Data representation

📌 Key points

1. Computers use binary (base 2) because they are made of switches and logic gates that have only two states (on/off, 1/0), so all data must be converted to binary before it can be processed.
2. Denary is base 10, binary is base 2 and hexadecimal is base 16 (digits 0–9 then A–F, where A = 10 and F = 15); one hex digit stands for exactly 4 bits (a nibble).
3. Hexadecimal is used because it is shorter and easier for humans to read, write and remember than binary, so fewer mistakes are made; uses include MAC addresses, HTML colour codes, error codes and memory dumps.
4. Binary addition rules: 0 + 1 = 1, 1 + 1 = 10 (write 0, carry 1), 1 + 1 + 1 = 11 (write 1, carry 1); an overflow error happens when the result needs more bits than the register can hold, e.g. a value greater than 255 in an 8-bit register.
5. A logical left shift of n places multiplies a positive binary integer by 2ⁿ and a logical right shift divides it by 2ⁿ; bits shifted off the end are lost and zeros fill the empty places.
6. Two's complement stores negative integers: in 8 bits the most significant bit has the place value −128, so the range is −128 to +127; to make a negative number, write the positive value, flip every bit and add 1.
7. Text is stored using a character set: ASCII uses 7 bits (128 characters, extended ASCII uses 8 bits for 256), while Unicode uses more bits per character so it can represent many more characters, including other alphabets and emojis.
8. Sound is stored by sampling the analogue wave: the sample rate is the number of samples taken per second and the sample resolution is the number of bits per sample; increasing either gives a more accurate recording but a larger file.
9. An image is made of pixels: the resolution is the number of pixels (width × height) and the colour depth is the number of bits per pixel; image file size in bits = width × height × colour depth.
10. Units: 4 bits = 1 nibble, 8 bits = 1 byte, 1 KiB = 1024 bytes, 1 MiB = 1024 KiB, then GiB, TiB, PiB, EiB; lossy compression permanently removes data (e.g. JPEG, MP3), lossless compression (e.g. run-length encoding) lets the original be rebuilt exactly.

📖 Key terms

Overflow error

When the result of a calculation is too large to be stored in the number of bits available in the register.

Most significant bit (MSB)

The leftmost bit, with the largest place value; in two's complement it has a negative value.

Character set

All the characters a computer can represent, each with its own unique binary code.

Sample rate

The number of sound samples taken per second, measured in hertz (Hz).

Colour depth

The number of bits used to store the colour of each pixel.

Lossy compression

Reduces file size by permanently removing data; the original file cannot be restored.

Run-length encoding (RLE)

Lossless compression that stores a run of identical values as the value and the number of times it repeats.