Key Learning Outcomes
By the end of this lesson, readers will:
- Understand the Concept of Number Systems
- Identify Different Types of Number Systems
- Perform Number System Conversions
- Understand Binary Arithmetic
- Understand Binary Arithmetic
- Recognize the Importance of Binary in Computers
- Solve Real-Life Problems
Click On The Name To Go To A Specific Topic:
Introduction
Numbers are an essential part of computing. Computers process and store data in a different number system than humans. In this chapter, we will learn about various number systems and their conversions.
What is a Number System?
A Number System is a way of representing numbers using a set of symbols and rules. It defines how numbers are written, stored, and computed in computers.
Types of Number Systems
There are four main types of number systems used in computers:
1. Binary Number System (Base 2)
- It is used in computers as it works on ON (1) and OFF (0) states.
- Each digit in a binary number is called a bit (Binary Digit).
Example:
- (1011)2 → Binary Number
Binary: 1011
Decimal Equivalent: 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 112. Decimal Number System (Base 10)
- The most common number system used by humans.
- Uses digits 0 to 9.
Example:
- (472)10 → Decimal Number
3. Octal Number System (Base 8)
- Uses digits from 0 to 7.
- Commonly used as a shortcut representation of binary numbers.
Example:
- (57)8
(57)₈ = (101111)₂4. Hexadecimal Number System (Base 16)
- Uses digits from 0 to 9 and letters A to F.
- Widely used in programming and memory addressing.
Example:
- (2F)16
(2F)₁₆ = (0010 1111)₂Conversion of Number Systems
1. Binary to Decimal
Multiply each binary digit by powers of 2 and sum them up.
Example: Convert (1010)2 to Decimal
= (1×2³) + (0×2²) + (1×2¹) + (0×2⁰)
= 8 + 0 + 2 + 0
= (10)₁₀2. Decimal to Binary
Divide the decimal number by 2 repeatedly and write down remainders in reverse order.
Example: Convert (13)10 to Binary
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Binary: (1101)₂
3. Decimal to Octal
Divide by 8 instead of 2.
Example: Convert (45)10 to Octal
45 ÷ 8 = 5 remainder 5
Octal: (55)₈
4. Decimal to Hexadecimal
Divide by 16 instead of 2.
Binary Arithmetic
Binary arithmetic is the mathematical operation on binary numbers (0 & 1). Since computers use the binary number system (Base-2), binary arithmetic is crucial for processing data and performing calculations.
Basic Binary Operations
Binary arithmetic includes the following operations:
- Binary Addition
- Binary Subtraction
- Binary Multiplication
- Binary Division
1. Binary Addition
Binary addition follows the same rules as decimal addition but is limited to 0 and 1.
Rules of Binary Addition:
Example 1: Add (101)₂ and (110)₂
101
+ 110
------
1011
Example 1: Add (1111)₂ and (1010)₂
1 1 1 1
+ 1 0 1 0
-------
1 1 0 0 1
2. Binary Subtraction
Binary subtraction is similar to decimal subtraction, but we use borrowing when needed.
Rules of Binary Subtraction:
Example: Subtract (1011)₂ from (1101)₂
1 1 0 1
-1 0 1 1
------
0010
3. Binary Multiplication
Binary multiplication is simpler than decimal multiplication because it follows these two rules:
Rules of Binary Multiplication:
Example: Multiply (101)₂ by (11)₂
101
× 11
------
101 (101 × 1)
+ 1010 (101 × 1, shift left)
------
1111
4. Binary Division
Binary division is similar to decimal division and follows the long division method.
Example: Divide (1010)₂ by (10)₂
101
------
10 | 1010
-10
----
01 (Bring down 0)
-10
----
00 Application of Number System
1. Digital Computing & Data Representation
- Computers use the Binary Number System (0s and 1s) for all operations.
- All types of data (text, images, videos, sounds) are stored and processed in binary format.
- Programming languages, machine code, and system-level operations depend on the binary system.
Example:
- ASCII & Unicode use binary values to represent text characters.
- RGB color codes use Hexadecimal (Base-16) to define colors (e.g.,
#FF5733).
2. Memory Addressing in Computers
- The Hexadecimal Number System (Base-16) is widely used for memory addresses in computer programming.
- It helps represent large binary numbers in a shorter and more readable format.
Example:
- A memory address like
101011101010in binary is written as0xAEAin hexadecimal.
