Introduction to DLD

Types of Number System in Computer

Number systems are the core foundation of computer science. They help computers represent, process, and store data in a structured format. Every operation in a computer—from simple calculations to complex processing—depends on number systems.

Binary Number System

Binary number system is the most important number system in computers. It uses only two digits: 0 and 1. These represent OFF and ON states in digital circuits.

  • Base: 2
  • Digits: 0, 1
  • Used in all digital systems
  • Represents machine-level data

Key Concept of Binary System

Binary system is the language of computers. All data including text, images, and instructions are converted into binary form for processing.

  • Works with electrical signals
  • Used in CPU operations
  • Foundation of all computing systems

Decimal Number System

Decimal number system is the standard number system used by humans in daily life. It uses digits from 0 to 9 and is also called base-10 system.

  • Base: 10
  • Digits: 0–9
  • Used in human calculations
  • Not directly used by computers

Key Concept of Decimal System

Decimal system is easy for humans to understand, but computers convert it into binary for processing.

  • Used in daily life mathematics
  • Input/output system in computers
  • Converted internally to binary

Octal Number System

Octal number system uses digits from 0 to 7. It is a compact way to represent binary numbers.

  • Base: 8
  • Digits: 0–7
  • Each digit = 3 binary bits
  • Used in digital systems

Key Concept of Octal System

Octal system reduces the length of binary numbers and makes them easier to read and manage.

  • Simplifies binary representation
  • Used in computer systems
  • Helpful in digital electronics

Hexadecimal Number System

Hexadecimal number system uses 16 symbols: 0–9 and A–F. It is widely used in programming and computer memory systems.

  • Base: 16
  • Digits: 0–9, A–F
  • Each digit = 4 binary bits
  • Used in memory addressing

Key Concept of Hexadecimal System

Hexadecimal system provides a compact representation of binary numbers, making it easier for programmers.

  • Used in coding and debugging
  • Represents large binary values
  • Important in system architecture

Number System Conversions

Number system conversions are the process of changing numbers from one base system to another. These conversions are very important in computer science because computers and humans use different number systems.

1. Decimal to Binary Conversion

Decimal to binary conversion changes base-10 numbers into base-2 format using repeated division by 2.

Example: Decimal 10 to Binary

  • 10 ÷ 2 = 5 remainder 0
  • 5 ÷ 2 = 2 remainder 1
  • 2 ÷ 2 = 1 remainder 0
  • 1 ÷ 2 = 0 remainder 1

Binary result: 1010

Example: Decimal 25 to Binary

  • 25 ÷ 2 = 12 remainder 1
  • 12 ÷ 2 = 6 remainder 0
  • 6 ÷ 2 = 3 remainder 0
  • 3 ÷ 2 = 1 remainder 1
  • 1 ÷ 2 = 0 remainder 1

Binary result: 11001

2. Decimal to Octal Conversion

Decimal to octal conversion uses repeated division by 8 to convert base-10 numbers into base-8 system.

Example: Decimal 64 to Octal

  • 64 ÷ 8 = 8 remainder 0
  • 8 ÷ 8 = 1 remainder 0
  • 1 ÷ 8 = 0 remainder 1

Octal result: 100

Example: Decimal 100 to Octal

  • 100 ÷ 8 = 12 remainder 4
  • 12 ÷ 8 = 1 remainder 4
  • 1 ÷ 8 = 0 remainder 1

Octal result: 144

3. Decimal to Hexadecimal Conversion

Decimal to hexadecimal conversion uses repeated division by 16 and converts remainders into hexadecimal symbols.

Example: Decimal 255 to Hexadecimal

  • 255 ÷ 16 = 15 remainder 15 (F)
  • 15 ÷ 16 = 0 remainder 15 (F)

Hexadecimal result: FF

Example: Decimal 1000 to Hexadecimal

  • 1000 ÷ 16 = 62 remainder 8
  • 62 ÷ 16 = 3 remainder 14 (E)
  • 3 ÷ 16 = 0 remainder 3

Hexadecimal result: 3E8

4. Binary to Octal Conversion

Binary to octal conversion is done by grouping binary digits into sets of three.

Example: Binary 110101 to Octal

  • Group: 110 101
  • 110 = 6
  • 101 = 5

Octal result: 65

Example: Binary 111100 to Octal

  • Group: 111 100
  • 111 = 7
  • 100 = 4

Octal result: 74

5. Binary to Decimal Conversion

Binary to decimal conversion uses positional weights of powers of 2.

Example: Binary 1010 to Decimal

  • (1×2³) + (0×2²) + (1×2¹) + (0×2⁰)
  • 8 + 0 + 2 + 0 = 10

Decimal result: 10

Example: Binary 1101 to Decimal

  • (1×2³) + (1×2²) + (0×2¹) + (1×2⁰)
  • 8 + 4 + 0 + 1 = 13

Decimal result: 13

6. Binary to Hexadecimal Conversion

Binary to hexadecimal conversion is done by grouping binary digits into sets of four.

Example: Binary 1111 to Hexadecimal

  • 1111 = F

Hexadecimal result: F

Example: Binary 10101010 to Hexadecimal

  • 1010 = A
  • 1010 = A

Hexadecimal result: AA

7. Octal to Binary Conversion

Octal to binary conversion replaces each octal digit with a 3-bit binary number.

Example: Octal 7 to Binary

  • 7 = 111

Binary result: 111

Example: Octal 52 to Binary

  • 5 = 101
  • 2 = 010

Binary result: 101010

8. Octal to Decimal Conversion

Octal to decimal conversion uses base 8 positional values.

Example: Octal 25 to Decimal

  • (2×8¹) + (5×8⁰)
  • 16 + 5 = 21

Decimal result: 21

Example: Octal 100 to Decimal

  • (1×8²) + (0×8¹) + (0×8⁰)
  • 64 + 0 + 0 = 64

Decimal result: 64

9. Octal to Hexadecimal Conversion

Octal to hexadecimal conversion is done using binary as an intermediate step.

Example: Octal 17 to Hexadecimal

  • 1 = 001
  • 7 = 111
  • Binary = 001111
  • Hex = F

Hexadecimal result: F

Example: Octal 64 to Hexadecimal

  • 6 = 110
  • 4 = 100
  • Binary = 110100
  • Hex = 34

Hexadecimal result: 34

10. Hexadecimal to Binary Conversion

Hexadecimal to binary conversion replaces each hex digit with a 4-bit binary equivalent.

Example: Hex A to Binary

  • A = 1010

Binary result: 1010

Example: Hex 2F to Binary

  • 2 = 0010
  • F = 1111

Binary result: 00101111

11. Hexadecimal to Decimal Conversion

Hexadecimal to decimal conversion uses base 16 positional values.

Example: Hex A to Decimal

  • A = 10

Decimal result: 10

Example: Hex 1F to Decimal

  • (1×16¹) + (15×16⁰)
  • 16 + 15 = 31

Decimal result: 31

12. Hexadecimal to Octal Conversion

Hexadecimal to octal conversion is done through binary conversion.

Example: Hex F to Octal

  • F = 1111
  • Group: 001 111
  • Octal result: 17

Example: Hex 2A to Octal

  • 2 = 0010
  • A = 1010
  • Binary = 00101010
  • Octal result: 52

Conclusion

Number systems and their conversions are essential in computer science. They help bridge the gap between human-readable numbers and machine-level data. Mastering binary, decimal, octal, and hexadecimal systems along with their conversions is crucial for programming, digital logic design, and computer architecture.