Binary / Hex / Decimal Converter

Binary (base-2) uses only 0 and 1 to represent numbers. Each position represents a power of 2, starting from the right: 1, 2, 4, 8, 16 and so on. This converter translates between binary, decimal (base-10), octal (base-8) and hexadecimal (base-16) in any direction, which covers most of the number-base work in UK A-Level Computer Science.

At the lowest level all digital computing is binary, where 0 and 1 stand for off and on, false and true, low and high voltage, and a processor holds billions of transistors flipping between those two states. Storage sizes follow from the same idea, because they are powers of 2. A byte is 8 bits and holds 256 values, or 2⁸. A short is 16 bits and holds 65,536. An int is 32 bits and reaches 4.3 billion, while a long is 64 bits and reaches 18 quintillion.

Two methods convert a decimal number into binary. Repeated division by 2 records the remainders and reads them bottom-up, so 42 ÷ 2 = 21 remainder 0, 21 ÷ 2 = 10 remainder 1, 10 ÷ 2 = 5 remainder 0, 5 ÷ 2 = 2 remainder 1, 2 ÷ 2 = 1 remainder 0 and 1 ÷ 2 = 0 remainder 1, which read upwards gives 101010. Subtraction is quicker once you know the powers, since you take the largest power of 2 that fits, mark a 1, and repeat, so 42 = 32+8+2 and again 101010.

Going the other way is positional arithmetic. Reading right to left, the column values double from 1 through 2, 4, 8, 16, 32 and 64 to 128, which are the powers 2⁰ up to 2⁷. So 10110 unpacks as (1×16) + (0×8) + (1×4) + (1×2) + (0×1) = 16+4+2 = 22, and 11111111 reads as 128+64+32+16+8+4+2+1 = 255, which is 377 in octal and FF in hexadecimal. Memorising the sequence 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048 and 4096 makes all of this faster.

Binary addition needs only four rules, since 0+0=0, 0+1=1, 1+0=1 and 1+1=10 with a carry of 1. Adding 1011 and 0110 bit by bit with the carry gives 10001. Subtraction mirrors decimal but borrows 2 rather than 10. Negative values use two's complement, where you flip every bit and add 1, so an 8-bit −5 starts as 5 = 00000101, flips to 11111010 and becomes 11111011. The point of two's complement is that the same circuit can then handle both addition and subtraction.

Hexadecimal exists because long binary strings are hard for people to read. Base-16 uses the digits 0-9 followed by the letters A to F, where A stands for 10 and F for 15, with B, C, D and E filling the gap, and each hex digit maps onto exactly four bits, a nibble. Split 11010110 into 1101 and 0110 and it becomes D and 6, or 0xD6, which is 214 in decimal. That compactness is why hex shows up in HTML colours, where #FFFFFF is white and equals 16,777,215 in decimal, as well as in MAC addresses, memory addresses such as 0x7FFE and file dumps.