Binary Calculator
Add, subtract, multiply, and divide binary numbers with step-by-step breakdowns. See the decimal equivalents, bit counts, and every carry or borrow in the arithmetic. Free, private, and runs entirely in your browser.
Binary Arithmetic
Enter two binary numbers, choose an operation, and click Calculate to see the result with a step-by-step breakdown.
Step-by-step breakdown
Summary
What does this Binary Calculator do?
This free online binary calculator performs the four basic arithmetic operations on binary numbers: addition, subtraction, multiplication, and division. Binary is the base-2 number system used by every computer on the planet, so understanding binary arithmetic is fundamental to computer science, digital electronics, and programming.
Beyond just giving you the answer, this calculator shows the full step-by-step process. You'll see each column-by-column addition with carries, each borrow in subtraction, each partial product in multiplication, and each subtraction step in long division. It also converts everything to decimal so you can verify the result intuitively.
Everything runs locally in your browser. The numbers you enter never leave your device — no server, no logs, no tracking. You can calculate as many binary operations as you want, with no limits and no signup.
How to use the Binary Calculator
- Enter the first binary number: Type a binary value (only 0s and 1s) into the "A" field. The input accepts up to 32 bits.
- Choose an operation: Click one of the four operator buttons — +, −, ×, or ÷.
- Enter the second binary number: Type the second binary value into the "B" field.
- Click Calculate: The result appears instantly, along with the decimal equivalents and a step-by-step breakdown.
- Copy if needed: Use the "Copy Results" button to grab a formatted summary.
That's it. No signup, no email, no waiting. The entire process takes seconds.
Binary arithmetic rules
Binary arithmetic follows simpler rules than decimal because there are only two digits. Here's the complete reference.
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (0, carry 1)
1 + 1 + 1 = 11 (1, carry 1)
1 − 0 = 1
1 − 1 = 0
0 − 1 = 1 (borrow 1 from the next higher bit)
0 × 1 = 0
1 × 0 = 0
1 × 1 = 1
Multiply the multiplicand by each bit of the multiplier (0 or 1), then add the partial products shifted left.
If divisor ≤ partial dividend → write 1, subtract
If divisor > partial dividend → write 0, bring down next bit
Continue until all bits are used.
The key insight is that binary arithmetic is exactly like decimal arithmetic — just with only two digits. Carries work the same way (1+1 produces a carry), borrows work the same way (0−1 requires borrowing), and multiplication by a single bit is trivial (multiply by 0 or by 1).
Binary to decimal conversion reference
Each position in a binary number represents a power of 2, starting from 2⁰ on the right. Here's the reference table for the first 16 values.
| Binary | Decimal | Power breakdown | Hex |
|---|---|---|---|
0000 | 0 | — | 0 |
0001 | 1 | 2⁰ | 1 |
0010 | 2 | 2¹ | 2 |
0011 | 3 | 2¹ + 2⁰ | 3 |
0100 | 4 | 2² | 4 |
0101 | 5 | 2² + 2⁰ | 5 |
0110 | 6 | 2² + 2¹ | 6 |
0111 | 7 | 2² + 2¹ + 2⁰ | 7 |
1000 | 8 | 2³ | 8 |
1001 | 9 | 2³ + 2⁰ | 9 |
1010 | 10 | 2³ + 2¹ | A |
1011 | 11 | 2³ + 2¹ + 2⁰ | B |
1100 | 12 | 2³ + 2² | C |
1101 | 13 | 2³ + 2² + 2⁰ | D |
1110 | 14 | 2³ + 2² + 2¹ | E |
1111 | 15 | 2³ + 2² + 2¹ + 2⁰ | F |
To convert binary to decimal: multiply each binary digit by its positional power of 2 and add the results. For example, 1011 = 1×8 + 0×4 + 1×2 + 1×1 = 11.
Where binary arithmetic is used in real life
Binary isn't just a classroom topic — it's the foundation of everything digital.
Computer processors
Every CPU, GPU, and microcontroller performs arithmetic in binary. The ALU (Arithmetic Logic Unit) inside a processor executes binary addition, subtraction, multiplication, and division using logic gates. When you add two numbers in a program, the CPU is doing binary arithmetic at the transistor level.
Digital electronics
Digital circuits like adders, subtractors, and multipliers are built from logic gates (AND, OR, NOT, XOR). Full adders and ripple-carry adders are standard building blocks in digital design, and they implement the same binary addition rules this calculator shows.
Networking and IP addresses
IPv4 addresses are 32-bit binary numbers. Subnet masks, CIDR notation, and network ranges all rely on binary arithmetic. Understanding binary is essential for anyone working with network configuration or cybersecurity.
Cryptography
Modern encryption algorithms (AES, RSA, elliptic-curve cryptography) operate on binary data. Bitwise operations, modular arithmetic, and binary exponentiation are core to cryptographic primitives.
Programming and data representation
Every integer, character, image, and video is stored as binary. Bitwise operations (AND, OR, XOR, NOT, shifts) are used in low-level programming, compression algorithms, hash functions, and graphics programming. Understanding binary makes these operations intuitive.
Why use this Binary Calculator?
Instant results
No server round-trip means your answer appears the moment you click Calculate. Zero waiting.
Step-by-step breakdown
See each column addition with carries, each borrow in subtraction, each partial product in multiplication, and every step of long division.
100% private
Everything runs in your browser. The numbers you enter never leave your device — no server, no logs, no tracking.
Decimal side-by-side
Every result also shows the decimal equivalent, so you can sanity-check your answer at a glance.
No limits
Calculate as many binary operations as you want. No daily caps, no watermarks, no "upgrade to pro" nonsense.
Works offline
Once the page loads, you can use the calculator without an internet connection. Bookmark it for exam prep.
Common mistakes to avoid
- Forgetting the carry in 1+1. The most common error in binary addition is writing 1+1=2. It's actually 10 in binary (0 with a carry of 1).
- Mixing up borrow and carry. Carries happen in addition (going left, adding 1). Borrows happen in subtraction (going left, subtracting 1). They look similar but are opposite operations.
- Assuming binary subtraction works like decimal. The rule 0−1 = 1 with borrow is counterintuitive at first. The borrowed 1 becomes 2 in the current position, so 0−1 becomes 10−1 = 1 with a borrow.
- Forgetting to align binary numbers by the least-significant bit. When adding or subtracting, always align from the right (the 2⁰ position), not the left. Otherwise the place values don't match.
- Dividing without bringing down bits. Binary long division requires bringing down the next bit each time, just like decimal long division requires bringing down the next digit.
- Ignoring the sign bit in signed arithmetic. This calculator handles unsigned (positive) binary numbers. Signed binary (two's complement) is a separate topic with different rules.