Number System Chapter Notes | Cyber Olympiad for Class 9 PDF Download

Introduction

The number system is a way to represent and work with numbers. It provides a framework for performing mathematical operations and is fundamental to various fields such as computer science, engineering, and mathematics.

Types of Number Systems

There are several types of number systems, each with its own set of rules and uses. The most common ones are:

• Decimal (Base 10)
• Binary (Base 2)
• Octal (Base 8)
• Hexadecimal (Base 16)

Decimal Number System (Base 10)

• Digits Used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Place Value: Each digit's position in a number represents a power of 10.
• Example: 352=3×10²+5×10¹+2×10⁰ 

Binary Number System (Base 2)

• Digits Used: 0, 1
• Place Value: Each digit's position in a number represents a power of 2.
• Example: 1101=1×2³+1×2²+0×2¹+1×2⁰ 

Conversion Between Decimal and Binary

• Decimal to Binary: Repeated division by 2, noting the remainders.
• Binary to Decimal: Summing the products of each binary digit and its corresponding power of 2.

Octal Number System (Base 8)

• Digits Used: 0, 1, 2, 3, 4, 5, 6, 7
• Place Value: Each digit's position in a number represents a power of 8.
• Example: 245=2×8²+4×8¹+5×8⁰

Hexadecimal Number System (Base 16)

• Digits Used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
• Place Value: Each digit's position in a number represents a power of 16.
• Example: 1A3=1×16²+A×16¹+3×16⁰ 
  Here, A = 10 in decimal.

Conversion Between Decimal and Hexadecimal

• Decimal to Hexadecimal: Repeated division by 16, noting the remainders.
• Hexadecimal to Decimal: Summing the products of each hexadecimal digit and its corresponding power of 16.

Conversions Between Number Systems

Binary to Octal: Group binary digits in sets of three, starting from the right. Convert each group to its octal equivalent.

Example:
Group the digits into sets of three, starting from the right: 101110
Convert each group to its octal equivalent:
101₂  =5₈ 
110₂  =6₈ 
Combine the octal digits:
101110₂  =56₈ 
So, the correct conversion is:  101110₂  =56₈ 

Binary to Hexadecimal: Group binary digits in sets of four, starting from the right. Convert each group to its hexadecimal equivalent.
Example: 
Group the digits into sets of four, starting from the right. If necessary, pad the leftmost group with zeros:
10111100
Convert each group to its hexadecimal equivalent:
1011₂  =B₁₆ 
1100₂  =C₁₆ 
Combine the hexadecimal digits:
10111100₂  =BC₁₆ 
So, the correct conversion is:
10111100₂  =BC₁₆ 

Octal to Binary:  Convert each octal digit to its 3-digit binary equivalent.
Example: Start with the octal number: 75₈.
Convert each octal digit to its 3-bit binary equivalent:
7₈=111₂
5₈=101₂
Combine the binary groups:
75₈=111101₂
So, the correct conversion is:
75₈=111101₂

Hexadecimal to Binary: Convert each hexadecimal digit to its 4-digit binary equivalent.
Example:
Start with the hexadecimal number:1F₁₆.
Convert each hexadecimal digit to its 4-bit binary equivalent:

Combine the binary groups:

So, the correct conversion is:

Applications of Number Systems

• Binary System: Used in computers and digital systems.
• Octal System: Sometimes used in computing.
• Hexadecimal System: Used in computer science for memory addresses, color codes, etc.

Interesting Facts

• Binary Code: The foundation of modern computing, representing all data in a form that computers can process.
• Hexadecimal System: Used in defining colors in web design, where #FFFFFF represents white and #000000 represents black.
• Decimal Dominance: Despite the presence of other number systems, the decimal system is the most widely used system in everyday life due to its simplicity and historical origins in human counting.

Latest Developments

• Quantum Computing: Utilizes quantum bits or qubits, which can represent 0 and 1 simultaneously, leading to potential advancements in processing power and speed.
• Blockchain Technology: Uses complex algorithms based on number systems for cryptography, ensuring secure and transparent transactions.