How binary stores data

A1.2.1 showed how to write numbers in binary. But a computer stores far more than numbers: text, photos, music, video. The trick is always the same: agree on a rule that turns the thing into numbers, then store the numbers in binary. The bits themselves carry no meaning. The same byte can be a number, a letter or a shade of grey, depending only on how a program reads it.

An unsigned integer is plain binary, as in A1.2.1: n bits store 0 to 2ⁿ − 1. To store negative numbers too, almost every computer uses two's complement. The idea is simple: the leftmost bit is worth minus its place value. In 8 bits it is worth −128 instead of +128, so:

• 00000101 = 4 + 1 = 5 (leftmost bit 0, an ordinary positive number)
• 11111011 = −128 + 64 + 32 + 16 + 8 + 2 + 1 = −5

So any number starting with 1 is negative, and 8 bits cover −128 to +127.

Why this odd rule? Because ordinary binary addition then just works for negative numbers too: the ALU needs no separate circuit for subtraction, it adds the two's complement instead. And there is only one zero. The cost is a lopsided range: −128 has no +128 to match it in 8 bits.

When a result needs more bits than are available, it wraps round to a wrong value: that is overflow. In 8-bit two's complement, 100 + 50 gives −106, because 150 does not fit.

To store text, every character is given a number by a character set. ASCII was the first widely used one: 7 bits, so 128 codes, covering English letters, digits, punctuation and invisible control codes. ‘A’ is 65, ‘a’ is 97, the digit ‘0’ is 48. Letters are in order, so ‘B’ is 66, which is how programs sort text alphabetically.

128 characters is nowhere near enough for the world's languages. Unicode gives every character in every writing system its own number, a code point, written like U+0041 (that is ‘A’ again: Unicode keeps the ASCII codes). Unicode now covers well over 100,000 characters, emoji included. To store those numbers, the web uses UTF-8: characters in the ASCII range take 1 byte, others take 2, 3 or 4.

A bitmap image is a grid of pixels, each storing a number for its colour. Two things fix the size: the resolution (width × height in pixels) and the colour depth (bits per pixel). With n bits per pixel there are 2ⁿ possible colours: 1 bit gives black and white, 8 bits give 256 colours, 24 bits (one byte each for red, green and blue) give about 16.7 million.

Sound is a smooth, continuously changing wave: it is analogue. To store it, the computer measures it at regular moments and stores each measurement as a number. How many times per second is the sample rate; how precisely each measurement is stored is the bit depth. Raise either and the stored sound is closer to the real one, and the file grows.

A video is a sequence of still images, called frames, shown quickly enough that the eye sees smooth movement: the frame rate, usually 24 to 60 frames per second. Each frame is stored like a bitmap image. Uncompressed, one second of full-HD video at 30 frames per second is 1920 × 1080 × 24 × 30 = 1,492,992,000 bits, about 187 MB, every second.

A video file holds more than frames. A container format such as MP4 or MKV packages several streams together: the video, one or more audio tracks (stored as in the sound section above), subtitles, and metadata such as the resolution, frame rate and length. The streams are interleaved in small chunks, so the sound for each moment arrives alongside its pictures and stays in sync during playback.

Because raw frames are so large, video is always compressed (A1.1.8), in two ways at once:

• Spatial compression (within a frame): each frame is compressed like a photo, removing detail the eye will not miss.
• Temporal compression (between frames): consecutive frames are usually almost identical, so only some frames are stored in full. For the frames in between, only what changed is stored, such as “this block moved 3 pixels left”. A talking head against a still background compresses enormously; fast action with lots of movement does not.

The way data is encoded decides how much space it takes, how quickly it can be found again, and whether it survives being stored or sent:

• Fixed width makes data quick to find. If every item takes the same number of bits (every integer 32 bits, every pixel 24 bits), the computer can jump straight to item number n by working out n × width, instead of reading everything before it. Variable-width encodings such as UTF-8 save space but must be read from the start.
• The program must know the format. Bits carry no meaning on their own (Figure 1). A file therefore starts with metadata, a header saying what kind of data follows and how it is laid out (an image's width, height and colour depth).
• Standard formats mean data can be shared. Because everyone agrees on ASCII, UTF-8, PNG or MP3, a file written on one computer can be read on any other.
• Extra bits can catch errors. Bits can be flipped by electrical noise or a damaged disc. A parity bit is one extra bit chosen so that the total number of 1s is even. For the 7 bits 1101011, which contain 5 ones, the parity bit is 1, giving 11010111. If any one bit later flips, the count of 1s becomes odd and the error is detected (though not where it is).