Binary basics
Binary is a base-2 system using digits 0 and 1. Each digit represents an increasing power of 2 from right to left. To convert binary to decimal, multiply each digit by its positional value and sum the results.
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Add, subtract, multiply, divide, and convert between binary and decimal representations. Includes reference tables, step-by-step examples, and formula explanations inspired by calculator.net's detailed guide.
Binary operations
Convert binary to decimal
Convert decimal to binary
Binary/decimal conversion table
| Decimal | Binary |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 10 |
| 3 | 11 |
| 4 | 100 |
| 5 | 101 |
| 6 | 110 |
| 7 | 111 |
| 8 | 1000 |
| 9 | 1001 |
| 10 | 1010 |
| 16 | 10000 |
| 32 | 100000 |
Converting between decimal and binary
Each binary place value represents 2ⁿ, just as decimal places represent 10ⁿ. Using 8 as an example: 8 × 10⁰ = 8 × 1 = 8. In decimal notation, 18 is (1 × 10¹) + (8 × 10⁰) = 10 + 8 = 18. In binary, 18 is represented as 10010, because reading right to left the first 0 represents 2⁰, the second 2¹, the third 2², and the fourth 2³ (which equals 8), giving 16 + 2 = 2⁴ + 2¹.
Bases are converted using: 18 = 16 + 2 = 2⁴ + 2¹ → 10010 = (1 × 2⁴) + (0 × 2³) + (0 × 2²) + (1 × 2¹) + (0 × 2⁰).
The step-by-step process to convert decimal to binary:
- Find the largest power of 2 that lies within the number.
- Subtract that value from the number.
- Find the largest power of 2 within the remainder.
- Repeat until there is no remainder.
- Enter 1 for each binary place value found, 0 elsewhere.
Example target 18: powers inside 18 are 2⁴ = 16 (leaves 2), then 2¹ = 2 (leaves 0). The table method below shows this visually—the 2⁴ and 2¹ columns get 1s, others get 0s.
Converting from binary to decimal is simpler: determine all place values where 1 occurs and sum them. EX: 10111 = (1 × 2⁴) + (0 × 2³) + (1 × 2²) + (1 × 2¹) + (1 × 2⁰) = 16 + 4 + 2 + 1 = 23.
Binary addition
Rules: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10 (carry 1). Example:
101 + 111
= 1100
Binary subtraction
Similar to decimal subtraction but borrowing occurs when a 0 must subtract 1. Borrow from the nearest 1 to the left, turning it into 0 and distributing 1s until reaching the needed column. Example:
1011 − 101
= 110
Binary multiplication
Binary multiplication is like decimal long multiplication. Multiply and shift partial products then sum. Example:
101 × 111 = 100011
Binary division
Works similar to long division. Compare the divisor to the current remainder, subtract when possible, and record 1s or 0s in the quotient. Example: 10111 ÷ 11 = 110 remainder 1.