“On the shoulders of giant numbers”
http://www.iteror.org/big/book/ch1/ch1_6.html
bigΨ
Ψ.1. Natural repetition
chapter 1.6, edit 0.2.9
published: 20101105
updated: 20111230
# 1.6. A small history of large numbers
tatraikastham jagat krtsnam
pravibhaktam anekadhâ
apasyad devadevasya
sarîre pândavas tadâ
At that time Arjuna could see in the universal form of the Lord the unlimited expansions of the universe situated in one place although divided into many, many thousands.
"...Arjuna could see in the body of Krsna many thousands of universes. As we learn from Vedic scriptures, there are many universes and many planets. Some of them are made of earth, some are made of gold, some are made of jewels, some are very great, some are not so great, etc. Sitting on his chariot, Arjuna could see all these universes. But no one could understand what was going on between Arjuna and Krsna."
–
Bhagavad Gita 11.13
A.C. Bhaktivedanta Swami Prabhupada
§1.6.1. Egyptian hieroglyph of a million (or not yet?)
The human ability to express large quantities in a terse manner has evolved from Egyptian hieroglyphs via Indian decimal numbers all the way to modern day operatorstyle exponentiation. Using only decimal signs for the sake of comparison, we can show how the word length for writing numbers gets shorter in the course of history.
Of course the numerals of the Egyptians were different,
but already they used powers of 10 as countable units
in a primitive number system without multiplication by digits.
The signs for 1 10 100 1000 10000 100000 1000000
appear to be five thousand petalled lotus years old, while some are inscribed
on a ceremonial
macehead
of the possessive conqueror Pharaoh
Narmer
(1st dynasty).
The hieroglyph of the numeral unit on the right – what
Georges Ifrah
calls a kneeling genie
(holding up the sky) –
is known to signify the megalithic figure of a million 10^6,
as well as infinity.
The sign of the sun for 10^7 that we find in
Gheverghese Joseph
is less well documented and of a later date.
Now it might be that the 3100 BC figure of a kneeling person
was misinterpreted to stand for a million by Egyptologists
as well as by the Egyptians of the 2nd millennium BC who adopted it as such.
Ifrah notices that, for this sign on the
macehead
to be a numeral, the usual order of the hieroglyphs must be jumbled up.
Surely, considering the count of 400,000 bulls in Pharaoh's war bounty,
a captured 1,422,000 goats is a ridiculously precise number!
We suggest that the figure later in use as a million, on Narmer's macehead still represents a subjugated cowboy (+cowgirls?), one of 2000+ herders who were needed to take care of the herds of the cows and some 420,000 goats. They clearly have their hands free, while the 120,000 manly prisoners of war sit with their arms tied behind their back.
§1.6.2. Greek powers of Archimedes (or not quite?)
The Greeks also used multiples of 1,10,100,1000
in their number system, each with 10 different letter signs.
Writing only four letters all whole numbers up to
a myriad or 10000 can be formed.
The same is true in our decimal system, but with all the variation in Greek letters
and auxiliary signs multiplication becomes troublesome.
In expressions such as
,ατλα.,εσιδ
= 13315214
the number of myriads is added to a smaller number by a dot
(in an example of Diophantus c.3d century AD).
Then myriad myriads or 10^8
could be named but not written.
Anyhow, the main topic of interest for the ancient Greeks was
geometry and trigonometry and there calculations rely on
ratios of lines, angles, areas and volumes.
Generously pursuing mathematics for its own virtue Euclid (c.300260 BC) wrote his
Elements,
the start of modern mathematics.
Euclid's foundation of geometry was rock solid.
Attempts to prove his postulate for
nonparallel lines
from other axioms failed and instead gave rise to
nonEuclidean geometry
in the 19th century.
With Archimedes of Syracuse (c.287212 BC)
the dispute over the scale of the physical universe versus
the capacity of arithmetical numbers
was decided once and for all in favour of pure mathematics.
In his
Sand number article
Archimedes
derives an upper bound for the number of sand grains 10^51
that fit inside the Greek universe (the sphere where the sun revolves around the earth)
from personal measurements and a clever geometrical proof.
He also discusses Aristarchus of Samos' heliocentric theory
where the sphere of fixed stars has a diameter that is 10000 times greater,
but calculates it contains no more than 10^63 sand grains.
Archimedes' system for large numbers is divided into 10^8 periods, each period subdivided in 10^8 orders, each order consisting of 10^8 numbers of units, each unit a copy of the highest number of the previous order.

Numbers from 1 up to (10^8)^(10^8)
in the 1st period:
 Call the numbers from 1 to a myriad myriads (10^8) numbers of the 1st order
 Then let 10^8 be unit of the 2nd order numbers up to (10^8)×(10^8)
 This 10^16 unit of 3d order numbers ending with (10^8)^3, and so on ...
 Until the 10^8th order numbers will end with (10^8)^^2 the 1st period.

