“On the shoulders of giant numbers”
http://www.iteror.org/big/book/ch1/ch1_1.html
bigΨ
Ψ.1. Natural repetition
chapter 1.1, edit 0.2.9
published: 20101105
updated: 20111230
# 1.1. Maths with makeup
at the entrance
of the heavenly circus
our garbage is a golden ticket
white clowns sing the “angelus”
and there’s a *salto mortale*
that’s not difficult after all
for ¥ou
– Franzes' poem
“The circle is close”
§1.1.1. Wild signs
Variables are written in lower case a,b,c,..
and can only be replaced by numbers or by expressions reducible to numbers.
The number substituted for a variable depends on the context –
parameters inside an expression where 1
is the only type of number unit
hold natural numbers {n≥1} larger than zero.
So when you choose to substitute a new type of number, such as pi
with its infinite digital expansion,
the justification for doing this will be a special number function,
most likely constructed from the negative 
unit.
A wildcard in upper case A,..,Z
holds the place of any word (sequence of characters)
allowed on that position in the expression.
Note that a wildcard can often also stand in for an empty string.
Accented wildcards X'
stand for a possible
next form of word X
after a first or intermediate
selection.
Wildcard R
(sometimes S,T
too)
is reserved to designate a row or part of a row of parameters.
In the following definition for a partial row R
with n
nonzero parameters, the empty row is substituted
when the
metarepetition
#n counts n=0 parameters.
The
incrementor
i
is repeated in the corresponding ellipsis
and incremented by 1
at each repetition.
The 3point ellipsis ...
is a wildcard for a sequence, specified by a
metarepetition
statement X#Y that follows after the expression.
A 2point ellipsis ..
can be used instead,
and if a metastatement doesn't follow this classic ..
relies on the imagination of the reader to continue its series
(common is a minimum of one item).
We notate repeating operator signs with a superscripted
ellipsis ×^{...}
or ×^{..}
An equation A=B
or A==B
states an exact reduction step of the left expression to the right.
An expression does not contain an
equality sign =
or an equal by iteration sign ==
or an approximation sign ~
or some comparison sign < > ≠
or substitution sign :=
or mixed sign thereof.
The inverse unit 
generating the negative integers *n
is treated in
chapter 2.2,
as is omega
ω
.
Inverse units are appended on the right n
of a number,
to tell them apart from subtraction n1.
Because we use e
as a 5th function parameter
and i
more as an incrementing index,
the classic signs e for the base of natural logarithms
and i = √1 for the square root of minus one,
are written as ê
and î = ^2^
You can find more special signs in the
Sign dictionary.
§1.1.2. Colour book
Metaexpressions
are written in violet font, and
backslash signs
can be too.
Within higher level subexpressions such as metaexpressions and subscripts
we prefer to use star operator notation (with addition as
zeroth star),
e.g. {t_{m1;}≥ab} specifies the comparison
a+b < t_{m+1;} as you'd usually write it.
Deeper
metastatements – owning others and therefore resolved first –
can get a deeper {{magenta}} colour.
To typify individual variables, wildcards, etc. these are subscripted as:
Sign,_{red,orange.;;}
To highlight an error and its consequences
you sometimes find red font and the
≠
not equal sign on the line.
Old style
expressions in brown font
use the operators +,×,^,^^,.. and apply old school
precedence.
Arrows and stars
a^^{...}b {^#m}
= a*^{...}b {*#m1}
both express
superpowers.
The two operations will result in the same number,
but the latter requires an extra star.
Operations with one or more arrows can be coloured
bluish simply to highlight them,
but formally this is done in a multiplicative context {0_{^}}
with countable arrows, where addition is the preceding operation
a+b
(of 1 arrows)
and writing a1
in the source is improper
(use a+1
outside of metaexpressions).
Also there's the issue of
precedence.
Operations with more arrows are resolved before those with less arrows
(majority precedence),
and then a smaller number of stars is evaluated first
(apply minority precedence to stars).

Minority star
operators have
text
colour*^{..}
and define the operatorial bigI. 
Countable
navy
majority arrows^^{..} {^#c}
can be written^_{c}
for bigE. 
bigA's plus
operations
+^{..}
green
are used in the construction of bigA. 
Sometimes an expression in a deviant context
^{ }
has the colourkryptonite
.
To improve the readability of bracketed expressions we've painted nested entries in marine colours.
We've provided for the headers of the subdivisions in this book to be helpful as anchors, stamped in various colours. A box will contain more advanced material, a comment is a note attached to the main text.
Part
Chapter
Subchapter
Section
Box
Comment
Paragraph
Code _{Sub}
note
_{
<source>}
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