Computers speak binary — the foundation of all digital logic, programming, CPUs, memory addressing, and data representation. Our completely free binary calculator lets you perform accurate binary addition, subtraction, multiplication, division, plus instant conversions to decimal (base-10) and hexadecimal (base-16). Save your calculation history locally, enjoy a clean dark-mode UI, error-proof inputs, and mobile responsiveness. Perfect for computer science students, software developers, digital electronics learners, competitive programmers, or anyone wanting to understand how machines really calculate.
How to Use the Binary Calculator Effectively
1. Enter Only Valid Binary Numbers (0s & 1s)
Type or paste binary strings into both input boxes. The tool strictly enforces binary rules — only digits 0 and 1 are accepted. Any invalid character (letters, 2–9, spaces, symbols) triggers an immediate clear error message.
Tip: Great for copying binary from code, registers, memory dumps, or logic simulators.
2. Select the Desired Operation
Pick one:
- Addition (+) — Bit-by-bit with carry
- Subtraction (−) — Bit-by-bit with borrow
- Multiplication (×) — Shift-and-add style
- Division (÷) — Integer division via repeated subtraction
3. Click “Calculate” – See Triple-Format Results
Results appear instantly in:
- Binary — Raw arithmetic result
- Decimal — Easy-to-read base-10 value
- Hexadecimal — Compact 0x format (programming favorite)
4. Reset for Instant New Calculations
One-click reset clears inputs and output — ideal for testing sequences of problems quickly.
Standout Features of This Free Binary Calculator
⚡ Instant & Precise Binary Operations
Handles large bit lengths without slowdown.
↔ Full Conversions (Bin ↔ Dec ↔ Hex)
One-click multi-base visibility.
📜 Local History via Browser Cookies
Review previous calculations anytime.
🧼 Modern Dark, Responsive Design
Comfortable for long sessions on phone, tablet, or desktop.
🔒 100% Free • No Ads • No Login
Unlimited use, privacy-focused.
🛡️ Smart Input Validation & Guidance
Teaches binary rules through helpful errors.
Need more number system tools?
Binary ↔ Decimal Conversions – Detailed with Multiple Examples
Binary to Decimal: Positional Power-of-2 Method
Every position in a binary number represents a power of 2. We always start counting positions from the rightmost bit (called the Least Significant Bit – LSB), which is 2⁰ = 1.
Formula:
Decimal = Σ (bit × 2^position), where position starts at 0 from the right.
Example 1: Convert 1011₂ (binary) to decimal (₁₀)
| Bit (from right to left) | Position | Power of 2 | Calculation | Value |
|---|---|---|---|---|
| 1 (LSB) | 0 | 2⁰ = 1 | 1 × 1 | 1 |
| 1 | 1 | 2¹ = 2 | 1 × 2 | 2 |
| 0 | 2 | 2² = 4 | 0 × 4 | 0 |
| 1 (MSB) | 3 | 2³ = 8 | 1 × 8 | 8 |
| Total Decimal Value | 11 | |||
Quick reminder: 1011₂ = 8 + 2 + 1 = 11₁₀
Only the 1s contribute to the final value — that's why binary is so efficient for computers!
Quick Reference: Decimal 0–20 and Their Binary Equivalents
| Decimal | Binary |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 10 |
| 3 | 11 |
| 4 | 100 |
| 5 | 101 |
| 6 | 110 |
| 7 | 111 |
| 8 | 1000 |
| 9 | 1001 |
| 10 | 1010 |
| 11 | 1011 |
| 12 | 1100 |
| 13 | 1101 |
| 14 | 1110 |
| 15 | 1111 |
| 16 | 10000 |
| 17 | 10001 |
| 18 | 10010 |
| 19 | 10011 |
| 20 | 10100 |
Simple binary representation (no leading zeros)
Example 2: Larger number — 110101₂ = 32 + 16 + 4 + 1 = 53₁₀
Decimal to Binary: Repeated Division by 2
Record remainders bottom → top.
Example: 29₁₀ → binary
- 29 ÷ 2 = 14 rem 1
- 14 ÷ 2 = 7 rem 0
- 7 ÷ 2 = 3 rem 1
- 3 ÷ 2 = 1 rem 1
- 1 ÷ 2 = 0 rem 1
Binary: 11101
Binary Arithmetic Operations – Full Examples & Rules
Binary Addition Rules & Example
0+0=0, 0+1=1, 1+0=1, 1+1=0 carry 1
Example: 1101₂ (13) + 1011₂ (11)
1 1 0 1 (carry: 1 1 0) 1 1 0 1 + 1 0 1 1 --------- 1 1 0 0 0 → 24₁₀
Binary Subtraction Rules & Example
0-0=0, 1-0=1, 1-1=0, 0-1=1 borrow 1 (becomes 10-1=1)
Example: 11010₂ (26) − 1011₂ (11)
1 1 0 1 0 - 1 0 1 1 --------- 1 0 0 1 1 → 19₁₀
Binary Multiplication Example
Like decimal — shift & add when bit=1
Example: 1011₂ (11) × 101₂ (5)
1011 × 101 ------ 1011 (×1) 0000 (×0 shifted) 1011 (×1 shifted twice) ------ 110111 → 55₁₀
Binary Division (Integer) Example
Repeated subtraction or long division style.
Example: 110110₂ (54) ÷ 11₂ (3)
Dividend: 110110 Divisor: 11 Quotient: 10010 (18₁₀) Remainder: 0
Note: Tool supports positive integers only. For signed/binary negative numbers, consider two's complement tools separately.
Results Format & Common Troubleshooting
Results shown in three bases for easy verification and learning:
- Binary: Direct computation output
- Decimal: Standard base-10 equivalent
- Hexadecimal: 0x format (memory, colors, bitwise debugging)
Fixing Common Issues
- Invalid Input Error: Only 0 & 1 allowed — remove other characters
- Division by Zero: Shows clear warning — change divisor
- Negative Results: Unsigned calculator only — for negatives use two's complement elsewhere
- Unexpected Result?: Verify inputs/operator → reset → re-calculate
Jump in now — practice binary arithmetic, conversions, and build real confidence in how computers think!