Why is hexadecimal used in computing?

Because one hex digit is exactly four bits. A byte is two hex digits, so a hex value maps onto the bits without arithmetic, which decimal cannot do.

Why not convert decimal fractions?

Because a fraction that terminates in one base usually repeats forever in another. A tenth is exact in decimal and infinitely repeating in binary, which is the source of most floating point surprises.

Math & Statisticsnumber theory

Number Base Converter

Whole numbers converted between binary, octal, decimal, hexadecimal and base 36, with the bit width and an optional zero padding. Hexadecimal exists because one hex digit is exactly four bits, which is why memory addresses are written in it and not in decimal.

Also called: binary converter, hex to decimal converter.

Converted value

11111111

255 in base 10 is 11111111 in base 2, which is 255 in decimal. The value needs 8 bits, so it fits in a single byte. In base 2 it takes 8 digits, against 3 in decimal. Bases that are powers of two convert to binary by regrouping digits. Decimal has no such relationship to the bits, which is why computing uses hex and octal instead.

In decimal

255

Binary

11111111

Octal

377

Hexadecimal

Bits required

Digits in the result

On the digits

The value needs 8 bits, so it fits in a single byte. In base 2 it takes 8 digits, against 3 in decimal.

On representation

Bases that are powers of two convert to binary by regrouping digits. Decimal has no such relationship to the bits, which is why computing uses hex and octal instead.

How this is calculated

The same value in several bases

Base	Value	Digits
2, binary	11111111	8
8, octal	377	3
10, decimal	255	3
16, hex	FF	2
36	73	2

Method and background

How this is calculated

A number in any base is the sum of its digits multiplied by powers of that base, so converting means evaluating that sum and then repeatedly dividing by the target base to read the new digits off the remainders. Bases that are powers of two convert to and from binary without arithmetic at all: one octal digit is three bits and one hexadecimal digit is four, so the conversion is regrouping rather than calculating. That property is the whole reason hexadecimal is used in computing, where decimal has no such relationship to the underlying bits.

each digit is worth its face value times the base raised to its position, which is the definition every base shares

d_k: Digit at position k
b: Base

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

255 into binary

Value: 255
From base: Decimal, base 10
To base: Binary, base 2
Pad binary to this many bits: 0

Converted value11111111

the largest value a single byte holds

Open this example

hex back to decimal

Value: FF
From base: Hexadecimal, base 16
To base: Decimal, base 10
Pad binary to this many bits: 0

Converted value255

boundary: the round trip must return the same value

Open this example

Written about this

Why 1,00,00,000 has commas in the wrong places · 1 min · The Indian numbering system groups digits three, then two, then two. It is not a variant of the international system, and neither has a word for the other one.

Method and limits

What it assumes

Whole numbers, since a fraction rarely terminates in a different base.

What it deliberately does not model

Fractional values are not converted, because a terminating fraction in one base often repeats forever in another.
Values above the safe integer range lose precision.

Formula version 1.0.0 · definition 1.0.0 · India · Report a problem with this calculator

Frequently asked questions

Why is hexadecimal used in computing?: Because one hex digit is exactly four bits. A byte is two hex digits, so a hex value maps onto the bits without arithmetic, which decimal cannot do.
Why not convert decimal fractions?: Because a fraction that terminates in one base usually repeats forever in another. A tenth is exact in decimal and infinitely repeating in binary, which is the source of most floating point surprises.