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Binary Calculator
Binary Arithmetic
Binary to Decimal
Decimal to Binary
How to Use This Binary Calculator: A Step-by-Step Guide
Our free online Binary Calculator is designed for quick and accurate base-2 operations, perfect for computer science students, programmers, and digital electronics enthusiasts. Handle arithmetic like addition, subtraction, multiplication, and division entirely in binary, plus seamless conversions between binary and decimal. It supports unsigned positive integers, validates inputs for errors, and runs client-side for privacy and speed—no downloads or logins needed. Responsive for all devices, with clear results in monospace font for easy reading.
- Binary Arithmetic: Enter two binary strings (e.g., 1010 and 1100) in the fields. Click Add, Subtract, Multiply, or Divide. Results appear instantly in binary; errors like division by zero are flagged clearly.
- Binary to Decimal Conversion: Input a binary number (e.g., 1101) and hit Convert. The tool parses positional values (2^0 to 2^n) to output the decimal equivalent, ideal for understanding bit weights.
- Decimal to Binary Conversion: Enter a positive decimal (e.g., 13) and convert. It uses repeated division by 2, reading remainders from bottom to top—great for learning the algorithm behind it.
- Handle Errors Gracefully: Invalid inputs (non-0/1 chars) or unsupported ops (negatives in unsigned mode) show helpful messages. Use Clear to reset and retry.
- Pro Tips for Efficiency: Pad short binaries with leading zeros for alignment in manual checks. Keyboard: Tab between fields, Enter to compute. Mobile? Touch-optimized buttons with visual feedback.
- Explore Deeper: Scroll to our comprehensive guide below, with derivations, examples, tables, real-world apps, and tips tailored for 2025's binary-heavy tech landscape like AI and blockchain.
Quick Start: Try 101 + 11 = 1000. For conversions: 1010 (binary) = 10 (decimal); 15 (decimal) = 1111 (binary). Bookmark for coding sessions!
Binary Operations and Conversions: Formulas, Examples, and In-Depth Guide
Binary, the backbone of computing since the 1940s ENIAC, uses base-2 for efficient electronic representation—0 for off, 1 for on. This calculator emulates that logic with precise arithmetic and conversions, grounded in Boolean algebra and positional notation. Below, we break down each function: formulas from number theory, historical derivations, step-by-step examples, computation tables, applications in modern tech, and pro tips. Ideal for grasping why binary powers everything from CPUs to cryptocurrencies in 2025.
Binary Addition: Formula, Derivation, and Detailed Examples
Binary addition mirrors decimal but with base-2: sum bits with carry (1+1=10). Formula: sum = a + b (mod 2, with carry). Derivation: From Leibniz's 1703 dyadic arithmetic, extended by Boole's logic—each bit as a half-adder gate. In hardware, full adders chain for multi-bit ops.
Addition is core to CPU arithmetic logic units (ALUs). In 2025, it's vital for quantum bit summation in error-corrected qubits.
1. Align: 101 + 011.
2. Right to left: 1+1=0 carry 1; 0+1+1=0 carry 1; 1+0+1=0 carry 1; carry=1.
3. Result: 1000. Tool auto-computes for speed.
| a (Binary) | b (Binary) | Sum (Binary) |
|---|---|---|
| 10 | 1 | 11 |
| 101 | 11 | 1000 |
| 1110 | 101 | 10011 |
- Tip: Watch carries—they propagate like dominoes; verify by converting to decimal.
- Common Error: Forgetting leading zero pads—always align bits.
- Advanced: Ripple-carry vs. carry-lookahead for faster hardware adds.
Binary Subtraction: Formula, Derivation, and Detailed Examples
Subtraction uses borrowing (0-1=1 borrow 1) or two's complement. Formula: diff = a - b (with borrow mod 2). Derivation: Inverse of addition, from von Neumann's 1945 stored-program concepts—complements simplify negatives in binary.
Subtraction computes differences in version control diffs. In 2025, it's key for differential privacy in AI data anonymization.
