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Number System:

A number system is a writing system for expressing numbers; it is a mathematical notation for representing numbers in a consistent manner. There are several different types of number systems that are commonly used in mathematics and computer science.

Here are the main types of Number Systems:

1. Decimal Number System (Base-10)

The Decimal system is the most widely used number system in the world. It is base-10, which means it uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
* Position and Place Value: Each position in a decimal number represents a power of 10.
* Example: In the number 345, the 3 is in the "hundreds" place, 4 is in the "tens" place, and 5 is in the "ones" place.

Example of Decimal Conversion:

  • Convert **(345)**₁₀ to binary:

  • Start by dividing by 2 and keeping track of remainders:

    • 345 ÷ 2 = 172, remainder 1
    • 172 ÷ 2 = 86, remainder 0
    • 86 ÷ 2 = 43, remainder 0
    • 43 ÷ 2 = 21, remainder 1
    • 21 ÷ 2 = 10, remainder 1
    • 10 ÷ 2 = 5, remainder 0
    • 5 ÷ 2 = 2, remainder 1
    • 2 ÷ 2 = 1, remainder 0
    • 1 ÷ 2 = 0, remainder 1
    • Reversed remainder sequence: **(101011001)**₂

2. Binary Number System (Base-2)

The Binary system is the number system used by all modern computers and digital systems because it is straightforward to implement with digital electronic circuitry, which has two states: ON (1) and OFF (0).

  • Digits: It uses only two digits, 0 and 1.

  • Place Value: Each digit (bit) represents a power of 2.

  • Example: The binary number 1011₂ is calculated as:

    • (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰) = 8 + 0 + 2 + 1 = 11 in decimal.

Example of Binary Conversion:

  • Convert **(1011)**₂ to decimal:

  • 1 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰ = 8 + 0 + 2 + 1 = 11₁₀

3. Octal Number System (Base-8)

The Octal system is a base-8 system, which means it uses eight digits: 0, 1, 2, 3, 4, 5, 6, 7. It is often used in computer science as a shorthand for binary.

  • Place Value: Each digit in an octal number represents a power of 8.

  • Example: The octal number 123₈ is calculated as:

    • (1 × 8²) + (2 × 8¹) + (3 × 8⁰) = 64 + 16 + 3 = 83 in decimal.

Example of Octal Conversion:

  • Convert **(123)**₈ to decimal:

  • 1 × 8² + 2 × 8¹ + 3 × 8⁰ = 64 + 16 + 3 = 83₁₀

4. Hexadecimal Number System (Base-16)

The Hexadecimal system is a base-16 system, which means it uses sixteen digits: 0-9 and A-F, where A=10, B=11, C=12, D=13, E=14, F=15.

  • Place Value: Each digit in a hexadecimal number represents a power of 16.

  • Example: The hexadecimal number 1A3₁₆ is calculated as:

    • (1 × 16²) + (A × 16¹) + (3 × 16⁰) = (1 × 256) + (10 × 16) + (3 × 1) = 256 + 160 + 3 = 419 in decimal.

Example of Hexadecimal Conversion:

  • Convert **(1A3)**₁₆ to decimal:

  • 1 × 16² + A × 16¹ + 3 × 16⁰ = 1 × 256 + 10 × 16 + 3 × 1 = 256 + 160 + 3 = 419₁₀

Number System Conversions

Common Number Systems

Number System Base Digits Used Example
Binary 2 0, 1 1010
Octal 8 0–7 745
Decimal 10 0–9 255
Hexadecimal 16 0–9, A–F 3F
BCD n/a 4-bit per digit 25 = 0010 0101

🔁 Most Used Conversion Paths

From → To Short Method Manual / Long Form
Decimal → Binary Divide by 2, reverse remainders 25 ÷ 2 → 12 R1 ...
Binary → Decimal Multiply by 2ⁿ 1×2⁴ + 0×2³ + ...
Decimal → Octal Divide by 8 83 ÷ 8 → 10 R3 → ...
Octal → Decimal Multiply by 8ⁿ 1×8² + 2×8¹ + 3×8⁰
Decimal → Hex Divide by 16 254 ÷ 16 → 15 R14 → F
Hex → Decimal Multiply by 16ⁿ F×16¹ + E×16⁰ = 240 + 14 = 254
Binary ↔ Octal Group 3 bits 110010 → 110 010 → 6 2
Binary ↔ Hex Group 4 bits 11011110 → 1101 1110 → D E
Octal ↔ Hex Convert via Binary Octal → Binary → Hex

