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Base System Conversion

Introduction

Every number can be represented in different bases - the most common being base 10 (decimal). In aptitude exams, you may need to convert numbers between bases such as binary (base 2), octal (base 8), or hexadecimal (base 16). Understanding these conversions helps in pattern recognition, digital computation, and logical reasoning questions.

Pattern: Base System Conversion

Pattern

Base system conversion means representing the same number using different bases by repeatedly dividing (for base conversion down) or multiplying (for base conversion up).

  • Formula 1 - Converting from Base 10 (Decimal) to Another Base (b):
    Divide the number repeatedly by b and write down the remainders in reverse order.
    Example: To convert 25₁₀ to base 2 → divide by 2 repeatedly.
    25 ÷ 2 = 12 R1 → 12 ÷ 2 = 6 R0 → 6 ÷ 2 = 3 R0 → 3 ÷ 2 = 1 R1 → 1 ÷ 2 = 0 R1 → Answer = 11001₂
  • Formula 2 - Converting from Any Base (b) to Base 10:
    Multiply each digit by its place value (base power) and add.
    Value = (dₙ × bⁿ) + (dₙ₋₁ × bⁿ⁻¹) + ... + (d₀ × b⁰)
    Example: Convert 1101₂ to base 10 → (1×2³) + (1×2²) + (0×2¹) + (1×2⁰) = 13₁₀
  • Formula 3 - Between Non-Decimal Bases (e.g., Binary ↔ Octal / Hexadecimal):
    Convert the number first to base 10 (decimal), then to the required base.
    OR, for binary ↔ octal/hex, use group-based shortcuts:
    • Binary to Octal → group 3 bits → convert each group to an octal digit.
    • Binary to Hexadecimal → group 4 bits → convert each group to a hexadecimal digit (A=10, B=11, …, F=15).
  • Formula 4 - Fractional Conversion (for advanced questions):
    For fractions, multiply repeatedly by the base and take integer parts in sequence.

Step-by-Step Example

Question

Convert 45₁₀ to binary (base 2).

Solution

  1. Step 1: Divide repeatedly by 2:

    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: Write remainders in reverse order:

    Reading from last to first remainder → 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. Convert binary to octal or hexadecimal directly using groupings (3 bits for octal, 4 for hex).

2. Convert octal or hex to binary by replacing each digit with its 3- or 4-bit equivalent.

3. Fractional base conversions by multiplying fractional parts repeatedly by target base.

Trick to Always Use

  • Step 1: For base ↓ conversion → use repeated division.
  • Step 2: For base ↑ conversion → use positional multiplication.
  • Step 3: Binary ↔ Octal/Hex → use group-of-3 or group-of-4 bit shortcuts.

Summary

Summary

  • Base system conversion is about representing the same value in different numeral systems.
  • For decimal → other base: divide repeatedly by base, read remainders backward.
  • For other base → decimal: multiply digits by powers of base and sum.
  • Binary ↔ Octal/Hex conversions are easiest with 3-bit/4-bit group patterns.

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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