Decimal To Binary Conversion Algorithms Explained

Step-by-Step Decimal To Binary Conversion Tutorial

Converting decimal (base-10) numbers to binary (base-2) is a fundamental skill in computing. This tutorial shows clear, actionable methods with examples and practice problems.

How binary works (quick)

• Binary digits (bits): only 0 or 1.
• Place values: rightmost bit = 2^0, next = 2^1, then 2^2, etc.
• Example: 1101₂ = 1·2^3 + 1·2^2 + 0·2^1 + 1·2^0 = 8+4+0+1 = 13₁₀.

Method 1 — Repeated division (standard)

1. Divide the decimal number by 2.
2. Record the remainder (0 or 1).
3. Set the quotient as the new number and repeat until quotient = 0.
4. Binary is the remainders read bottom-to-top (last remainder is MSB).

Example: Convert 37₁₀ to binary

• 37 ÷ 2 = 18 remainder 1
• 18 ÷ 2 = 9 remainder 0
• 9 ÷ 2 = 4 remainder 1
• 4 ÷ 2 = 2 remainder 0
• 2 ÷ 2 = 1 remainder 0
• 1 ÷ 2 = 0 remainder 1 Read remainders bottom-to-top: 100101₂

Method 2 — Subtract highest power of two

1. Find largest 2^k ≤ N.
2. Put 1 in that position, subtract 2^k from N.
3. For next lower power, put 1 if it fits, else 0. Repeat until 2^0.

Example: Convert 37₁₀

• Largest power ≤37 is 32 (2^5) → 1, remainder 5
• 2^4=16 >5 → 0
• 2^3=8 >5 → 0
• 2^2=4 ≤5 → 1, remainder 1
• 2^1=2 >1 → 0
• 2^0=1 ≤1 → 1 Result: 100101₂

Method 3 — Using bitwise operations (programming)

• In many languages: repeatedly take N & 1 for least significant bit, then N >>= 1 until N = 0. Collect bits and reverse.
• Example (Python):

python
Copy CodeCopied
def dec_to_bin(n):
if n == 0:
        return “0”
    bits = []
    while n:
        bits.append(str(n & 1))
        n >>= 1
    return “.join(reversed(bits))

Fractional decimals (optional)

To convert fractional part (0.) to binary:

1. Multiply fractional part by 2.
2. Integer part of result is next binary digit (0 or 1).
3. Keep fractional remainder and repeat until remainder = 0 or desired precision reached.

Example: 0.625

• 0.625×2 = 1.25 → bit 1, remainder 0.25
• 0.25×2 = 0.5 → bit 0, remainder 0.5
• 0.5×2 = 1.0 → bit 1, remainder 0 Result: 0.101₂

Quick checks and tips

• Verify by converting binary back to decimal using place values.
• Powers of two are single 1 followed by zeros (e.g., 8 = 1000₂).
• For large numbers use programming methods to avoid error.
• Some decimals yield repeating binary fractions (like 0.1₁₀).

Practice problems

1. Convert 19₁₀ → 10011₂
2. Convert 255₁₀ → 11111111₂
3. Convert 6.75₁₀ → 110.11₂

Answers: 1) 10011₂, 2) 11111111₂, 3) 110.11₂

Done.