📌 Fractional Number Conversions

From → To Integer Part Method Fraction Part Method Example
Decimal → Binary ÷2 and reverse ×2, take integer parts 10.625 → 1010.101
Binary → Decimal ∑ digit × 2ⁿ ∑ digit × 2⁻ⁿ 1010.101 = 10.625
Decimal → Octal ÷8 ×8, take integer parts 25.375 → 31.3
Octal → Decimal ∑ digit × 8ⁿ ∑ digit × 8⁻ⁿ 31.3₈ = 25.375
Decimal → Hex ÷16 ×16, take integer parts 26.5625 → 1A.9
Hex → Decimal ∑ digit × 16ⁿ ∑ digit × 16⁻ⁿ 1A.9 = 26.5625

💡 Tricks & Tips

Trick Use Case
Group 3 bits (from right) Binary → Octal
Group 4 bits (from right) Binary → Hex
A–F = 10–15 Hex → Decimal
Use left/right 0-padding Before/after decimal in groupings
Invalid digits in Octal Only 0–7 allowed (no 8/9)
Decimal Fraction → Binary Multiply fraction ×2 repeatedly

🔍 Decimal Fraction to Binary (Step-by-Step)

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

🔍 Binary Fraction to Decimal

Convert 0.101 to decimal:

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

🧾 Quick Cheatsheet

From → To Integer Part Fractional Part
Decimal → Binary ÷2 + reverse remainders ×2 + take integer parts
Binary → Decimal ∑ digit × 2ⁿ ∑ digit × 2⁻ⁿ
Decimal → Octal ÷8 ×8
Octal → Decimal ∑ digit × 8ⁿ ∑ digit × 8⁻ⁿ
Decimal → Hex ÷16 ×16
Hex → Decimal ∑ digit × 16ⁿ ∑ digit × 16⁻ⁿ
Binary ↔ Octal Group 3 bits Group 3 after decimal
Binary ↔ Hex Group 4 bits Group 4 after decimal

🔠 BCD (Binary-Coded Decimal)

BCD = Write each decimal digit as a 4-bit binary.

Digit BCD
0 0000
1 0001
2 0010
3 0011
4 0100
5 0101
6 0110
7 0111
8 1000
9 1001

🧠 Example: 25 in BCD → 0010 0101

🗺️ Conversion Map (All Paths)

       Decimal (10)
      /    |     \
  Binary  Octal  Hex
    |       |     |
   Octal   Hex   Octal
    \       |    /
         Binary

🧪 Binary ⇄ Decimal

Binary to Decimal

  1. 1101₂ = 13₁₀
  2. 10101.11₂ = 21.75₁₀
  3. 100000.101₂ = 32.625₁₀

Decimal to Binary

  1. 23₁₀ = 10111₂
  2. 45.625₁₀ = 101101.101₂
  3. 0.8125₁₀ = 0.1101₂

🧪 Decimal ⇄ Octal

Decimal to Octal

  1. 63₁₀ = 77₈
  2. 100.375₁₀ = 144.3₈
  3. 15.5₁₀ = 17.4₈

Octal to Decimal

  1. 127₈ = 87₁₀
  2. 3.4₈ = 3.5₁₀
  3. 75.23₈ = 61.296875₁₀

🧪 Decimal ⇄ Hexadecimal

Decimal to Hex

  1. 254₁₀ = FE₁₆
  2. 26.5625₁₀ = 1A.9₁₆
  3. 1023.875₁₀ = 3FF.E₁₆

Hex to Decimal

  1. 1A₁₆ = 26₁₀
  2. 3F.8₁₆ = 63.5₁₀
  3. C9.4F₁₆ = 201.30859375₁₀

🧪 Binary ⇄ Octal

Binary to Octal

  1. 110101₂ = 65₈
  2. 100111.011₂ = 47.3₈

Octal to Binary

  1. 745₈ = 111100101₂
  2. 21.3₈ = 10001.011₂

🧪 Binary ⇄ Hex

Binary to Hex

  1. 11011110₂ = DE₁₆
  2. 1010.1111₂ = A.F₁₆

Hex to Binary

  1. 3F₁₆ = 00111111₂
  2. A9.B₁₆ = 10101001.1011₂

🧪 BCD (8421)

Decimal to BCD

  1. 27₁₀ = 0010 0111
  2. 304₁₀ = 0011 0000 0100

BCD to Decimal

  1. 0010 1001 = 29₁₀
  2. 0100 0110 0111 = 467₁₀

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