616 Appendix
1. Dimensions
1.1 Definition of dimension, repeated from GC 12.
dimension : = (base-unit: unit)½ [dimension-expression]
1.2 Basic dimensions
time / [t] : = dimension(base-unit: second)
length / [l] : = dimension(base-unit: meter)
cost / [$] : = dimension(base-unit: dollar)
weight : = dimension(base-unit: kilogram)
power : dimension(base-unit: watt)
temperature : = dimension(base-unit: degree-centigrade)
voltage : = dimension(base-unit: volt)
current : = dimension(base-unit: ampere)
component / [c] : = dimension
operation / [o] : = dimension
information / [i] : = dimension(base-unit: bit)
state : = dimension(base-unit: state)
2. General units
2.1 Definition of unit, repeated from CC 12.
unit : = (dimension; conversion-list)½ unit-name : = multiplier unit½ simple-name
conversion : = number-name unit ½ number-name / unit½ arithmetic-expression (unit)
2.2 We give the basic units, but no variations with multipliers.
second / sec / s : = unit(dimension: time)
minute / min : = unit(dimension: time; conversion: 60 s)
meter / m : = unit(dimension: length)
foot / ft : = unit(dimension: length; conversion: 3.28 / meter, 12 in)
inch / in : = unit(dimension: length; conversion: 39.37 / meter, 12 / ft)
dollar / $ : = unit(dimension: cost)
operation / o : = unit(dimension: operation)
watt / w : = unit(dimension: power)
volt / v : = unit(dimension: voltage)
ampere / amp / a : = unit(dimension: current)
kilogram / kg : = unit(dimension: weight; conversion: 2.2 / lb)
pound / lb : = unit(dimension: weight; conversion: 2.2 kg)
3. Information units
3.1 Units
state : = unit(dimension: state; conversion: 2x bits)
binary-digit / bit / b : = unit(dimension: [i]; conversion: log2(x) states)
octal-digit / od : = unit(dimension: [i]; conversion: 3 bits)
decimal-digit / digit / d / dit rare: unit(dimension: [i]; conversion:
log2(10) bits, 1og10(x) states)
hexa-decimal-digit / hex : = unit(dimension: [i], conversion: 4 bits)
character / char / ch : = unit(dimension: [i]; conversion: 4 ~ 8 bits)
byte / by : = unit(dimension: [i]; conversion: 8 bits)
COMMENT The byte is almost standardized at 8 bits; occasional use otherwise, although not in this book.
3.2 I-units
i-unit : = base-unit ½
length X i-unit½
i-unit-name ½
(base-unit; length- list; content: product(length-list) base-unit; level:number(length-list))
i-unit-name : = i-unit-prefix i-unit-name ½
simple-name
i-unit-prefix : = + integer½
multiple/m½
quadruple/q½
triple/t½
double/d ½
*single/s ½
half/h ½
fractional/fr
base-unit : = unit(dimension: [i])
length : = + integer
The i-unit is a hierarchically organized information structure, in which each level consists of a number of subunits, all identically organized. The number of subunits in a level is called its length. Units eventually occur that cannot be decomposed further. These are called base-units and are some unit of information-e.g., the bit or the character. Thus, if the lengths are L1, L2, . . . , Ln and the base unit is the bit, then the total amount of information (the content of the i-unit) is L1 x L2 x . . . x Ln bits and the number of levels is n. The i-unit may be likened to an n-dimensional rectangular volume of information (except that the "dimensions"¾ the lengths¾ occur in a fixed order).
COMMENT
Almost all information in computer systems is organized in terms of i-units¾ e.g., a memory consists of a number of words, each of a number of characters, each of a number of bits. More exotic data structures are invariably encoded into i-units and are not reflected in the hardware.