Number system

 


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📘 Number Systems Study Guide

🔢 1. What is a Number System?

A number system defines a way to represent numbers using a specific base (or radix). It includes a set of symbols (digits) and rules for representing and manipulating numbers.

🌐 2. Common Number Systems

NameBaseDigits UsedExample
Binary20, 11010₂ = 10₁₀
Octal80–717₈ = 15₁₀
Decimal100–9123₁₀
Hexadecimal160–9, A–F (A=10...F=15)1A₁₆ = 26₁₀

🔁 3. Conversion Between Number Systems

✅ Decimal to Binary

Method: Divide by 2, record remainder, repeat with quotient until 0.
Example:
13₁₀ → 1101₂

✅ Binary to Decimal

Method: Multiply each bit by 2ⁿ (right to left) and sum.
Example:
1101₂ = 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 13₁₀

✅ Decimal to Hexadecimal

Method: Divide by 16 and record remainders.
Example:
47₁₀ → 2F₁₆

✅ Hexadecimal to Binary

Method: Convert each hex digit to 4-bit binary.
Example:
2F₁₆ = 0010 1111₂

🧠 4. 1’s and 2’s Complement (Binary Only)

✅ 1’s Complement

Invert each bit (0→1, 1→0).
1010 → 0101

✅ 2’s Complement

Add 1 to the 1’s complement.
0101 + 1 = 0110

Used to represent negative binary numbers.

🔡 5. Binary Arithmetic

OperationRule
Addition0+0=0, 1+0=1, 1+1=10 (carry), 1+1+1=11
SubtractionUse 2’s complement of the subtrahend
MultiplicationSimilar to decimal, use shifting
DivisionRepeated subtraction or long division method

🧾 6. BCD (Binary-Coded Decimal)

Each decimal digit is represented separately in 4-bit binary.

DecimalBCD
50101
91001
120001 0010

📍 7. Applications in Computer Science

  • Memory Addressing (Binary)

  • Machine-level instructions

  • Data Representation

  • Digital Circuit Design

  • Error Detection (e.g., parity bits)

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