Binary, explained from zero

Binary looks alien and is actually the same number system you already use, with the dial turned down. Decimal has ten digits and places worth powers of ten; binary has two digits and places worth powers of two. Every skill transfers. Ten minutes here and the binary translator stops being a magic box and becomes a checker for conversions you can do yourself.

Place value is the whole trick

In decimal, 4,072 means 4×1000 + 0×100 + 7×10 + 2×1 — each place is worth ten times the one to its right. Binary works identically with two: places are worth 1, 2, 4, 8, 16, 32, 64, 128… doubling leftward. So 1011 means 1×8 + 0×4 + 1×2 + 1×1 = 11.

Why two? Because computers are built from switches, and a switch has two reliable states. Ten voltage levels per wire would be fragile; on/off is bulletproof. Binary isn't computers being clever — it's engineering choosing the alphabet hardware can't misread.

Counting in binary

Counting works exactly like decimal: run out of digits, carry left. 0, 1 — out of digits — 10, 11 — carry — 100, 101, 110, 111, 1000. The first sixteen numbers are worth recognising on sight:

Patterns worth noticing: odd numbers always end in 1; powers of two are a 1 followed by zeros; and all-ones is one less than the next power of two — 1111 is 15, one shy of 10000's 16, the binary cousin of 999 + 1 = 1000.

Decimal to binary by hand

The subtraction method reads best: repeatedly take out the largest power of two that fits. For 77: 64 fits (13 left), 32 and 16 don't, 8 fits (5 left), 4 fits (1 left), 2 doesn't, 1 fits. Mark the places you used — 64+8+4+1 — and you get 1001101.

The division method is more mechanical: halve repeatedly, noting remainders — 77→38 r1, 38→19 r0, 19→9 r1, 9→4 r1, 4→2 r0, 2→1 r0, 1→0 r1 — then read the remainders bottom-up: 1001101. Same answer, and the translator confirms either in a keystroke.

Binary to decimal by hand

Write the doubling places over the digits, add the ones under a 1. For 110101: places 32 16 8 4 2 1, ones at 32, 16, 4 and 1 → 53. With practice you stop writing the places and just walk the doubling in your head.

Bits, bytes and why 8

One binary digit is a bit; eight of them make a byte, which can hold 2⁸ = 256 values (0–255). Eight won for practical reasons — it was enough for a full character set, it's a power of two itself, and once IBM's System/360 standardised on it in the 1960s, the industry followed. Bytes are why so many computing numbers are 255, 256, 65,535 or 4,294,967,295 — they're all "largest value in N bytes" in disguise. It's also why hex color codes pair so naturally with bytes: one hex digit is exactly four bits, so a byte is always two hex characters.

How text becomes binary

Text is numbers wearing letters. ASCII, standardised in 1963, assigns each character a number: A is 65 (01000001), a is 97, the space is 32. "Hi" is 72 then 105 — 01001000 01101001. Modern text uses UTF-8, which keeps every ASCII character identical in one byte and extends to the rest of the world's writing (and emoji — 😀 is four bytes) with multi-byte sequences. Type a sentence into the translator and you're looking at exactly what your computer stores; the leading 010 on every capital letter, or lowercase letters sitting exactly 32 above their capitals (one flipped bit), stop being trivia and start being visible.