# Start to count
A superstar? Well right you are!
– John Lennon
(1st Beatle)
How do people define a largest number,
when she can always count 1
more?
# Only ones
Start with 1
(the unit one)
which is next to nothing
(zero)
and then add up all natural numbers as a series
1...
of units one.
Childlike finger counting is our final system of choice,
specially for writing
numbers that have become unwieldy large.
From input to output –
all expressions in this article are (in theory) to be reduced to
a string 1..
(of characters 1
repeated n
times).
So the final evaluation is a string (or word),
and this word (as well as its size) is the venerable
variable number value n
itself.
n = 1.. :n
mn = 1..1.. 1:m 1:n = m+n
That :n
right after the expression
indicates the 1
is to be written n
times.
Simple counting shows that a maximum number m
is hard to establish.
Indeed you can easily add some number n
to topple the
World Record
through mn
.
A truly democratic pursuit!
In this article we will climb the most majestic mathematical heights together. Standing upon the shoulders of gaint number enthusiasts, and the work they have done in the direction we want to go, which is… “Big, Bigger, Biggest!”
# Big
The Big part is about the construction of natural numbers. The normal quantities people use are not that big. Fresh up on your old school arithmetic and you are all ready there.
# Decimal numbers
The familiar decimal system defines small strings
{0,1,2,3,4,5,6,7,8,9,10,...}
in a unique number code that is humanly readable.
This number system with its
arabic numerals
is not so easy to learn. (you've done it, congratulations!)
For writing Big numbers its rules are not
helpful
at all,
but for people with small needs and ten fingers it has its benefits.
Even science journalists tend to get confused and make
mistakes
when calculating with large numbers.
The quantities dealt with in science and finance,
such as a million
1000000
= 1E6
,
a billion 1,000,000,000
= 1E9
and a trillion 1E12
have many decimals, but are often estimates.
It is easier to count decimal zeros,
than to write large numbers precisely.
For example: the
public debt
of the United States is about
2E13
U.S. dollar,
a 2
with 13
zeros,
as it grows multiplied by a percentage plus one,
no one knows it exactly.
# Scientific E
Scientific notation keeps track of the number of decimals. A scientist simply adds the number of zeros (power of ten) of two numbers to multiply them.
mEn = 10..*m 0:n
= 10*..m :n = m*10^n
mEn*pEq = (m+p)*10^(n+q) = mp*(1111111111**nq)
Some extreme examples of scientific and exponential notation:
There is a physical limit to the amount of information
we can manage. In the present universe
(8E60
Planck moments after the Big Bang)
the estimated
number of quantum bits
is 1E120
,
therefore
the binary state of our cosmos expresses at most a size
2^10^120
power tower
(power of a power).
And the history of this universe can be stored on a disk
as a random number smaller than 1E1E180
(the laws of nature should act to compress this number).
Somewhere inside this small universal number, a subsequence of quantum bits that is fighting entropy expresses a very much larger number. Can this be the work of God?
They help us deal with some “larger than life”
numbers, but decimal notation, its scientific extension by
E
and the elementary operators
+
*
^
are small sidesteps in our quest.
Sbiis'
cascading E
makes a jolly good attempt (we reviewed his
hyper E
notation) of extending the function of E
to a workable system for Big numbers.
But when we discuss some older number records
we prefer the common extension of powers ^
to
Knuth's arrows.
# Boundary of arithmetic
 We colour old school expressions brown, using the standard operator signs:

+
plus to adda+b
(takea
, add1
, repeatb
times), 
*
star to multiplya*b
(start with0
, adda
, repeatb
times), 
^
or**
for powersa^b
or exponentiation (init1
, multiply bya
, repeatb
times), 
^^
or***
for tetrationa^^b
(init1
, lifta
to this power, repeat thisb
times). 
Note that
10^n = 1En
and let10^10^n =
10^1En = 1E1En
.
The higher superpower operators
just repeat the preceding operators.
For example, tetration a***b
repeats a**..
a:b
(a power tower of exponents **a
).
Here, when a=10
we have 10^^b = 1E..1
:b
= E1#b
in
hyperE.
Higher uparrows ^^{...}
are reduced first (majority precedence),
then come superstars *^{...}
lowest first (minority precedence).
Equal operators are resolved from right to left.
Addition without an operator ab
immediately takes effect.
The plus +
postpones addition (without brackets)
until the other operations are done.
Instead of writing c
arrowheads in
a^^{..}b ^:c
we can count uparrows
a↑{c}b
by RepExp (our simple
RegExp for maths)
or in a 3entry array a→b→c
like chained arrows.
John H. Conway's
chained arrow notation
harnesses a very powerful yet natural
algorithm
(a row of double recursions), that we shall relate to on occassion.
Arithmetic has no boundaries, as long as we know how to expand its rules. The principle of recursion theory is to count a class of functions and to operate on its index directly. Then expand the function array and eventually the rule system, so that we can recursively increase the previous index and explode the array structure.