Decimal, Binary and Hex, and Why Bit Width Matters

Quick Answer

Type a number and you get its bits. Paste bits and you get the number.

For example, Input: 2026. Output: 11111101010, alongside the hex value 0x7EA, the octal 0o3752 and a count of 11 bits. Those extras sit under the result because the question after a conversion is almost never just what is the value.

The binary converter reads the direction from what you paste. It also offers to strip a 0b prefix or the spaces in a nibble-grouped value, because that is the shape binary actually arrives in.

What is base 2?

Base 2 is the counting you already do, with two digits instead of ten.

In decimal, each place is worth ten times the place to its right: 209 is two hundreds, no tens and nine units. In binary, each place is worth twice the one to its right. For example, 1011 is one eight, no fours, one two and one unit, therefore 11.

That is the whole system. Everything else about binary follows from having only two symbols, which is convenient because a wire is either carrying current or it is not.

How does the conversion work?

Going to binary is repeated halving. Halve the number, write the remainder, repeat until nothing is left, then read the remainders backwards.

Take 11. Halving gives 5 remainder 1, then 2 remainder 1, then 1 remainder 0, then 0 remainder 1. Read upward: 1011. Going the other way is a single pass that doubles a running total and adds each bit.

This converter runs that arithmetic on arbitrary-precision integers rather than ordinary numbers. Unlike a converter built on plain JavaScript numbers, it stays exact above 2^53, so a 64-bit register value or a key fragment survives the round trip digit for digit. The ceiling is 512 bits, and anything past it gets refused instead of truncated.

Why use hexadecimal instead of binary?

Because one hex digit is exactly four bits, so translating between them needs no arithmetic at all.

1111 is F. 1010 is A. 11111111 is FF. A 32-bit value takes 32 binary characters, or eight hex ones, and eight characters can be read aloud, compared by eye and typed without losing your place halfway. That is why memory dumps, colour codes and MAC addresses come in hex while the hardware underneath works in bits.

Octal follows the same logic with three bits per digit. It survives mainly in Unix file permissions, where 755 maps cleanly onto three groups of three bits. Both values sit under every result here for that reason.

What does bit width mean, and why keep leading zeros?

Bit width is how many places your context reserves for the value, and it is separate from the value itself.

65 is 1000001, which is seven bits. Written as a byte it becomes 01000001. Written into a 16-bit register it picks up nine more zeros. None of those change the number, but they change whether two values line up when you put them one above the other.

Leading zeros are not decoration, and the common mistake is dropping them and then trying to compare two byte values by eye. Do not do that: 1000001 above 11111111 tells you nothing, while 01000001 above 11111111 shows the difference instantly. The converter reports the natural bit count so you know how far to pad.

Binary numbers vs binary text

These are two different jobs that produce confusingly similar output.

This tool turns the number 65 into 1000001. A binary text converter turns the letter A into 01000001. Those are almost the same bits, and that is a coincidence of the character set rather than a connection: A sits at code point 65.

The difference between them is what the input means. Numbers convert by place value and carry no fixed width. Text converts through a character encoding, pads to eight bits per character, and produces one group per letter. Instead of guessing, ask what you are holding. A count, an address or a mask belongs here. A word, a name or a message belongs in the binary text converter.

Why did my pasted binary need cleaning up?

Because binary is hardly ever written bare.

Source code writes 0b1010. Datasheets write 1010 1100 in nibbles so a human can read it. Numeric literals write 1010_1100. Every one of those characters is a parse failure in a tool that only accepts zeros and ones, which means the tool rejects most of the binary anyone actually has to hand.

So the converter offers the cleaned value in one tap rather than reporting a fault and stopping. Decimal input gets the same treatment for commas, spaces and apostrophes. A value that is already clean triggers nothing at all.

For the same bits in hex, use the hex encoder and decoder. Everything here runs in your browser, and nothing you paste is uploaded.