3. Networking & IP Addressing
- The Octal (Base-8) and Hexadecimal (Base-16) systems are used in networking for IP addresses and MAC addresses.
- IPv4 addresses use the Decimal System (e.g., 192.168.1.1), while IPv6 uses Hexadecimal.
Example:
- IPv6 address:
2001:0db8:85a3:0000:0000:8a2e:0370:7334
4. Electronics & Digital Circuits
- The Binary Number System is essential in logic gates, circuits, and microprocessors.
- Microcontrollers and processors process binary data for input, processing, and output.
Example:
- Logic gates (AND, OR, NOT) work using binary inputs (0 & 1).
5. Banking & Financial Transactions
- The Decimal Number System (Base-10) is widely used in banking, accounting, and finance.
- Credit card numbers, transactions, and currency calculations rely on decimal operations.
Example:
- ATM machines and online banking systems store and process financial data in decimal and binary format.
6. Barcodes, QR Codes & Digital Encoding
- Binary and Decimal systems are used in barcode scanners and QR code technology.
- Each barcode stores numerical data in binary format for quick scanning.
Example:
- UPC barcodes use a combination of binary and decimal encoding for product identification.
7. Cryptography & Cybersecurity
- The Number System (Binary, Hexadecimal) is used in encryption and decryption algorithms.
- Cryptographic keys and hashing algorithms work with binary and hex data to ensure security.
Example:
- MD5 Hash (Message Digest Algorithm) uses Hexadecimal Representation for secure passwords.
8. Robotics & AI (Artificial Intelligence)
- Robots and AI systems use the binary number system for decision-making and processing commands.
- Sensor data and AI models rely on numerical processing using binary and hexadecimal numbers.
Example:
- AI-based image recognition models convert pixel data into binary values for processing.
Important Summary Points
- Computers use the binary number system (0 and 1).
- The decimal system is the most used by humans.
- Octal and hexadecimal are used in computing for memory management.
- Conversions between number systems are important in programming.
- Binary Addition follows simple rules; carry is generated when 1+1 = 10.
- Binary Subtraction requires borrowing when subtracting a larger digit from a smaller one.
- Binary Multiplication follows simple multiplication rules (1×1=1, others=0).
- Binary Division is similar to decimal division using the long division method.
Interesting Facts About Integers
💡 A single binary digit is called a “bit“.
💡 A group of 4 bits is called a “Nibble“.
💡 A group of 8 bits is called a “Byte“.
💡 1 KB (Kilobyte) = 1024 Bytes
💡 1 MB (Megabyte) = 1024 KB
Activity (Exercise):
A. Fill in the Blanks.
- The base of the binary number system is ___.
- The decimal equivalent of (101)2 is ___.
- In hexadecimal, the letter ‘A’ represents the decimal number ___.
- The octal number system has a base of ___.
B. Convert the Following.
- Convert (10101)2 to Decimal.
- Convert (78)10 to Binary.
- Convert (3D)16 to Decimal.
- Convert (67)10 to Octal.
C. Solve the Following.
- Add (1011)₂ and (1001)₂.
- Subtract (1100)₂ from (10110)₂.
- Multiply (110)₂ by (101)₂.
- Divide (10010)₂ by (10)₂.
Quiz:
Coming Soon…
Supplementary Materials:
Provide downloadable materials for learners to review:
- – PDF Guide: “Coming Soon”
- – Cheat Sheet: “Coming Soon”
- – Video Source: “JNG ACADEMY“
- – Articles: “Blog Page“
FAQs:
Q1: Why do computers use binary numbers?
Computers use binary because they work with electronic circuits that have two states: ON (1) and OFF (0).
Q2: What is the difference between octal and hexadecimal?
Octal uses a base of 8 (0-7), whereas hexadecimal uses a base of 16 (0-9, A-F).
Q3: How do I convert a binary number to a decimal manually?
Multiply each digit by powers of 2 and sum them up.
Q4: Where is the hexadecimal system used?
It is widely used in programming, color codes, and memory addressing.
Q5: Why is binary arithmetic important?
Binary arithmetic is essential for computer processing, logic circuits, and digital systems.
Q6: How is binary addition different from decimal addition?
Binary addition only uses 0 and 1, whereas decimal addition involves 0-9.
Q7: What happens when 1+1 in binary?
The sum is 0, and a carry of 1 is added to the next column.