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Base System Conversion

Introduction

हर number को अलग-अलग bases में represent किया जा सकता है-सबसे common base 10 (decimal) है। Aptitude exams में आपको numbers को binary (base 2), octal (base 8) या hexadecimal (base 16) में convert करना पड़ सकता है। इन conversions को समझना pattern recognition, digital computation और logical reasoning में मदद करता है।

Pattern: Base System Conversion

Pattern

Base system conversion का मतलब है एक ही number को अलग base में दिखाना-base कम करने पर repeated division और base बढ़ाने पर repeated multiplication का उपयोग करें।

  • Formula 1 - Base 10 (Decimal) से किसी Base (b) में Conversion:
    Number को बार-बार b से divide करें और remainders को reverse order में लिखें।
    Example: 25₁₀ को base 2 में convert करें:
    25 ÷ 2 = 12 R1 → 12 ÷ 2 = 6 R0 → 6 ÷ 2 = 3 R0 → 3 ÷ 2 = 1 R1 → 1 ÷ 2 = 0 R1 → Answer = 11001₂
  • Formula 2 - किसी Base (b) से Base 10 में Conversion:
    हर digit को उसके place value (base power) से multiply करें और जोड़ें।
    Value = (dₙ × bⁿ) + (dₙ₋₁ × bⁿ⁻¹) + ... + (d₀ × b⁰)
    Example: 1101₂ → (1×2³) + (1×2²) + (0×2¹) + (1×2⁰) = 13₁₀
  • Formula 3 - Non-Decimal Bases के बीच Conversion (जैसे Binary ↔ Octal / Hexadecimal):
    पहले decimal में convert करें, फिर target base में।
    OR, group-based shortcuts use करें:
    • Binary → Octal → 3 bits को group करें → हर group को octal digit में बदलें।
    • Binary → Hex → 4 bits को group करें → हर group को hexadecimal digit में बदलें (A=10, B=11 … F=15)।
  • Formula 4 - Fractional Conversion (advanced):
    Fractions को target base से multiply करते जाएँ और integer parts को sequence में लिखें।

Step-by-Step Example

Question

45₁₀ को binary (base 2) में convert करें।

Solution

  1. Step 1: बार-बार 2 से divide करें

    45 ÷ 2 = 22 R1
    22 ÷ 2 = 11 R0
    11 ÷ 2 = 5 R1
    5 ÷ 2 = 2 R1
    2 ÷ 2 = 1 R0
    1 ÷ 2 = 0 R1

  2. Step 2: Remainders को reverse order में पढ़ें

    Last से first तक पढ़ने पर → 101101

  3. Final Answer:

    45₁₀ = 101101₂

  4. Quick Check:

    (1×32) + (0×16) + (1×8) + (1×4) + (0×2) + (1×1) = 32 + 8 + 4 + 1 = 45 ✅

Quick Variations

1. Binary → Octal/Hex → direct groupings (3 bits for octal, 4 bits for hex).

2. Octal/Hex → Binary → हर digit का 3-bit या 4-bit binary लिखें।

3. Fractional conversions → fractional part को बार-बार base से multiply करें।

Trick to Always Use

  • Step 1: Base ↓ conversion में repeated division use करें।
  • Step 2: Base ↑ conversion में positional multiplication use करें।
  • Step 3: Binary ↔ Octal/Hex में group-of-3/group-of-4 सबसे आसान method है।

Summary

Summary

  • Base system conversion एक ही value को अलग numeral systems में represent करना है।
  • Decimal → other base: divide repeatedly, remainders backward पढ़ें।
  • Other base → decimal: digits × base powers का sum।
  • Binary ↔ Octal/Hex सबसे आसान group-based patterns से convert होते हैं।

Practice

(1/5)
1. Convert 25₁₀ to binary.
easy
A. 11001₂
B. 11010₂
C. 10101₂
D. 11100₂

Solution

  1. Step 1: Divide repeatedly by 2:

    25 ÷ 2 = 12 R1 → 12 ÷ 2 = 6 R0 → 6 ÷ 2 = 3 R0 → 3 ÷ 2 = 1 R1 → 1 ÷ 2 = 0 R1.

  2. Step 2: Write remainders in reverse order:

    Reading upward → 11001.

  3. Final Answer:

    25₁₀ = 11001₂ → Option A.

  4. Quick Check:

    (1×16) + (1×8) + (0×4) + (0×2) + (1×1) = 25 ✅
Hint: Divide by 2 repeatedly and reverse the remainders.
Common Mistakes: Forgetting to reverse the remainders after division.
2. Convert 1101₂ to decimal.
easy
A. 12₁₀
B. 13₁₀
C. 14₁₀
D. 15₁₀

Solution

  1. Step 1: Multiply each binary digit by its base power:

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

  2. Final Answer:

    1101₂ = 13₁₀ → Option B.

  3. Quick Check:

    Binary weights 8 + 4 + 1 = 13 ✅
Hint: Multiply each bit by its positional power of 2 and add them.
Common Mistakes: Forgetting to assign correct powers of 2 (rightmost bit = 2⁰).
3. Convert 57₁₀ to octal.
easy
A. 71₈
B. 72₈
C. 70₈
D. 73₈

Solution

  1. Step 1: Divide by 8 repeatedly:

    57 ÷ 8 = 7 R1 → 7 ÷ 8 = 0 R7.

  2. Step 2: Read remainders in reverse order:

    → 71.

  3. Final Answer:

    57₁₀ = 71₈ → Option A.

  4. Quick Check:

    (7×8¹) + (1×8⁰) = 56 + 1 = 57 ✅
Hint: Use division by 8 and read remainders from bottom to top.
Common Mistakes: Reading remainders in forward order instead of reverse.
4. Convert 3F₁₆ to decimal.
medium
A. 61₁₀
B. 62₁₀
C. 63₁₀
D. 64₁₀

Solution

  1. Step 1: Replace hexadecimal digits:

    3 = 3, F = 15.

  2. Step 2: Multiply by powers of 16:

    (3×16¹) + (15×16⁰) = 48 + 15 = 63.

  3. Final Answer:

    3F₁₆ = 63₁₀ → Option C.

  4. Quick Check:

    16×3=48, +15=63 ✅
Hint: In hex, A=10, B=11, … F=15. Multiply digits by 16 powers and add.
Common Mistakes: Confusing hexadecimal digit values or forgetting F = 15.
5. Convert 110110₂ to hexadecimal.
medium
A. 35₁₆
B. 36₁₆
C. 2E₁₆
D. 1B₁₆

Solution

  1. Step 1: Group binary digits into sets of 4 (from right):

    110110 → 0011 0110.

  2. Step 2: Convert each group:

    0011 = 3, 0110 = 6 → combine as 36.

  3. Final Answer:

    110110₂ = 36₁₆ → Option B.

  4. Quick Check:

    3×16 + 6 = 48 + 6 = 54₁₀; binary 110110 = 54₁₀ ✅
Hint: Group 4 bits per hex digit (right to left).
Common Mistakes: Grouping binary digits incorrectly or skipping leading zeros.

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