Number Base converter

Binary, octal, decimal, hexadecimal, and every other base in this list are all positional notation systems — the same idea applied with a different number of digit symbols. In any base b, each digit's contribution to the total value is the digit multiplied by b raised to that digit's position, counting from zero at the rightmost digit; decimal is simply the base-10 case everyone learns first, and the other bases work by exactly the same rule with a different b.

To read a number in any base as a decimal value, multiply each digit by the base raised to its position and add the results. Binary 1011 is 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11, and hexadecimal 2F is 2×16¹ + 15×16⁰ = 32 + 15 = 47 (the hex digit F stands for the value 15).

Going the other way — decimal to another base — repeatedly divide by the target base and record the remainder at each step, then read the remainders from last to first. Converting decimal 47 to binary: 47÷2=23 r1, 23÷2=11 r1, 11÷2=5 r1, 5÷2=2 r1, 2÷2=1 r0, 1÷2=0 r1; reading the remainders bottom-to-top gives 101111, which checks out against the hex example above since 47 is the same value as hexadecimal 2F.

Binary and hexadecimal convert into each other without passing through decimal at all, because 16 is exactly 2⁴: group the binary digits into sets of four from the right and convert each group independently. Binary 101111 padded to 00101111 splits into 0010 and 1111, which are hex 2 and F — giving 2F, the same answer as converting through decimal.

Binary and hexadecimal are the two bases built into how computers represent and display data — memory addresses, color codes, and byte values are almost always shown in hex because it packs four bits into one readable character — while octal once served a similar compact role and still appears in Unix file-permission notation, and higher bases like base 32 or base 36 are used in short-code schemes such as URL shorteners and ID generators, where every character needs to carry as much information as possible.

A base only has digit symbols for values from 0 up to one less than the base — base 8 has no digit "8" or "9," for instance — so bases above 10 borrow letters (A for 10, B for 11, and on up to Z for 35 in base 36) to represent the extra values, and forgetting that letters are standing in for numbers is a common source of errors when converting by hand. The other frequent mistake is reading the remainders from a decimal-to-base conversion in the wrong order: they come out last-first, not first-last, so the very first remainder calculated is actually the last digit of the answer.

Leading zeros can also be added or removed freely in any base without changing a number's value, which is why binary numbers are often padded out to a fixed width — 8, 16, or 32 digits — when working with computer memory; the padding is purely a formatting convention for lining values up consistently, not a change to the underlying number being represented.