2.1 Concept of Number System
2.1 Concept of Number System:
Definition of Number systems, Application of Number system conversion
2.2 Binary Calculation: Addition, Subtraction
2.3 Number Conversion
2.3.1 Decimal to Binary, Octal, Hexadecimal.
2.3.2 Binary, Octal, Hexadecimal to Decimal
2.3.3 Binary to Hexadecimal and vice versa , Binary to octal and vice versa
The
concept of numbers developed early in human history when people used fingers,
sticks, pebbles, knots of rope, and symbols to count objects and perform
simple calculations like addition and subtraction. As human needs increased,
different calculating methods and devices were invented. Today, numbers
are used in daily life for various purposes such as counting, measuring, and
calculating.
A number system is a method
of writing numbers using a specific set of symbols and rules. Different number
systems use different symbols and follow different rules for representing
numbers and performing calculations.
For
example, the decimal number system is the most commonly used number
system in daily life. It consists of 10 symbols (0 to 9). Using these
symbols, we can represent any number. Due to its simplicity and ease of
understanding, the decimal number system is widely used around the world.
a)
What is a digit? What is a computer word?
A
digit is a single symbol used to represent a number in a number system,
such as 0–9 in the decimal system.
A computer word is a fixed-sized group of bits that a computer processes
as a single unit, usually representing data or instructions.
b)
Define the base or radix of the number system.
The
base or radix of a number system is the total number of unique digits or
symbols used in that system.
For example, the decimal number system has base 10, while the binary number
system has base 2.
c)
Which language is used by computer systems, smartphones, and tablets?
Computer
systems, smartphones, and tablets use binary language, which consists of
only two digits: 0 and 1.
Types
of Number Systems (Definition with Example)
Decimal
Number System:
The decimal number system is a number system that uses 10 digits (0–9)
and has a base of 10. It is commonly used in daily life.
Example: 456₁₀
Binary
Number System:
The binary number system is a number system that uses two digits (0 and 1)
and has a base of 2. It is used by computers and digital devices.
Example: 1011₂
Octal
Number System:
The octal number system is a number system that uses 8 digits (0–7) and
has a base of 8. It is used as a compact form of binary numbers.
Example: 157₈
Hexadecimal
Number System:
The hexadecimal number system is a number system that uses 16 symbols (0–9
and A–F) and has a base of 16. It is widely used in computer
programming.
Example: 2F₁₆
Application
of Number System Conversion
Number
system conversion has important applications in computer science and digital
technology. Since computers use the binary number system to process
data, it is necessary to convert numbers between binary, decimal, octal, and
hexadecimal systems.
In
digital logic design, electronic circuits and components work using
binary and hexadecimal values, making number system conversion essential. In computer
networking, IP addresses are often represented in dotted-decimal or hexadecimal
form, so conversion helps in understanding and configuring networks.
Number
system conversion is also widely used in fields such as cryptography, computer
graphics, and signal processing to represent and manage data
efficiently. Therefore, understanding number system conversion is crucial for
anyone working with digital devices and computer systems.
2.2
Binary Calculation
Binary
calculation refers to performing arithmetic operations using the binary
number system, which consists of only two digits: 0 and 1. Since
computers work internally using binary numbers, binary calculations are
essential in computer science.
There
are four main types of binary calculation:
i.
Binary Addition
ii. Binary Subtraction
iii. Binary Multiplication
iv. Binary Division
Binary
Addition
Binary
numbers are added in a similar way to decimal numbers, but using only the
digits 0 and 1.
Rules
of Binary Addition
|
A |
B |
A + B |
|
0 |
0 |
0 |
|
0 |
1 |
1 |
|
1 |
0 |
1 |
|
1 |
1 |
10 (0 with carry 1) |
Steps
of Binary Addition
Step
1: Align the
binary numbers properly, just like decimal addition.
Step 2: Start adding from the rightmost bit.
Step 3: Add the digits using the rules of binary addition.
Step 4: Carry the extra bit if the result is 10.
Step 5: Repeat the process until all columns are added.
Example
of Binary Addition
1100
+ 0101
--------
10001
Hence, 1100₂ + 0101₂ = 10001₂
Binary
Subtraction
Binary
subtraction is also similar to decimal subtraction and follows specific rules
using borrowing when needed.
Rules
of Binary Subtraction
|
A |
B |
A − B |
|
0 |
0 |
0 |
|
1 |
0 |
1 |
|
1 |
1 |
0 |
|
0 |
1 |
1 (with borrow) |
Example
of Binary Subtraction
10111
- 00101
--------
10010
Hence, 10111₂ − 00101₂ = 10010₂
Binary
Multiplication
Binary
multiplication is simpler than decimal multiplication because it involves only 0
and 1.
