Decimal-Binary Conversion: Division-by-2, Weight Expansion & Quick Estimation

Decimal to Binary: Division-by-2 Remainder Method

The most common method: repeatedly divide the decimal number by 2, record each remainder, stop when the quotient reaches 0, then read the remainders from bottom to top.

Example: Convert 25 to Binary

25 ÷ 2 = 12 ... remainder 1
12 ÷ 2 = 6  ... remainder 0
6  ÷ 2 = 3  ... remainder 0
3  ÷ 2 = 1  ... remainder 1
1  ÷ 2 = 0  ... remainder 1

Reading remainders bottom-up gives 11001 . Verification: 1×16 + 1×8 + 0×4 + 0×2 + 1×1 = 25 ✓

Example: Convert 100 to Binary

100 ÷ 2 = 50 ... remainder 0
50  ÷ 2 = 25 ... remainder 0
25  ÷ 2 = 12 ... remainder 1
12  ÷ 2 = 6  ... remainder 0
6   ÷ 2 = 3  ... remainder 0
3   ÷ 2 = 1  ... remainder 1
1   ÷ 2 = 0  ... remainder 1

Remainders reversed: 1100100 . Verification: 64 + 32 + 4 = 100 ✓

Mnemonic: Divide by 2, take remainder, read from bottom up.

Binary to Decimal: Weight Expansion Method

Each binary position has a weight: from right to left, 1, 2, 4, 8, 16, 32… (2⁰, 2¹, 2²…). To convert, sum the weights of positions that contain 1.

Example: Convert 10110 to Decimal

Bit:    1   0   1   1   0
Weight: 16  8   4   2   1

Add weights for 1-bits: 16 + 4 + 2 = 22 . So binary 10110 = decimal 22.

Power-of-2 Quick Reference (Positions 1–8 from right)

Position 1: 1

Position 2: 2

Position 3: 4

Position 4: 8

Position 5: 16

Position 6: 32

Position 7: 64

Position 8: 128

Continuing left: 256, 512, 1024, 2048… each step doubles.

Alternative: Subtraction (Make-Up) Method

Instead of division, repeatedly subtract the largest power of 2 not exceeding the remaining value. The used powers indicate which bits are 1.

Example: Convert 57 to Binary

Largest power ≤ 57: 32 (2⁵). 57 − 32 = 25.

Largest power ≤ 25: 16 (2⁴). 25 − 16 = 9.

Largest power ≤ 9: 8 (2³). 9 − 8 = 1.

1 = 2⁰.

Used powers: 5, 4, 3, 0 → bits 5,4,3,0 are 1; others 0.

Position: 6 5 4 3 2 1 0
Bits:     0 1 1 1 0 0 1

Result: 111001 . Verification: 32 + 16 + 8 + 1 = 57 ✓. Advantage: only subtraction and power-of-2 memory needed.

Fractional Decimal to Binary: Multiply-by-2 Take-Integer Method

For the fractional part, repeatedly multiply by 2, take the integer part (0 or 1) as the next binary digit, continue with the new fractional part until it becomes 0 or desired precision is reached. Read the integer parts top-down.

Example: Convert 0.625 to Binary

0.625 × 2 = 1.25 → take 1
0.25  × 2 = 0.5  → take 0
0.5   × 2 = 1.0  → take 1

Reading top-down: 0.101 . Verification: 1×0.5 + 0×0.25 + 1×0.125 = 0.625 ✓

Important: Many decimal fractions become infinite repeating binaries. Example: decimal 0.1 = binary 0.0001100110011… (repeating). This is why floating-point arithmetic has precision issues — 0.1 cannot be represented exactly in binary.

Common Decimal-Binary Reference Table

0 = 0

1 = 1

5 = 101

10 = 1010

15 = 1111

16 = 10000

32 = 100000

64 = 1000000

100 = 1100100

127 = 1111111

128 = 10000000

255 = 11111111 (8 ones) — maximum value of one byte

256 = 100000000

1024 = 10000000000

Quick Estimation: Highest-Set-Bit Technique

To estimate a binary number's decimal range, locate the highest 1-bit. Its weight gives the lower bound; adding the next lower weight narrows the range.

Example: 10010110 — highest bit is position 8 (weight 128), so value is 128–255. Second bit (position 7, weight 64) is also 1 → 128+64=192, so value is 192–255. Continue to refine.

Conclusion

Decimal-binary conversion is a foundational computer-science skill. The methods require only elementary division, addition, subtraction, and memorization of powers of 2. Practice with numbers like 42, 100, 255 to build fluency.