Convert a binary number to decimal by summing the place values – each 1 bit contributes its power of two. The steps panel shows every power for your exact input.
Show calculation steps
Processed privately in your browser — nothing you paste is uploaded, logged or stored.
From bits to a value
A binary numeral is a compact sum: each digit multiplies a power of two, and the number is the total. Converting to decimal just evaluates that sum — which is why the method is called place-value expansion, and why the calculation-steps panel can show every term for your exact input.
Binary is base 2 because it uses two digit symbols. Decimal is base 10 with ten. Nothing else differs: both are positional systems, and the same digits-times-powers rule governs each.
Worked example
Input: 1101101 1×64 + 1×32 + 0×16 + 1×8 + 1×4 + 0×2 + 1×1 = 64 + 32 + 8 + 4 + 1 = 109
| Power | Value |
|---|---|
| 2^0 | 1 |
| 2^1 | 2 |
| 2^2 | 4 |
| 2^3 | 8 |
| 2^4 | 16 |
| 2^5 | 32 |
| 2^6 | 64 |
| 2^7 | 128 |
| 2^8 | 256 |
| 2^9 | 512 |
| 2^10 | 1024 |
Doing it by hand, quickly
Only the 1-bits contribute, so skilled readers scan for them and add the corresponding powers from the table above. A useful bound to remember: n bits can never reach 2ⁿ, so a 7-digit binary number is always under 128 — instant sanity checking for any result.
Where the conversion appears
Subnet mask arithmetic (11111111 11100000 → /19 → how many hosts?), reading sensor registers bit by bit, decoding flag fields in protocols, and every introductory computer-science course. It is the single most practised conversion in computing.
Accuracy and privacy
Arithmetic is exact at any length — a 300-bit pattern converts perfectly, thanks to BigInt. Signed interpretations are deliberately out of scope: a leading 1 here is just a digit, not a sign; when the pattern is a two’s-complement register, the Signed Binary to Decimal tool applies the sign conventions instead. Conversion runs locally — your bits are never transmitted.
How to use the Binary to Decimal Converter
- Type or paste the binary number (spaces and 0b prefix are accepted).
- Click "Convert to decimal".
- Open "Show calculation steps" to see every place value: the power of two behind each digit of YOUR number.
- Copy the decimal result with one click.
Frequently asked questions
How does binary to decimal conversion work?
Every binary digit has a place value that doubles right to left: 1, 2, 4, 8, 16… To convert, multiply each digit by its place value and add. For 1101101 that is 64+32+8+4+1 = 109 – and the steps panel writes this out for whatever you enter.
What is the largest binary number this handles?
There is no practical cap: BigInt arithmetic converts a 300-bit number exactly. Converters that rely on ordinary JavaScript numbers corrupt values past 53 bits – this one does not.
Can it convert a binary number with a fractional point?
Not this page – integers only, by design. The Binary Fraction to Decimal converter handles values like 101.101 using negative powers of two, with its own steps.
What happens if my input has an invalid character?
You get an error naming the character and its position – "Invalid binary digit "2" at position 3" – rather than a silently wrong answer. Spaces and underscores used as grouping are fine and simply ignored.
Does a leading zero change the value?
No. 000101 and 101 are both five: leading zeros affect only the written width, which matters in fixed-width contexts like registers, but never the value.
Is my number sent anywhere when I convert?
No – the sum of place values is computed in your browser. Nothing is uploaded, and the site's analytics record only that a conversion happened.
How do I check the result manually?
Add the place values of the 1 bits only. A quick sanity check: a binary number with n digits is always less than 2ⁿ, so 8 digits must give a result under 256. If your answer violates that bound, recount the digits.
