Representing Data in a Fixed Number of Bits

Reasons for representing data in binary: computer hardware (CPU registers) works with a fixed word size — a fixed number of bits — which has real consequences for the range of values that can be stored.

Representing integers:

Overflow error

Occurs when a calculation produces a result too large (or too small) to fit within the fixed number of bits allocated — the value 'wraps around' or is truncated incorrectly, producing an unexpected/incorrect result.

Representing real numbers uses a mantissa and exponent (similar to scientific notation): the mantissa stores the significant digits, and the exponent stores the scale/position of the decimal point. This format can also overflow, and introduces accuracy trade-offs:

Rounding

Adjusting a number to the nearest representable value, which can round up or down.

Truncation

Simply cutting off the extra digits without rounding, always resulting in a value equal to or smaller than the original in magnitude.

Possible combinations of a fixed number of bits show up throughout IT:

• IP addressing — IPv4 uses 32 bits (giving ~4.3 billion addresses), IPv6 uses 128 bits (a vastly larger address space) — see 11.2.1.
• MAC address — typically represented using hexadecimal digits for compactness.
• Number of colours a pixel can display and screen resolution — more bits per pixel (colour depth) means more possible colours; e.g. 8 bits per colour channel gives 256 shades.

Shortening techniques: hexadecimal digits are used to represent binary values more compactly (since each hex digit represents exactly 4 bits) — this is why MAC addresses and colour codes (e.g. #FF5733) are written in hex rather than long binary strings.