Number Base Converter

The base of a numeral system is simply how many digits it has before it must start a new column. Decimal (base 10) uses 0-9; binary (base 2) uses only 0 and 1; octal (base 8) uses 0-7; hexadecimal (base 16) runs 0-9 and then borrows letters, A=10 through F=15. What stays the same across all of them is positional notation: each column is worth the base raised to a power, counting from zero on the right. In decimal, 255 means 2×10² + 5×10¹ + 5×10⁰. In binary the same quantity is 11111111 — 128+64+32+16+8+4+2+1. In hexadecimal it is FF — 15×16 + 15. Three very different-looking strings, one number. Nothing about the quantity changes when you change base; only the notation does. Base 10 has no mathematical privilege at all — it is an accident of having ten fingers, and computers, which have none, settled on something else.

Binary place values double as you move left: 1, 2, 4, 8, 16, 32, 64, 128. To convert a binary number to decimal, add up the place values wherever there is a 1. So 1010 is 8 + 2 = 10, and 1101 is 8 + 4 + 1 = 13. Going the other way, divide repeatedly by 2 and read the remainders from the bottom up. Or work greedily from the largest power that fits: 200 = 128 + 64 + 8, which gives 11001000. The reason any of this exists is physical. A circuit reliably distinguishes two states — voltage present or absent — and distinguishing ten would be far more error-prone. So every digital device stores everything in binary, and every other base on this page is just a more readable way of writing the same bits. Eight bits form a byte, which is why 255 comes up so often: 11111111 is the largest value eight bits can hold, giving 256 distinct values from 0 to 255.

Binary is correct and unreadable. A single 32-bit value is thirty-two characters of ones and zeros, and no human reliably spots a transposition in that. Hexadecimal fixes it with one clean property: 16 is 2⁴, so each hex digit maps to exactly four bits — a group called a nibble. The mapping is fixed and needs no arithmetic. F is 1111, A is 1010, 7 is 0111. Convert digit by digit and you are done: 0x2A is 0010 1010, which is 42. That makes two hex digits exactly one byte, and eight hex digits exactly one 32-bit word. A dump of memory in hex is four times shorter than the same dump in binary and converts back without a calculator. Decimal offers no such shortcut, because 10 is not a power of 2 — converting decimal to bits requires real division every time. The same trick is why octal exists: 8 is 2³, so each octal digit is exactly three bits. It was widely used on older machines whose word sizes were multiples of three bits, and hexadecimal displaced it once 8-, 16- and 32-bit architectures made groups of four the natural unit.

Web colors are the most common encounter. A CSS color like #FF0000 is three bytes written as three pairs of hex digits — red, green, blue, each from 00 to FF, that is 0 to 255. So #FF0000 is full red, #FFFFFF is white, #000000 is black, and #808080 is the midpoint gray. Three channels of 256 levels give 256³ = 16,777,216 colors, the "16 million" in a display spec. Memory addresses and pointers appear in hex because the digits align with the underlying bits, which matters when reading a crash dump. The 0x prefix is the standard marker for hexadecimal, mirrored by 0b for binary and 0o for octal in most modern languages. MAC addresses are six hex-encoded bytes; IPv6 addresses are written as eight groups of four hex digits precisely because a 128-bit address in decimal would be unreadable. Unicode code points use hex too — the U+ in U+1F600 introduces a hexadecimal number. And Unix file permissions are the classic octal survival. chmod 755 works because each permission group is exactly three bits — read, write, execute — and three bits is exactly one octal digit. 7 is 111 (rwx), 5 is 101 (r-x). The notation is not a coincidence; it is the base doing its job.

Three techniques cover almost everything. To decimal: multiply out the place values. Hex 1F4 is 1×256 + 15×16 + 4 = 500. From decimal: divide repeatedly by the target base and read the remainders bottom to top. For 500 into hex: 500 ÷ 16 = 31 remainder 4; 31 ÷ 16 = 1 remainder 15 (F); 1 ÷ 16 = 0 remainder 1. Bottom-up gives 1F4. Between binary, octal and hex: never go through decimal. Because all three bases are powers of two, you regroup bits directly — four at a time for hex, three at a time for octal, padding on the left. 11111111 splits as 1111 1111 = FF in hex, and as 11 111 111 = 377 in octal. A useful sanity check: the more digits a base has, the shorter the number. Any value will always be longest in binary, shorter in octal, shorter still in decimal, and shortest in hex. If a conversion produced a longer hex string than the binary original, something went wrong.