Number base converter, binary, decimal, hex.

A number base converter rewrites one number using a different set of digits. Binary uses two digits, octal eight, decimal ten and hexadecimal sixteen, but the number itself does not change. This converter shows binary, octal, decimal and hex together, handles any base from 2 to 36, keeps every digit of very large numbers, and can show a number as a two's complement bit pattern of the length you choose. It answers binary to decimal, decimal to hex, hex to binary and every other pair in one box.

How to use the converter

1. Type or paste the number into Number. The default is 255.
2. Choose what you typed under From base: Binary, Octal, Decimal, Hexadecimal, or Other base for anything from 2 to 36. A 0x, 0b or 0o in front sets the base for you.
3. Read the rows. Binary, Octal, Decimal and Hexadecimal are always there. The fifth row follows Other base row, which starts at 36.
4. Press Copy beside a row for that answer, Copy all for every row, or Copy link for a link that opens with your number.
5. Tick Group the digits to space binary and hex in fours and octal and decimal in threes. Untick Capital letters for a to f instead of A to F.
6. Tick Fractions (a point) to convert a number such as 10.8 or 0.1, and set Places after the point.
7. Set the Bit length to see two's complement, and open The steps to see one conversion by hand. Choose which with Show the steps to.

Binary to decimal

Each binary digit is worth a power of two, counting from the right: 1, 2, 4, 8, 16, 32 and so on. Add the values where the digit is 1. Take 101101: that is 32 + 8 + 4 + 1, which is 45. The steps panel lays this out as a table of digit, place value and digit times place value.

Decimal to hex, and decimal to binary

Going the other way, divide by the new base again and again and keep the remainders. For 45 to binary: 45 / 2 = 22 remainder 1, 22 / 2 = 11 remainder 0, 11 / 2 = 5 remainder 1, 5 / 2 = 2 remainder 1, 2 / 2 = 1 remainder 0, 1 / 2 = 0 remainder 1. Read the remainders from the bottom up and you get 101101. The same method gives hex with 16: 4660 / 16 = 291 remainder 4, 291 / 16 = 18 remainder 3, 18 / 16 = 1 remainder 2, 1 / 16 = 0 remainder 1, so 4660 is 0x1234.

Hex to binary, and back

One hex digit is exactly four binary digits, so you can convert in groups. 1101 is D and 0110 is 6, so 11010110 is D6 in hex, which is 13 x 16 + 6 = 214 in decimal. Octal works the same way with three bits per digit. That is why hex is the usual way to write binary compactly.

Two's complement

Computers store negative whole numbers in two's complement. Pick a bit length, then flip every bit of the positive number and add 1. For -5 in 8 bits: 5 is 00000101, flipped it is 11111010, and adding 1 gives 11111011, which is 0xFB. The same pattern read as unsigned is 251. The converter shows both readings and the range for the length you chose, and says so when a number does not fit.

Fractions and numbers that never end

A number with a point can be exact in one base and endless in another. 0.1 in decimal becomes 0.000110011001... in binary, a pattern that repeats forever, which is why 0.1 + 0.2 is not exactly 0.3 in most programming languages. The converter cuts the digits off after the number of places you set and tells you when it did. Multiplying the fraction by the new base and keeping the whole part, one digit at a time, is the method shown in the steps.

Very large numbers

Whole numbers are not rounded. 18446744073709551616, which is 2 to the power of 64, becomes 1 followed by sixteen zeros in hex. A number up to 20,000 digits long is accepted.

Letters, text and other bases

Bases above 10 use letters for the digits after 9: a is 10, b is 11 and z is 35, so base 36 uses every letter. In base 36, zz is 35 x 36 + 35, which is 1295. When the number is also a character code, the Character row shows it, so 65 is A. When a long hex number is made of UTF-8 bytes, the Bytes as text row shows the words. For a full text translator, use the binary translator.

Common problems

• A digit is not allowed. The message names the character and its place. A 2 cannot appear in binary, and a g cannot appear in hex.
• The number has a point and nothing happens. Tick Fractions first.
• The hex answer looks wrong. Check the From base. The text 10 is 2 in binary, 8 in octal, 10 in decimal and 16 in hex.
• A negative number. A minus sign works in any row. For the bit pattern a computer stores, use the two's complement box.

For colours, the hex to RGB converter turns a hex code into red, green and blue.