Binary to Decimal Converter
This free binary to decimal converter translates binary numbers into decimal — and converts decimal back into binary — instantly, right in your browser. Type into either box and the other side updates as you type, with the hexadecimal value shown as a bonus and every calculation step worked out so you can check the math.
Enter a number in either box
Type in either box — the other converts instantly. Spaces are ignored.
Your results
Type a binary or decimal number and the conversion will appear here.
How to use this calculator
Type your binary number into the top box — only the digits 0 and 1 are allowed. Every keystroke converts instantly, so there is nothing to submit; the Convert button just re-runs the conversion if you want. Spaces in the input are ignored, so 1010 1010 converts exactly like 10101010.
To go the other way, type a whole decimal number into the second box. The binary equivalent appears instantly, with the hexadecimal value shown as a bonus. Negative numbers, fractions, commas and letters are rejected with a clear message rather than a guess, so you always know what to fix.
Below the results you get the step-by-step working: converting from binary shows each bit's place value (powers of 2), and converting from decimal shows every division step so you can follow the math line by line.
How it works
Binary is a base-2 positional system: each position holds a power of 2 instead of a power of 10. Reading a binary number from right to left, the place values are 1, 2, 4, 8, 16, 32, and so on — that is, 20, 21, 22, 23, 24, 25, …
To convert to decimal, multiply each 1-bit by its place value and add the results: 1101 = 1×23 + 1×22 + 0×21 + 1×20 = 8 + 4 + 0 + 1 = 13. Bits that are 0 contribute nothing, which is why 1010 = 10 while 11111111 = 255.
To convert decimal back to binary, divide by 2 repeatedly and record each remainder: 42 gives the remainders 0, 1, 0, 1, 0, 1. Read those remainders from the last division back up to the first and you get 101010. The process stops when the quotient reaches 0, and the first remainder you wrote down becomes the rightmost bit.
Hexadecimal appears as a bonus: each group of four bits maps to one hex digit, so binary 11111111 is hex 0xFF and decimal 255. The calculator accepts up to 53 bits — decimal values from 0 to 9,007,199,254,740,991 — so every conversion uses exact integer math with no rounding. Anything outside that range is rejected with an explanation.
Assumptions: inputs are non-negative whole numbers only. Binary input may contain nothing but the digits 0 and 1 (spaces are ignored). Results are exact values, not estimates.
Frequently asked questions
How do I convert binary to decimal by hand?
Read the bits right to left and give each one a power of 2: the rightmost bit is 2⁰ = 1, then 2¹ = 2, 2² = 4, 2³ = 8, and so on. Add up the powers sitting under each 1. For example, 1101 = 1×8 + 1×4 + 0×2 + 1×1 = 13. This calculator shows that exact breakdown in the results.
How do I convert decimal to binary by hand?
Divide the number by 2 repeatedly and write down each remainder. For example: 42 ÷ 2 = 21 remainder 0, 21 ÷ 2 = 10 remainder 1, 10 ÷ 2 = 5 remainder 0, 5 ÷ 2 = 2 remainder 1, 2 ÷ 2 = 1 remainder 0, 1 ÷ 2 = 0 remainder 1. Read the remainders from the last division back to the first: 42 = 101010. This calculator shows every division step in the results.
Why does the calculator reject numbers like 102?
Binary only uses two digits, 0 and 1, so any other digit — like the 2 in 102 — makes the input invalid. The calculator flags exactly what's wrong instead of guessing. Remove the invalid digits and convert again; spaces are ignored automatically.
Is there a limit on how long the binary number can be?
Yes — this calculator accepts up to 53 bits, which covers decimal values from 0 to 9,007,199,254,740,991. That limit keeps every conversion exact, with no rounding. Longer binary strings are rejected with a clear message.