Rules
of Binary Multiplication
|
A |
B |
A × B |
|
0 |
0 |
0 |
|
0 |
1 |
0 |
|
1 |
0 |
0 |
|
1 |
1 |
1 |
Steps
of Binary Multiplication
- Write
the numbers properly aligned.
- Multiply
each digit of the multiplier with the multiplicand.
- Shift
left for each next digit (like decimal multiplication).
- Add
all partial products using binary addition.
Example
of Binary Multiplication
101
× 11
-------
101
+ 1010
-------
1111
Hence, 101₂ × 11₂ = 1111₂
Binary
Division
Binary
division is similar to decimal division but is performed using binary digits 0
and 1.
Steps
of Binary Division
- Compare
the divisor with the leftmost bits of the dividend.
- If
the divisor is smaller or equal, subtract it and write 1 in the
quotient.
- If
smaller, write 0 and bring down the next bit.
- Repeat
until all bits are processed.
Example
of Binary Division
1010
÷ 10 = 101
Hence, 1010₂ ÷ 10₂ = 101₂
Exam
Practice Numericals (With Carry & Borrow)
A.
Binary Addition (With Carry)
1011
+ 1101
--------
11000
Answer: 1011₂ + 1101₂ = 11000₂
B.
Binary Subtraction (With Borrow)
10010
- 00111
--------
01011
Answer: 10010₂ − 00111₂ = 01011₂
C.
Binary Multiplication (Practice)
110
× 10
------
1100
Answer: 110₂ × 10₂ = 1100₂
D.
Binary Division (Practice)
1100
÷ 11 = 100
Answer: 1100₂ ÷ 11₂ = 100₂
2.3
Number Conversion
People
commonly use the decimal number system in daily life, whereas computers
use binary, octal, and hexadecimal number systems for processing
information. Since humans and computers use different number systems, number
system conversion is required to convert numbers from one system to another
so that both humans and machines can understand them.
There
are different methods to convert numbers between number systems.
A.
Decimal to Binary Conversion
Steps:
- Divide
the given decimal number by 2 and write the remainder.
- Divide
the quotient again by 2 and note the remainder.
- Repeat
the process until the quotient becomes 0.
- Write
the remainders from bottom to top.
Example:
Convert
(13)₁₀ to binary.
2
| 13 → 1
2
| 6
→ 0
2
| 3
→ 1
2
| 1
→ 1
0
Therefore,
(13)₁₀ = (1101)₂
B.
Decimal to Octal Conversion
Steps:
- Divide
the decimal number by 8 and write the remainder.
- Divide
the quotient repeatedly by 8.
- Stop
when the quotient becomes 0.
- Write
the remainders from bottom to top.
Example:
Convert
(345)₁₀ to octal.
8
| 345 → 1
8
| 43 → 3
8
| 5 → 5
0
Therefore,
(345)₁₀ = (531)₈
C.
Decimal to Hexadecimal Conversion
Steps:
- Divide
the decimal number by 16 and write the remainder.
- Divide
the quotient repeatedly by 16.
- Replace
remainders above 9 with A–F.
- Write
the remainders from bottom to top.
Example:
Convert
(88)₁₀ to hexadecimal.
16
| 88 → 8
16
| 5 → 5
0
Therefore,
(88)₁₀ = (58)₁₆
D.
Binary to Decimal Conversion
Steps:
- Multiply
each binary digit by its place value (powers of 2).
- Add
all the products.
Example:
Convert
(10011)₂ to decimal.
Therefore,
(10011)₂ = (19)₁₀
E.
Octal to Decimal Conversion
Steps:
- Multiply
each digit by its place value (powers of 8).
- Add
all the products.
Example:
Convert
(157)₈ to decimal.
Therefore,
(157)₈ = (111)₁₀
F.
Hexadecimal to Decimal Conversion
Steps:
- Multiply
each digit by its place value (powers of 16).
- Replace
A–F with values 10–15.
- Add
all the products.
Example:
Convert
(1AC)₁₆ to decimal.
Therefore,
(1AC)₁₆ = (428)₁₀
G.
Binary to Hexadecimal Conversion
Steps:
- Group
binary digits into sets of four from right to left.
- Convert
each group into its hexadecimal equivalent.
- Write
the result with base 16.
Example:
Convert
(100000110101)₂ to hexadecimal.
1000
0011 0101
8 3 5
Therefore,
(100000110101)₂ = (835)₁₆
H.
Hexadecimal to Binary Conversion
Steps:
- Write
the 4-bit binary equivalent of each hexadecimal digit.
