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Binary Calculator | Convert Bases And Perform Binary Arithmetic

Convert instantly between binary and decimal, and perform binary arithmetic and bitwise logic like AND, OR and XOR. Each result includes the place-value breakdown, showing why binary place values are powers of two — ideal for computer science students and low-level debugging.

Last updated: June 2026 · Free · No sign-up required

Results

Enter values above and click Calculate to see your result instantly.

Quick reference

BinaryDecimal
11
102
1015
10008
111115
1000016

functions Shows the working, not just the answer

For students and teachers, the final number is only half the value. Every math tool here exposes the formula it applied, the intermediate steps, and the rounding rule, so you can follow along, check your homework, or use the answer in a proof or report with confidence.

calculate Accurate to the spec

Calculations use 64-bit floating point with sensible rounding for the domain (currency to 2 decimals, percentages to 4 decimals, algebra to 6 significant figures). Where exact rational arithmetic matters — fractions, factorials, simplification — we use a dedicated BigNumber path so 1/3 + 1/6 returns ½, not 0.49999.

school Free for classroom use

Educators are welcome to link to any math calculator on CalcaTools from a class site, Google Classroom, or worksheet. The pages are mobile-friendly, free, ad-supported (so we can keep them free) and have no sign-up wall — students just click and use them in class or at home.

tips_and_updates Pair with the spoke articles

Below the calculator we link a small set of plain-English explainer pages — "What is a percentage?", "Why does PEMDAS matter?", and so on. They cover the underlying concept in 4–6 short paragraphs. Read those before the calculator if the topic is new, or after if you want the extra context.

Interpretation guide

OperationBinary rule
AdditionCarry when sum reaches 2
SubtractionBorrow 2 from next place
ConversionSum powers where bit = 1

lightbulb Decoding a sensor register: 11010110

A microcontroller status register reads 11010110. To find which flags are set, convert it to decimal: write each bit over its place value (128, 64, 32, 16, 8, 4, 2, 1) and add the values where the bit is 1.

Result: 128 + 64 + 0 + 16 + 0 + 4 + 2 + 0 = 214. In hexadecimal that's 0xD6 (13×16 + 6), and in octal 326.

What this means: The decimal 214 gives the register's numeric value, while the bit pattern shows exactly which of the 8 flags are active — bits 7, 6, 4, 2, and 1. That's the real payoff of binary: one glance at the pattern reveals the hardware state.

Formula & methodology

Formula: Binary place values are powers of 2

How binary conversion works

Binary is base 2: each digit is 0 or 1, and each place value doubles from right to left (…, 32, 16, 8, 4, 2, 1). To convert to decimal, add the place values under every 1 bit — 1101₂ = 8 + 4 + 0 + 1 = 13. To convert decimal to binary, repeatedly divide by 2 and read the remainders bottom-to-top.

  • AND: 1 only where both bits are 1 — used to mask flags.
  • OR: 1 where either bit is 1 — used to set flags.
  • XOR: 1 where the bits differ — used to toggle flags.

Signed numbers use two's complement, where the top bit carries negative weight: 11111111₂ reads as −1, not 255. One hexadecimal digit always equals exactly four bits, which is why programmers write 0xD6 instead of 11010110.

Authoritative source: Wolfram MathWorld

Frequently asked questions

How do I use the binary calculator?
Enter a binary number using only 0s and 1s in the first field, or a decimal number in the other, and the tool converts between them instantly. You can also choose an operation such as AND, OR, or addition to combine two binary values. Every result includes the place-value breakdown, which shows why binary place values are powers of 2.
How does binary-to-decimal conversion actually work?
Each bit sits on a power of two — 1, 2, 4, 8, 16, doubling right to left. Add the place values where the bit is 1: 11010110 = 128 + 64 + 16 + 4 + 2 = 214. The page shows this breakdown for every conversion so the math is never a black box.
How do I read 11111111 as a negative number?
As unsigned 8-bit it's 255, but in two's complement — the standard signed representation — it's −1. Computers store negatives this way so addition circuits work unchanged. The converter notes both readings so a register dump can't mislead you.
Why do my results differ from another calculator?
The Binary Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. This binary calculator converts between binary, decimal, hexadecimal, and octal, and performs addition, subtraction, multiplication, and division on binary numbers directly. Enter values in any base and the tool shows the full conversion and arithmetic There is no limit on usage, and it works on any device.
What do the AND, OR, and XOR operations do?
They compare two bit patterns position by position: AND keeps a 1 only where both bits are 1, OR where either is 1, XOR where they differ. Masking a register with AND isolates specific flags, which is why these ops are bread and butter for embedded debugging.
Can it convert hexadecimal and octal too?
Yes — binary, decimal, hex, and octal all convert between each other. One hex digit is exactly 4 bits, so 0xD6 splits into 1101 0110; one octal digit is 3 bits. Programmers use hex as shorthand because long binary strings are hard to read.
Is base conversion exact or approximate?
Exact — converting between integer bases involves no rounding at all. 214 becomes 11010110 and converts back bit-for-bit, because each place value is a precise power of two. What you see is mathematically exact, not a floating-point estimate.

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