Binary conversion

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 What is a Number System?

A number system is a way to represent numbers using a consistent set of symbols or digits. The base (or radix) of a number system determines how many unique digits it uses.

Types of Number Systems

1. Binary Number System (Base-2)

  • Digits Used: 0, 1

  • Base: 2

  • Application: Widely used in digital electronics, computers, and communication systems.

2. Octal Number System (Base-8)

  • Digits Used: 0 to 7

  • Base: 8

  • Application: Sometimes used in computing as a compact form of binary.

3. Decimal Number System (Base-10)

  • Digits Used: 0 to 9

  • Base: 10

  • Application: Most commonly used in daily human activities and calculations.

4. Hexadecimal Number System (Base-16)

  • Digits Used: 0–9 and A–F (where A=10, B=11, ..., F=15)

  • Base: 16

  • Application: Used in programming and debugging as a human-friendly representation of binary-coded values.

Binary Conversion

1. Decimal to Binary (Integral Part)

Steps:

  1. Divide the decimal number by 2.

  2. Record the remainder.

  3. Repeat the process until the quotient becomes 0.

  4. The binary number is the remainder sequence read in reverse.

Example: Convert 13 to binary

13 ÷ 2 = 6 → R1  

6 ÷ 2 = 3 → R0  

3 ÷ 2 = 1 → R1  

1 ÷ 2 = 0 → R1  

Binary = 1101

2. Decimal to Binary (Fractional Part)

Steps:

  1. Multiply the fractional part by 2.

  2. Record the whole number part of the result.

  3. Repeat with the new fractional part.

  4. Stop when the fraction becomes 0 or repeats.

  5. The binary result is read in sequence after the decimal point.

Example: Convert 0.625 to binary

0.625 × 2 = 1.25 → 1  

0.25 × 2 = 0.5 → 0  

0.5 × 2 = 1.0 → 1  

Binary = .101

3. Decimal to Binary (Integral + Fraction)

Example: Convert 13.625 to binary

  • Integer part (13) → 1101

  • Fractional part (.625) → .101
    Binary = 1101.101

4. Binary to Decimal Conversion

For Integer: Multiply each bit by 2ⁿ, where n is its position from the right (starting from 0).
For Fraction: Multiply each bit after the decimal by 2⁻ⁿ (starting from 1).

Example: Convert 1101.101 to decimal

= 1×2³ + 1×2² + 0×2¹ + 1×2⁰ → 8 + 4 + 0 + 1 = 13  

+ 1×2⁻¹ + 0×2⁻² + 1×2⁻³ → 0.5 + 0 + 0.125 = 0.625  

Decimal = 13.625



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