- Combine
all binary groups.
Example:
Convert
(9A3)₁₆ to binary.
9
→ 1001
A
→ 1010
3
→ 0011
Therefore,
(9A3)₁₆ = (100110100011)₂
Binary
to Octal and Octal to Binary Conversion
Clean
steps + clear examples. You can write this exactly as it is.
Binary
to Octal Conversion
Binary
to octal conversion is done by grouping binary digits into sets of three (3)
from right to left, because 8 = 2³.
Steps:
- Group
the binary digits into sets of three from right to left.
- If
needed, add leading zeros to complete a group.
- Convert
each group into its octal equivalent.
- Write
the result with base 8.
Example:
Convert
(1101011)₂ to octal.
Binary
number: 1 101 011
Add
zero → 001 101 011
001
= 1
101
= 5
011
= 3
Therefore,
Octal
to Binary Conversion
Octal
to binary conversion is done by replacing each octal digit with its 3-bit
binary equivalent.
Steps:
- Write
the binary equivalent (3 bits) of each octal digit.
- Combine
all binary groups.
- Write
the result with base 2.
Example:
Convert
(753)₈ to binary.
7
→ 111
5
→ 101
3
→ 011
Therefore,
✅ Quick Reference Table (Exam-friendly)
|
Octal |
Binary |
|
0 |
000 |
|
1 |
001 |
|
2 |
010 |
|
3 |
011 |
|
4 |
100 |
|
5 |
101 |
|
6 |
110 |
|
7 |
111 |
3.
Calculate the following as indicated
a)
Perform the following binary addition
i.
(11110)₂ + (1001)₂
ii. (1011)₂ + (1001)₂
iii. (101011)₂ + (11011)₂
iv. (1010)₂ + (110)₂
v. (101001)₂ + (1110)₂
vi. (100001)₂ + (100011)₂
vii. (100111)₂ + (11010)₂
viii. (110001)₂ + (100101)₂
b)
Perform the following binary subtraction
i.
(1100)₂ − (1001)₂
ii. (1001)₂ − (110)₂
iii. (11101)₂ − (1010)₂
iv. (101100)₂ − (10011)₂
v. (11111)₂ − (10110)₂
vi. (110011)₂ − (10100)₂
vii. (100100)₂ − (1110)₂
viii. (1000001)₂ − (10101)₂
4.
Convert the given numbers as indicated
a)
Decimal to Binary Conversion
i.
(56)₁₀
ii. (78)₁₀
iii. (123)₁₀
iv. (345)₁₀
v. (540)₁₀
vi. (572)₁₀
vii. (546)₁₀
viii. (1098)₁₀
ix. (2103)₁₀
x. (445)₁₀
b)
Binary to Decimal Conversion
i.
(1101)₂
ii. (1010)₂
iii. (10010)₂
iv. (10110)₂
v. (101001)₂
vi. (11100111)₂
vii. (111100)₂
viii. (10010011)₂
ix. (1011100)₂
x. (100110)₂
c)
Decimal to Octal Conversion
i.
(69)₁₀
ii. (216)₁₀
iii. (767)₁₀
iv. (79)₁₀
v. (443)₁₀
vi. (413)₁₀
vii. (765)₁₀
viii. (1334)₁₀
ix. (1825)₁₀
x. (2783)₁₀
d)
Octal to Decimal Conversion
i.
(124)₈
ii. (242)₈
iii. (333)₈
iv. (763)₈
v. (103)₈
vi. (451)₈
vii. (3401)₈
viii. (1045)₈
ix. (438)₈
x. (611)₈
e)
Decimal to Hexadecimal Conversion
i.
(55)₁₀
ii. (540)₁₀
iii. (225)₁₀
iv. (880)₁₀
v. (2046)₁₀
vi. (2024)₁₀
vii. (6678)₁₀
f)
Hexadecimal to Decimal Conversion
i.
(56)₁₆
ii. (67)₁₆
iii. (558)₁₆
iv. (B74)₁₆
v. (20D3)₁₆
vi. (DEF)₁₆
vii. (6E3)₁₆
viii. (63F)₁₆
g)
Binary to Hexadecimal Conversion
i.
(1000110)₂
ii. (11001)₂
iii. (1111000)₂
iv. (11110000111)₂
v. (101010110)₂
vi. (1110010110)₂
vii. (11011001)₂
viii. (1001100)₂
h)
Hexadecimal to Binary Conversion
i.
(D4)₁₆
ii. (643)₁₆
iii. (189)₁₆
iv. (2BF)₁₆
v. (A9F)₁₆
vi. (FACE)₁₆
vii. (FB4)₁₆
viii. (1B2)₁₆
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