42

Therefore, (101010)2 = (42)10

## Convert the following to Decimal numbers

#### Question 1

(510)8

Octal
No
PowerValueResult
0 (LSB)8010x1=0
18181x8=8
5 (MSB)82645x64=320

Equivalent decimal number = 8 + 320 = 328

Therefore, (510)8 = (328)10

#### Question 2

(ABC)16

Number
PowerValueResult
C (12)160112x1=12
B (11)1611611x16=176
A (10)16225610x256=2560

Equivalent decimal number = 12 + 176 + 2560 = 2748

Therefore, (ABC)16 = (2748)10

#### Question 3

(1001011)2

Binary
No
PowerValueResult
1 (LSB)2011x1=1
12121x2=2
02240x4=0
12381x8=8
024160x16=0
025320x32=0
1 (MSB)26641x64=64

Equivalent decimal number = 1 + 2 + 8 + 64 = 75

Therefore, (1001011)2 = (75)10

#### Question 4

(CD7)16

Number
PowerValueResult
716017x1=7
D (13)1611613x16=208
C (12)16225612x256=3072

Equivalent decimal number = 7 + 208 + 3072 = 3287

Therefore, (CD7)16 = (3287)10

#### Question 5

(101001)2

Binary
No
PowerValueResult
1 (LSB)2011x1=1
02120x2=0
02240x4=0
12381x8=8
024160x16=0
1 (MSB)25321x32=32

Equivalent decimal number = 1 + 8 + 32 = 41

Therefore, (101001)2 = (41)10

#### Question 6

(1100111)2

Binary
No
PowerValueResult
1 (LSB)2011x1=1
12121x2=2
12241x4=4
02380x8=0
024160x16=0
125321x32=32
1 (MSB)26641x64=64

Equivalent decimal number = 1 + 2 + 4 + 32 + 64 = 103

Therefore, (1100111)2 = (103)10

## Convert the following to binary numbers

#### Question 1

(342)8

Octal
Number
Binary
Equivalent
2010
4100
3011

Therefore, (342)8 = ($\bold{\underlinesegment{011}}\medspace\bold{\underlinesegment{100}}\medspace\bold{\underlinesegment{010}}$)2

#### Question 2

(203)8

Octal
Number
Binary
Equivalent
3011
0000
2010

Therefore, (203)8 = ($\bold{\underlinesegment{010}}\medspace\bold{\underlinesegment{000}}\medspace\bold{\underlinesegment{011}}$)2

#### Question 3

Number
Binary
Equivalent
D (13)1101
A (10)1010
91001

Therefore, (9AD)16 = ($\bold{\underlinesegment{1001}}\medspace\bold{\underlinesegment{1010}}\medspace\bold{\underlinesegment{1101}}$)2

#### Question 4

(157)8

Octal
Number
Binary
Equivalent
7111
5101
1001

Therefore, (157)8 = ($\bold{\underlinesegment{001}}\medspace\bold{\underlinesegment{101}}\medspace\bold{\underlinesegment{111}}$)2

#### Question 5

(ABC)16

Number
Binary
Equivalent
C (12)1100
B (11)1011
A (10)1010

Therefore, (ABC)16 = ($\bold{\underlinesegment{1010}}\medspace\bold{\underlinesegment{1011}}\medspace\bold{\underlinesegment{1100}}$)2

#### Question 6

(DE)16

Number
Binary
Equivalent
E (14)1110
D (13)1101

Therefore, (DE)16 = ($\bold{\underlinesegment{1101}}\medspace\bold{\underlinesegment{1110}}$)2

## Convert the following to their hexa-decimal equivalent

#### Question 1

(110011101111)2

Grouping in bits of 4:

$\underlinesegment{1100} \quad \underlinesegment{1110} \quad \underlinesegment{1111}$

Binary
Number
Equivalent
1111F (15)
1110E (14)
1100C (12)

Therefore, (110011101111)2 = (CEF)16

#### Question 2

(11010111100)2

Grouping in bits of 4:

$\underlinesegment{0110} \quad \underlinesegment{1011} \quad \underlinesegment{1100}$

Binary
Number
Equivalent
1100C (12)
1011B (11)
01106

Therefore, (11010111100)2 = (6BC)16

#### Question 3

(89392)10

16QuotientRemainder
16893920
1655873
16349D (13)
16215
1611
0

Therefore, (89392)10 = (15D30)16

#### Question 4

(100101101110)2

Grouping in bits of 4:

$\underlinesegment{1001} \quad \underlinesegment{0110} \quad \underlinesegment{1110}$

Binary
Number
Equivalent
1110E (14)
01106
10019

Therefore, (100101101110)2 = (96E)16

#### Question 5

(9894)10

16QuotientRemainder
1698946
16618A (10)
16386
1622
0

Therefore, (9894)10 = (26A6)16

#### Question 6

(4966)10

16QuotientRemainder
1649666
163106
16193
1611
0

Therefore, (4966)10 = (1366)16

#### Question 1

What are the different types of number systems that a computer deals with?

The different types of number systems are:

1. Binary Number System
2. Octal Number System
3. Decimal Number System

#### Question 2

What is meant by the following terms? Give an example of each.

(a) An octal number
(b) A hexa-decimal number

(a) An Octal number — An octal number uses 8 types of digits — 0, 1, 2, 3, 4, 5, 6, 7. It is represented with base 8.

(b) A hexa-decimal number — A Hexa-decimal number uses 16 types of digits (0 to 15). To represent digits from 10 to 15 it uses letters from A to F respectively. It is represented with base 16.

#### Question 3a

Give two differences between Binary number and Decimal number

Binary numberDecimal number
It uses 2 digits — 0 and 1.It uses 10 digits — 0 to 9.
It uses base 2.It uses base 10.

#### Question 3b

Give two differences between Octal number and Binary number