Information encoding and the binary system

A computer represents any information — a number, text, an image, sound — with just two symbols, 0 and 1. The reason is simple: in an electronic device it is more reliable to distinguish and store two states (for example, voltage present or absent). One binary digit is called a bit, and 8 bits make a byte. n bits can encode 2^n different states, so 1 byte expresses 256 values (from 0 to 255).

The binary number system is positional: the value of a digit depends on its place in the number. In the decimal system the places correspond to powers of 10 (1, 10, 100…), and in the binary system to powers of 2 (1, 2, 4, 8, 16…). For example, the number 1101 equals 1 × 8 + 1 × 4 + 0 × 2 + 1 × 1 = 13. To convert a decimal number to binary, divide it by 2 repeatedly and read the remainders in reverse order: for 13 the remainders are 1, 0, 1, 1, and in reverse order they give 1101.

To shorten a long binary notation, the hexadecimal system is used: each hexadecimal digit (0–9, A–F) corresponds to exactly 4 bits, for example 1010 1111 = AF. Python has these operations built in: bin(13) gives 0b1101, int("1101", 2) gives 13, and hex(175) gives 0xaf.

Text is also encoded with numbers: every character gets a numeric code. The ASCII table covers 128 characters (Latin letters, digits, punctuation and control characters); for example, the letter A has code 65. Unicode assigns a separate code to the characters of almost all writing systems in the world, and the UTF-8 encoding uses 1 to 4 bytes for each character: ASCII characters take 1 byte and Cyrillic letters take 2. An image is encoded as a grid of pixels: with 24-bit color each pixel takes 3 bytes (red, green and blue components).

Key takeaways

• A computer represents all information as a sequence of 0 and 1, that is, bits.
• 8 bits = 1 byte; n bits encode 2^n different values.
• The binary system is positional: each place corresponds to a power of two (1, 2, 4, 8…).
• When converting from decimal to binary, the remainders of division by 2 are read in reverse order.
• Text is encoded with ASCII and Unicode (UTF-8) codes, and an image with numbers giving the color of pixels.