#
*“On the shoulders of giant numbers”*

http://www.iteror.org/big/book/ch1/ch1_5.html
big**Ψ**

http://www.iteror.org/big/book/ch1/ch1_5.html

## Ψ.1. Natural repetition

#####
chapter 1.5, *edit 0.2.9*

published: *2010-11-07*

updated: *2011-12-30*

### # 1.5. Elementary powers

At the ninth hour Jesus cried with a loud voice,
"Eloi! Eloi, lama sabahkthani?"

Which means, "God, my God, why have you left me?"

#### §1.5.1. Exponentiation is a hypercube

With addition `+`

as the
empty
operation, any number of operators `+..`

or `+`

will be empty too.
Without operation there is just separation,
where all pluses function as spaces _{n;}`0`

adding nothing but _{*}`0`

in effect.

Such meaningless separators that represent the `0`

concept
can be used to define the ground state of an ordered structure.
This helps to show, for example that
multiplication `a*b`

contains an extra repetition concealed in its left item `a`

,
simply because this is a number variable.

3*2 = 1+1+1++

1+1+1++
(rectangle)

3*3 = 1+1+1++

1+1+1++

1+1+1++
(square)

a_{i}*... `{a _{i}#n}` =
1+....

`{+..#n #a`(box)

_{i}}
Single `+`

divide the space inside a natural number,
separating the units `1`

that count that number.
Lower constructs can be ordered visually in a higher dimension
by multiple plus signs or with comma `,`

separators.
We think of constructions like these in terms of dimensions,
occupying a *discrete space* (not continuous).

This way a rectangle with more than 2 dimensions can be formed,
which is called a hyperrectangle or simply a box,
when the size of any dimension may differ from that of other dimensions.

Special boxes are the *hypercube* with a higher dimensional array where
all dimension sizes `a`

are equal, the initial number value
_{i}`a`

_{0} = 1+..`{1#a}` included,
though the number of dimensions may differ.

And then the *dimension box*
where the number of dimensions equals the size of each subdimension, but where
(as specified) a different parameter value `a`

is allowed.

Usually exponentiation is pictured as a series of (equal) numbers `a`

connected with operators `*`

of (the preceding operation of) multiplication.
But work this out completely and exponentiation in its *ground state*
is just the symmetric ordering of units `1`

in a multidimensional cube or hypercube.

The example below is kept brief by omitting unit level separators
from the
dual repetition.
Of the two power patterns, the first reduces to a row of multiplications,
the second directly fills a hypercube with ones.

3**4 = 3*3*3*3
(4-dimensional cube or
tesseract)

= 1+.... `{+..#4 #3}` =
111,.... `{,..#3 #3}`

= 111,..,..,.. `{#3 #3 #3}`

=
111,111,111,, 111,111,111,, 111,111,111,,,

111,111,111,, 111,111,111,, 111,111,111,,,

111,111,111,, 111,111,111,, 111,111,111,,,*
= 81*

a**b = a*.. `{a#b}`
(repetition over a row is actually..

= 1+.... `{+..#b #a}`
..repetition over dimensions..

= a,.... `{,..#b- #a}`
..in a hypercube)

*Exponentiation* expressed as sequences of `1..`

separated by empty `,`

is equally void an operation as *addition*.
However, its virtue lies not in the power to make things happen

,
but in the sheer economy of writing numbers in the exponential format
`a^b`

when its *ground state*
of repeated or added groups of ones had caused a lot of overhead.

This chapter meant to remind you of the true nature
of the exponential function – it creates *hypercubes!*

#### §1.5.2. Googol world

number | US-name | SI-prefix | |
---|---|---|---|

10^3 | thousand | k | kilo |

10^6 | million | M | mega |

299792458 m light-second | |||

10^9 |
billion | G | giga |

13.75 Ga age of our universe | |||

10^12 |
trillion | T | tera |

10^15 |
quadrillion | P | peta |

9460730472581 km light-year | |||

10^18 |
quintillion | E | exa |

10^21 |
sextillion | Z | zetta |

10^24 |
septillion | Y | yotta |

10^27 |
octillion | H | hella * |

6.187151E34 length units/meter | |||

1.854860E43 time units/second |

In decimal notation we use `10^n` powers of ten
`10...` `{0#n}` to express numbers,
and round off larger numbers to `10^(3×n)`
powers of a thousand `1000,...` `{000#m}`
and convert these to names.
But the words differ, past a million `1000,000`
people have to make a choice between the short scale

American system
or the long scale

old European.

