# endless iteror Ψ #2

# Numeration

##
Frances
K^{*},
2016

Learn to create ever bigger numbers

in an article series on
site and
blog,

dedicated to Chelsea Manning hero.

© Kreative commons

## #2 Numeration

We will unravel how decimal notation works, see how large numbers are handled in history, and explore the boundaries of mathematics.

### 2.1 Radix notation

The numbers you use daily are in the decimal system.
This is a type of radix notation
with number base `10`

called ten

.
Although in any radix its base is written as
one zero

`10`

.

Here we show numbers in decimal base on an
*orange* background.

You've forgotten (internalized) how this actually works, it's complicated.

The base determines a set of
digits
in the range
`0`

to multiply the powers of the base with
and add that in series.**≤**d**<**10

This way a word composed of digits equals a unique number `n`

.

Show the popular radices, both in decimal notation (orange) and by simple counting of ones (yellow).

#### @

`binary`

2=:11

{0,1}`octal`

8=:11111111

{0,1,2,3,4,5,6,7}`decimal base`

10=:1111111111

{0,1,2,3,4,5,6,7,8,9}`hexadecimal`

16=:1111111111111111

{0,1,2,3,4,5,6,7,8,9,a,b,c,d,e,f}

We use the sign `=:`

to change between formats.

Click @ to put the digits of each radix on the board.

After you agree on a base, apply repeated multiplication to it.
Index each position `p`

by a radix power, starting right with
`10^0`

`=`

`1`

and `10^1`

`=`

`10`

and put larger powers `10^p`

on the left
(Arabic writing direction).

On each numeral place you can write a digit, that multiplies its radix' power. Then add up that series to get your number, as below.

#### @

2016= 1*6+`10`

*1+`10^2`

*0+`10^3`

in base*2`10`

2015= 1*7+`8`

*3+`8^2`

*7+`8^3`

in base*3=:3737`8`

2020= 1*0+`2`

*0+`2^2`

*1+`2^3`

*0+`2^4`

*0+`2^5`

*1+`2^6`

*1+`2^7`

*1+`2^8`

in base*1+`2`

`2^9`

*1+`2^`

10*1=:11111100100

2018= 1*2+

16*14+

16^2*7`=:`

in base7e2`16`

Radix notation is optimal,
because we can uniquely express all natural numbers up to any
`n`

within a minimum word space.
Any other system with a set of characters of radix size
will do worse.

This goes for all natural number bases,
but real bases are possible too. Try them in the above App:
number `10`

in base pi

for example.

Radices below `2`

show increasing overlap
as they approach the lower limit of `1`

,
A growing proportion of their digit series expresses the same numbers.
So radices in the range
`(`

could model the overlap produced by arithmetical systems
with a **1**,**2**)basic

number.

### 2.2 Trends in history

The mumbo-jumbo of radix notation soon became automated, else society would now be crippled. What luck that children in the ages before social media were able to learn their elementary operations in decimals. We stand on the shoulders of midgets, as well as giants!

The ancient Egyptians had eight numeral hieroglyphs
for the powers of ten up to `10000000`

that were directly added in any position to depict numbers
and baffle the crowd.

The Romans some `MDCLXVI`

years ago had just those seven letters to work with.
Yet the digit concept was already present in speech.
`D`

our `500`

was quingenti

in latin,
and `8000`

horses octo milia equorum

.

The left position of decimal digits is most important.
But powers of ten elude the human psyche after a few
`1000000000`

billion.

Bankers who believe
Reaganomics
will last, choose to ignore the impact of compound interest
and the large debt that produces.

Any growth must stop somewhere,
but exponential growth rapidly becomes unsustainable.

For example. If a family of two keeps growing at a general annual rate of
`1.1`

for **%**`2016`

years,
it will be larger than the current world population.
Enter that on a calculator.

`2`

*****1.011^2016`≈`

`7.6`

**E**9

Exponent

in scientific notation
means that you multiply the decimal factor on its left by
**E**9

to get the number, or often an approximation.*****10^9

A percentage sign `%`

is equivalent to an exponent

on the factor.
Multiply the current quantity with that number to get the increase.
Add the increase to the quantity, to get the next total.**E**-2

### 2.3 Physical limits

Radix notation seems more economical than
unary
notation, but the extra digit signs employed
for number input**/**output
do not help the operations in the throughput.
If your goal is to generate extreme numbers,
without regard for the unnumbered gaps in between:
then using just the character `1`

is most frugal,
an easy win.

Take the
tetration
`10^^`

for example.
A unary number **r1**`1..`

is written in exactly that many places.
In radix notation this space is a power of
`10`

smaller, that is
`10^^r`

digit places.

This reduction in word size is sufficient to express only the simplest tetrations, given the resources of our physical universe.

#### @

`3^^3 = 3^3^3 = 3^`

27`=:`

111**111111111111111111111111111`=`

7625597484987`≈ 8`

E12`4^^3 = 4^4^4 = 4^`

256`=`

10^(log(4)*256)`≈`

10^154.13`≈ 1`

exponential notationE154`2^^5 = 2^2^2^2^2 = 2^`

65536`=`

10^(log(2)*65536)`≈`

10^19728`≈`

double exponential notationE2E4`3^^4 = 3^`

7625597484987`=`

10^(log(3)*8)E12`≈`

E4E12`≈`

multi-exponential notationEEE1

Basically, a power of `10`

less subtracts `1`

from the tetration iterator.

But even in exponential notation,
to approximate moderate input size tetrations,
your output becomes too long.
Worse is, by reducing your system's precision
most numbers will escape the net.

Because we are physical beings we cannot uniquely express the majority of the numbers in the arithmetical sea between the scattered islands of tetration. Even with the help of computers we can only point out a small portion of these illusive numbers in the rather random choice of systems we can make.

According to
Seth Lloyd
the universe as a computer has at most
`10^90`

qubits,
less than `2^300`

binary places.
Then the radix representation of tetrations like
`4^^4`

and `2^^6`

by far defeats any known physical data capacity.

The human mind can envision even larger constructs: check out the
ancient record
`10^^(`

in buddhist poetry.**10^( 5*2^120)**)

Moving higher, we can still contemplate the strength of our rules, but the big numbers they produce lie wholly beyond imagination…

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