Binary Calculator & Converter
Convert between binary, decimal, hexadecimal, and octal, perform binary addition, subtraction, and multiplication with carries and borrows shown bit by bit, or run bitwise AND/OR/XOR/NOT and shifts — with a place-value grid, two's complement explained visually, and a selectable 8, 16, or 32-bit register width.
Background
Computers store every number as a fixed-width string of bits, each one worth a power of two. The position of a bit — not just whether it's a 1 or a 0 — determines its value, exactly like place value in the decimal system you already know. Signed numbers add one more layer: two's complement lets the same bit pattern represent negative numbers without a separate sign symbol.
How to use this calculator
- Pick a bit width and whether numbers are signed first — these settings apply to every mode.
- Choose Convert to translate a number between binary, decimal, hexadecimal, and octal, with a place-value breakdown.
- Choose Arithmetic to add, subtract, multiply, or divide two numbers and see the binary carries, borrows, or partial products.
- Choose Bitwise & Shifts to run AND, OR, XOR, NOT, or a left/right shift directly on bit patterns.
How binary numbers work
Each bit position is worth a power of two, doubling from right to left: 1, 2, 4, 8, 16, 32... The rightmost bit is the least significant bit (LSB); the leftmost is the most significant bit (MSB).
A number's value is the sum of the place values of every bit that's a 1 — exactly like decimal place value, just with powers of 2 instead of powers of 10.
Hexadecimal groups bits in 4s (one hex digit = 4 bits); octal groups them in 3s. Both exist purely to write long bit patterns more compactly.
A register only has a fixed number of bits, so numbers that need more bits than that simply don't fit — this calculator flags that as an overflow rather than silently giving a wrong answer.
Two's complement lets the same bits represent negative numbers: flip every bit of a number and add 1 to get its negative. The most significant bit acts as a sign bit — 0 for non-negative, 1 for negative.
Formula & Equations Used
Place value: value = Σ (bit_i × 2^i), summed over every bit position i from 0 (LSB) up to n−1 (MSB)
Unsigned n-bit range: 0 to 2ⁿ − 1
Signed n-bit range (two's complement): −2ⁿ⁻¹ to 2ⁿ⁻¹ − 1
Two's complement negation: −x = (NOT x) + 1 — flip every bit, then add 1
Binary subtraction via addition: a − b = a + (−b), computed with the same bit-by-bit addition as a + b
Shift & add multiplication: for each 1-bit in the multiplier at position i, add the multiplicand shifted left by i places
Example Problems & Step-by-Step Solutions
Example 1 — Decimal to binary
Convert 170 to 8-bit binary.
Step 1: 170 fits under the largest power of two available, 128 (2&sup7;).
Step 2: 170 − 128 = 42; 42 − 32 = 10; 10 − 8 = 2; 2 − 2 = 0.
Result: 170 = 10101010 (128 + 32 + 8 + 2).
Example 2 — Addition with a carry
Add 77 (01001101) + 45 (00101101) in 8 bits.
Step 1: Add bit by bit from the right, carrying a 1 whenever a column sums to 2 or more.
Result: 122 = 01111010. No overflow — 122 fits comfortably in 8 unsigned bits (max 255).
Example 3 — Subtraction via two's complement
Compute 45 − 77 with signed 8-bit numbers.
Step 1: Negate 77 (01001101): flip to 10110010, add 1 → 10110011 (−77).
Step 2: Add 45 (00101101) + 10110011 using ordinary binary addition.
Result: 11100000 = −32 in signed 8-bit — matching 45 − 77 = −32.
Example 4 — Bitwise AND
Compute 11001010 AND 10110110.
Step: Compare each column — the result bit is 1 only where both inputs are 1.
Result: 10000010 — only bit positions 7 and 1 had a 1 in both numbers.
Frequently Asked Questions
Why do computers use binary instead of decimal?
Electronic circuits are built from switches that are most reliably read as two states — on or off, high voltage or low voltage. Binary maps perfectly onto that; representing ten distinct decimal digits electronically would be far less reliable.
What's the difference between signed and unsigned?
Unsigned numbers use every bit for magnitude, so an 8-bit unsigned register holds 0 to 255. Signed numbers (via two's complement) sacrifice the top bit as a sign indicator, so an 8-bit signed register instead holds −128 to 127 — same number of bit patterns, different meaning assigned to them.
Why is two's complement used instead of just a sign bit?
A plain "sign bit + magnitude" scheme needs separate circuitry for addition and subtraction, and has two representations of zero. Two's complement lets the exact same addition circuit handle subtraction automatically (by adding a negated number), and there's only one representation of zero — which is why virtually every modern processor uses it.
What's the difference between a logical and an arithmetic right shift?
A logical right shift always fills the vacated bits on the left with 0 — appropriate for unsigned numbers. An arithmetic right shift instead fills them by copying the sign bit, which preserves the number's sign and correctly divides a signed number by two (rounding toward negative infinity).
What happens when a calculation overflows the bit width?
The extra high-order bits are simply discarded, so the register shows a wrapped-around value that no longer matches the true mathematical result — this is exactly the bug class behind real-world integer overflow errors. This calculator flags it rather than hiding it.
Why do hex and octal exist if computers only use binary?
They're purely a convenience for humans reading and writing bit patterns. A 32-bit binary string is unwieldy to read; the same value in hexadecimal is a quarter as long (since each hex digit maps to exactly 4 bits) and much easier to work with by hand.
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