General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

mapping ψb. This is the the same process we did for the logics; we are now doing it for number
systems. However, here, we also want to ensure that the meaning mapping ψb is a bijection, since
we want to define the arithmetics on the encodings by reference to The arithmetical operators on
the unary natural numbers.
We can look at this as a bootstrapping process, where the unary natural numbers constitute
the seed system we build up everything from.
Just like we did for string codes earlier, we build up the meaning mapping ψb on characters from
Db first. To have a chance to make ψ bijective, we insist that the “character code” ϕb is is a
bijection from Db and the first b unary natural numbers. Now we extend ϕb from a character code
to a string code, however unlike earlier, we do not use simple concatenation to induce the string
code, but a much more complicated function based on the arithmetic operations on unary natural
numbers. We will see later 14 that this give us a bijection between Db + and the unary natural EdNote:14
numbers.
Commonly Used Positional Number Systems
Example 408 The following positional number systems are in common use.
name set base digits example
unary N1 1 / ///// 1
binary N2 2 0,1 01010001112
octal N8 8 0,1,. . . ,7 630278
decimal N10 10 0,1,. . . ,9 16209810 or 162098
hexadecimal N16 16 0,1,. . . ,9,A,. . . ,F F F 3A1216
Notation 409 attach the base of N to every number from N . (default: decimal)
Trick: Group triples or quadruples of binary digits into recognizable chunks
(add leading zeros as needed)
1100011010111002 = 01102 00112 01012 11002 = 635C16
616
316
516
C16
1100011010111002 = 1102 0012 1012 0112 1002 = 615348
68
18
58
38
F 3A16 = F16
316
A16
= 1111001110102, 47218 = 48 78 28 18 = 1001110100012
11112 00112 10102 1002 1112 0102 0012 48
c○: Michael Kohlhase 239
We have all seen positional number systems: our decimal system is one (for the base 10). Other
systems that important for us are the binary system (it is the smallest non-degenerate one) and
the octal- (base 8) and hexadecimal- (base 16) systems. These come from the fact that binary
numbers are very hard for humans to scan. Therefore it became customary to group three or four
digits together and introduce we (compound) digits for them. The octal system is mostly relevant
for historic reasons, the hexadecimal system is in widespread use as syntactic sugar for binary
numbers, which form the basis for circuits, since binary digits can be represented physically by
current/no current.
Now that we have defined positional number systems, we want to define the arithmetic operations
on the these number representations. We do this by using an old trick in math. If we have
an operation fT : T → T on a set T and a well-behaved mapping ψ from a set S into T , then
we can “pull-back” the operation on fT to S by defining the operation fS : S → S by fS(s) :=
ψ −1 (fT (ψ(s))) according to the following diagram.
14 EdNote: reference
131