Adding binary: 1011 + 1101
1011₂ = 11, 1101₂ = 13. Sum: 11 + 13 = 24 = 11000₂. Verify by column addition with carries: 1011 + 1101 = 11000. Used in CPU adder circuits.
Binary Calculator - Free Online Calculator
Add, subtract, multiply or divide binary numbers - and instantly see the result in binary, decimal, octal and hex. Built on BigInt so you can compute large binary values without precision loss.
- Formula shown on the page
- Runs in your browser
- Your inputs stay private
Digits 0 and 1 only
Digits 0 and 1 only
A in decimal
11
B in decimal
6
Result (binary)
10001
Decimal
17
Octal
21
Hex
11
Convert: decimal ↔ binary / hex / octal
Type a binary number above and read its decimal, hex and octal in the result row. To go decimal → binary, enter a decimal in our Scientific Calculator then convert.
128 in binary: 10000000 · 255 in binary: 11111111
Understanding the Binary Calculator
Binary (base-2) is the native language of computers - every byte of storage, every signal in a CPU, every bit transmitted over the internet is ultimately a sequence of 0s and 1s. Understanding binary is essential for computer science students, JEE Advanced aspirants (numerical methods section), embedded systems engineers, networking professionals, and anyone learning bit-level programming. This calculator handles addition, subtraction, multiplication, division and modulo on binary numbers of any length, using BigInt internally so precision is never lost regardless of magnitude.
Beyond pure binary arithmetic, the calculator simultaneously shows the result in decimal (base 10, our everyday system), octal (base 8, legacy Unix file permissions), and hexadecimal (base 16, used everywhere in programming for memory addresses, colour codes, MAC addresses). One hex digit equals exactly 4 binary bits - that's why hex is the preferred compact notation for byte-level work. Indian engineering curricula (B.Tech CSE, ECE, IT) include binary arithmetic in digital electronics, computer organisation, and operating systems courses; this calculator helps verify homework and understand cross-base relationships.
Using the Binary Calculator in 3 steps
Enter two binary numbers (digits 0 and 1 only).
Pick an operation: +, −, ×, ÷, mod.
See the result in all four bases.
How the Binary Calculator works
Binary inputs are parsed digit by digit and converted to BigInt for arithmetic. Negative numbers use a leading minus sign. Result is displayed back in binary, decimal, octal and hex for cross-reference.
Formula
Binary uses base 2: each digit represents a power of 2. 101₂ = 1×2² + 0×2¹ + 1×2⁰ = 5
- 0, 1Only valid binary digits
- Position nWorth 2ⁿ
How the Binary Calculator gets its answer
Binary representation: each position represents a power of 2. From right to left: position 0 is worth 1 (2⁰), position 1 is worth 2 (2¹), position 2 is worth 4 (2²), position 3 is worth 8, etc. So 1011₂ = 1×8 + 0×4 + 1×2 + 1×1 = 11 in decimal. Reading binary becomes second nature with practice.
Arithmetic in binary works just like decimal but with only two digits. Addition: 0+0=0, 0+1=1, 1+1=10 (write 0, carry 1). Subtraction uses borrowing. Multiplication is shift-and-add (multiplying by 2 just shifts left). Division by 2 shifts right. This shift-based arithmetic is much faster on hardware than decimal arithmetic - which is why computers use binary internally even though we display results in decimal.
Base conversions: 1 hex digit = exactly 4 binary bits (16 = 2⁴). So 11010110₂ = D6₁₆. 1 octal digit = exactly 3 binary bits (8 = 2³). So 11010110₂ groups as 11.010.110 = 326₈. These exact mappings make hex and octal natural 'compression' notations for binary - a 32-bit machine register reads as 8 hex digits, much easier than 32 bits. Decimal doesn't map cleanly to any power of 2, which is why programmers prefer hex over decimal for byte-level work.
Worked examples
Convert decimal 255 to binary, hex
255₁₀ = 11111111₂ = FF₁₆ = 377₈. This is the maximum value of an unsigned 8-bit byte. Useful for understanding why colour values 0-255 are used in RGB (8 bits per channel).
Subnet mask: 255.255.255.0 in binary
Each 255 = 11111111. Each 0 = 00000000. Full mask: 11111111.11111111.11111111.00000000. The leading 1s define the network portion; the trailing 0s the host portion. This is a /24 CIDR mask.
When to use the Binary Calculator
- B.Tech computer science homework - digital electronics, computer organisation.
- Software engineers debugging bit-level operations.
- Networking professionals computing subnet masks and IP addresses.
- Cryptography students working with bitwise operations.
- Embedded systems engineers programming microcontrollers.
- Game developers using bit flags for state management.
- Database engineers using bitmap indexes.
- Educational - understanding how computers represent information.
Common mistakes and gotchas
- Confusing binary 10 (decimal 2) with decimal 10. Context matters.
- Forgetting that binary digits are 0 and 1 only - 2 is not valid.
- Off-by-one in bit positions. Position 0 (rightmost) is 2⁰=1, not 2.
- Mixing signed and unsigned representations. Negative binary requires 2's complement convention.
- Confusing one hex digit (4 bits) with one byte (8 bits = 2 hex digits).
Get more out of this tool
- Memorise powers of 2 up to 2¹⁶: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536.
- Convert binary to hex by grouping 4 bits from the right. Each group becomes one hex digit.
- Convert binary to octal by grouping 3 bits from the right. Used in legacy Unix permissions (rwx = 421).
- For mental conversions, remember that 2¹⁰ = 1024 ≈ 10³, so a 10-bit number can hold ~1024 values.
- Pair with the hex, decimal-octal-hex and IP subnet calculators on this site for related base conversion work.
Questions about the Binary Calculator
Why does computing use binary?
Transistors have two stable states (off/on) which map naturally to 0/1. All computer storage and logic uses binary at the lowest level.
How do I convert decimal to binary?
Repeatedly divide by 2 and read remainders bottom-up. Or use this calculator - type a decimal and read the binary output.
What are 1's complement and 2's complement?
Methods for representing negative binary numbers. 2's complement is the modern standard - flip all bits then add 1. This calculator uses signed decimal style with explicit minus.
Why are bytes 8 bits?
Historical convention - early IBM computers used 8-bit chunks for character encoding, and the standard stuck. 1 byte = 8 bits = values 0-255 unsigned.
What's the largest binary number I can compute?
BigInt has no fixed limit. You can compute thousands of digits with no precision loss, though display becomes impractical past a few hundred bits.
How is hex related to binary?
One hex digit = exactly 4 binary bits. That's why programmers use hex - it's a compact notation that maps cleanly to binary.
Math in your browser
From simple percentages to scientific notation, statistics, geometry and big-number arithmetic - every common math problem you'd reach for a calculator for, runs in your browser with no installs. The fraction calculator preserves exact rationals, the big-number calculator uses BigInt for hundreds of digits without precision loss, and the scientific calculator's expression evaluator is a hand-written shunting-yard parser (no eval) that supports trig, log, factorial and parentheses.
If you're a student, double-check your work without paying for a graphing calculator. If you're a professional, use these tools as a sanity check on spreadsheet formulas. Every result is shown alongside its formula and inputs, so the math is always auditable.
The Binary Calculator is part of the Math suite on Online Calculator Tools, alongside 6 related calculators. All tools share the same UI patterns, the same privacy-first rendering (nothing is sent to a server), and the same commitment to showing the formula and assumptions behind every result.