1. Align: 110 - 010.
2. Borrow: 0-0=0; 1-1=0; 1-0=1 (no borrow).
3. Result: 100. Tool flags negatives as errors in unsigned mode.
| a (Binary) | b (Binary) | Diff (Binary) |
|---|---|---|
| 100 | 1 | 11 |
| 110 | 10 | 100 |
| 1010 | 11 | 111 |
- Tip: For negatives, switch to two's complement: invert +1 then add.
- Common Error: Borrow chains—trace from LSB to MSB.
- Advanced: Booth's algorithm for efficient multi-bit subtracts.
Binary Multiplication: Formula, Derivation, and Detailed Examples
Multiplication shifts and adds (AND for partial products). Formula: product = a × b (shifted ORs). Derivation: From Wallace trees in 1960s VLSI—parallel reduction of partial sums via carry-save adders.
Multiplication scales matrices in graphics. In 2025, it's optimized for neuromorphic chips in edge AI.
1. 11 shifted by 1 (for b=10): 110.
2. Add partials: Just 110 (no other bits).
3. Result: 110. Tool handles large shifts seamlessly.
| a (Binary) | b (Binary) | Product (Binary) |
|---|---|---|
| 10 | 10 | 100 |
| 11 | 10 | 110 |
| 101 | 11 | 1111 |
- Tip: Count set bits in b for partial adds—fewer ops.
- Common Error: Shift misalignment—start from LSB.
- Advanced: Karatsuba for logarithmic time on bigints.
Binary Division: Formula, Derivation, and Detailed Examples
Division restores via shifts and subtracts. Formula: quotient = a / b (shift-subtract loop). Derivation: Non-restoring algorithm from 1950s—avoids add after subtract for speed in hardware dividers.
Division computes quotients in file compression ratios. In 2025, it's in floating-point units for ML inference.
1. 10 into 10 (first bits)=1, subtract=0, bring down 0=0.
2. 10 into 00=0, bring down 0=100 /10=10 (quotient 100).
3. Result: 100 (remainder 0). Tool errors on /0.
| a (Binary) | b (Binary) | Quotient (Binary) |
|---|---|---|
| 100 | 10 | 10 |
| 1000 | 10 | 100 |
| 1101 | 11 | 10 |
- Tip: Align divisor length; track remainder for fractions.
- Common Error: Early shifts—restore if oversubtract.
- Advanced: SRT division for radix-4 speedup.
Binary to Decimal Conversion: Formula, Derivation, and Detailed Examples
Conversion sums powers of 2. Formula: dec = Σ (bit_i * 2^i). Derivation: Positional notation from al-Khwarizmi's 9th-century works, adapted to binary by Leibniz.
Conversions debug hex dumps in forensics. In 2025, they're in blockchain for wallet address verification.
1. Positions: 1*2^3=8, 0*2^2=0, 1*2^1=2, 1*2^0=1.
2. Sum: 8+0+2+1=11.
3. Result: 11. Tool automates for long strings.
| Binary | Decimal |
|---|---|
| 1010 | 10 |
| 1111 | 15 |
| 10000 | 16 |
- Tip: Rightmost bit is 2^0—double each position leftward.
- Common Error: Off-by-one powers—start from 0.
- Advanced: Horner's method: ((b_n*2 + b_{n-1})*2 + ... ) for efficiency.
Decimal to Binary Conversion: Formula, Derivation, and Detailed Examples
Conversion divides by 2, reads remainders. Formula: bin = remainders of dec / 2 (LSB first). Derivation: Euclidean algorithm variant, from ancient division methods digitized in Babbage's engines.
Conversions generate machine code opcodes. In 2025, they're in NFT metadata encoding.
1. 13/2=6 rem 1; 6/2=3 rem 0; 3/2=1 rem 1; 1/2=0 rem 1.
2. Read up: 1101.
3. Result: 1101. Tool reverses automatically.
| Decimal | Binary |
|---|---|
| 10 | 1010 |
| 15 | 1111 |
| 16 | 10000 |
- Tip: Stop at quotient 0; remainders are bits from bottom.
- Common Error: Reversing order—LSB is last remainder.
- Advanced: Binary search for bit positions in sparse data.
These binary fundamentals unlock deeper tech insights—from gates to GPUs. Practice here to master the code of computers. For bitwise ops, try advanced tools.