# 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

Gothic queen with crown and crucifix bends over in forest

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 add a+b (take a, add 1, repeat b times),
* star to multiply a*b (start with 0, add a, repeat b times),
^ or ** for powers a^b or exponentiation (init 1, multiply by a, repeat b times),
^^ or *** for tetration a^^b (init 1, lift a to this power, repeat this b times).
Note that 10^n = 1En
and let 10^10^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 hyper-E.

Higher up-arrows ^... 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 up-arrows a↑{c}b by RepExp (our simple RegExp for maths) or in a 3-entry 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.

by Giga Gerard

People can read this article plain or choose the hard & fast & open version.