Our advice is to follow the Americans and continue with a billion
and a trillion, but after that to give up on names
and use calculator speak

instead!
It's real easy to parrot E fifteen

`1E15` (a
quadrillion)
or two E twenty

`2E20`
(200 quintillion)
from the calculator screen and also handy,
given that we multiply by adding up exponents.

So the folks on planet Earth express large numbers with individual names
that have some translation to a power of ten.
What's important then is to reach an international consensus on a system of construction,
which is transparent and practical in its application.

The table on the right lists the current conventions for integer magnitudes
and throws in some astronomical sizes and Planck units.
A computer geek would prefer to write a number in its binary form
and then count the required bits or file size in bytes, where
`1 KB` `= 8×2^10 bit`

Because this is a book about Big numbers we owe you the
googol and the googolplex.

The two numbers became part of popular maths culture when
the founders of the world's main search engine adopted the name
Google
in 1997 to reflect their mission to organize
a seemingly infinite amount of information on the web

.
The Google index stores nearly `100PB` of web page data as of June 2010.
Its company building and campus is called the *Googleplex*.

Both names were originally put forward by a 9 year old Jewish boy
_{
<Milton Sirotta>}
in 1920 (with the traditional dessert
Gogl-Mogl as a likely inspiration)
to be used for the new Big numbers of his uncle, the mathematician Edward Kasner,
who popularized the family *googols*
with an article (1938) and in a
book
(1940).

We've expanded these two numbers to a system dubbed the `googolmultiplex`

which may become the height of fashion of tomorrow
(every Big number aficionado his own claim to fame nay?-).

We'll use Knuth's arrows `^^..`
with an occassional `\` *backslash*,
which naturally continues from the lines before –
in the
next chapter
these new signs will be explained.

Now if you want even more money for your *googols* then
click here!

googol = 1111111111**1111111111**11
*= 10^100 = 333 bit*

googolplex = *10^*googol
*= 10^10^100 ~ 3.32E100 bit*

googolduplex = *10^*googolplex
*= 10^10^10^100*

googoltriplex = *10^*googolduplex
*= 10^10^10^10^10^2*

googol-4-plex = *10^*googol-3-plex
*= 10^^6\^2*

googolmultiplex : googol-n-plex
*= 10^^n\^*googol

googolgoogolplex = *10^^*googol
*~ 10^^(2+10^100)\^2*

googolgoogolduplex = *10^^*googolgoogolplex
*= 10^^10^^*googol

googolgoogoltriplex
*= 10^^10^^10^^*googol

googolgoogol-n-plex
*= 10^^^n\^^*googol

googol-3-flex = googolgoogolgoogolplex = *10^^^*googol

googol-3-flex-n-plex *= 10^^^^n\^^^*googol

googolmultiflex : googol-m-flex =
*10^ ^{...}*googol

`{^#m}`

googoldiflux = googolgoogolflex =

*10^*googol

^{...}`{^#googol}`

googolflux = googolflex =
googolplex *= 10^10^100* = gog**100**

gog(**n**) *= 10^10^n*

ma(**m**)gog(**n**) *= m^m^n*

gog *= 10^10*
~ maêgog(**3.14**)

We end with a teaser.
All of human experience can be positioned in the
data range
between *googol* and *googolplex*,
approximately between `gog`

and **5**`gog`

in double-exponential
plexcentages
and more accurately in the range between **27**`maêgog`

and
**13**`maêgog`

with natural double-exponents.
**64**What is your

Consider this.*plexenta*
and range of mind?

## Humanity's hard disk

Your brain is a smart computer that mostly interprets information from the senses. With the visual cortex receiving

20 Mbit/sfrom both eyes, it processes about80 KBin a30 Hzblink, enough bits to put2000googolsin a row.The data stream of all the senses combined plus thoughts and emotions, shouldn't be more than

50 Mbit/sfor the average waking state. So a long life will fit on500 TBof disk space (without compression), which is affordable.With an estimated total of 108 billion people

_{ <Grünwald 2008>}to date, the story of mankind takes no more than1E28 bitto express a mathematically random and physically fuzzy number around10^10^27– far smaller than agoogolplex.