Numbers from P = 10^(8×10^8) up to P^2
in the 2nd period:
 Call the numbers from unit P to P×10^8 of the 1st order in the 2nd period
 Let P×10^8 be the unit of its 2nd order up to P×(10^8)^2, and so on ...
 Until its 10^8th order numbers end with P×(10^8)^(10^8) the 2nd period.

Rest from P^2 up to 10^(8×10^16)
in the 10^8th and last period:
 The 3d period then will cover the numbers up to P^3
 The 4th period the numbers up to P^4, and so on ...
 Until the 10^8th order in the 10^8th period ends with P^(10^8)
 Last octad (10^8)^(10^8)^2 = 10^(8×10^16)
In this system natural numbers are precisely expressed
(and uniquely, without the myriad myriads terms)
as the total of the numbers of different units.
But usually an approximation is more practical, Archimedes recognized this
and divided up the orders of his system in octads,
with as terms just the order's unit and the next 7 powers of ten.
So the quantity 10^63 = 10^(7+8×7)
= 10^7×(10^8)^7 of sand grains is expressed as
a thousand myriad units of the eighth order –
serving to illustrate how small this first period number is,
compared to those in the later periods of Archimedes' system.
Archimedes wrote a series of octads^{ }
with indices from A_{1}
to A_{8}
expressing the numbers 1 to 10^7,
and defined his law of indices as
A_{m}*A_{n}
= A_{m+n1}
much like our law of exponents
a^{m}×a^{n} = a^{m+n}
but not quite!^{ }
Because index value 0
is missing,
this hides the fact that a^{0} = 1 and the second law
A_{m}^n
= A_{m*nn+1}
will become less obvious.^{ }
Archimedes multiplied two octads^{ }
by adding the distance of their indices
from base index A_{1}
^{ }
and in this context the multiplication
of indices escapes from view.^{ }
With the luxury of an exponential operator ^
we now define the power laws.^{ }
2nd power law: (a^m)^n = a^(m*n)
In this chapter we met the largest number 10^{8×1016} of Archimedes and hence of ancient Greece.
Although
Archimedes'
system covers all whole numbers up to this limit,
a binary number this size that's randomly chosen would consume 30213 TB
of computer disk space.
Our universe has expanded since the time of Aristarchus,
nowadays (2010) it is estimated to contain 3E23 stars –
a number bigger than the sand grains on all the world's beaches,
but still less than the sands in the Sahara desert.
§1.6.3. Indian decimal expansions (oh really?)
About that time in
India
the void (Sanskrit: šünyatä = emptiness) got a good press
and eventually was rewarded (circa 650 AD) with its own numeric sign 0
for no cipher
.
This nought had the practical use of filling in
the empty spaces in decimal numbers such as 108
but came too late for the early religious devotees who used large numbers
to explain the greatness of their Gods or to count a multitude of enlightened beings.
Centuries after the historical Buddha was gone, in the
Lalitavistara Sutra
a literary Buddha explains a number system to the mathematician Arjuna.
Starting from a koti 10^7
moving in multiples of 100 up to 10^53
(tallakshana = a mark), calling every odd power by its own name.
Building on this first kotisatottara numeration
follow 8 numerations of the powers up to 10^(7+46×9)
provided these are listed as before (something of a modern tradition).
The way the sands of Ganges rivers come into the equation
makes this interpretation very awkward, but we don't have any better,
so we'll follow up on it.
The last number
10^421 (uttaraparamânurajahpraveša)
is often cited as a Sutra record, but if numerations line up as presumed,
it would mark the initial atom of a highest numeration ending with
108470495616000 atoms in a yojana.
Even bigger is the mass
of a yojana,
calculated at about 1E28 atoms.
When a world of four continents contains 34000 yojana
and there are 1E9 such worlds, the total mass would be
about 3.4E41 atoms, equal to a record 3.4E462 in number
(the physical entity translates to a pure Big number).
Assuming that all 30 realms of the
three thousand great thousands of worlds in the Lalitavistara
are quantatively similar
(a tall order), and given that the Buddha declares their universal mass as
incalculable
, our fuzzy Sutra record 1E464
sets a new bound on the
asamkhyeya,
the number the Indians conjectured to be at infinity
.
With this lesson in enumeration the young Buddha, prince of Šâkyas,
passes his maths test. But still an error went unnoticed.
Usually a threethousandgreatthousandfoldworld counts a billion,
a 1000×1000×1000 worlds, times 3
to extend this cube over the trinity of desire, formation and formlessness.
But the
Lalitavistara
lists 30 realms, each with 100 kotis of worlds. So either a
koti is a million (instead of ten million)
or the Sutra writers didn't have the skill to compare such numbers.
Here we've assumed the latter.
Large Indian numbers can be lengthy compound words,
or in a list mostly imagined names, where they were recited as mantras,
perhaps meant to develop one's arithmetical prowess.
Never the series of decimal zeros (made in India) modern man expects here.
Zero
was
not a word
yet and the concept of a 1 followed by a row of empty 0..
didn't ring a (mindfulness) bell, which explains the error above.
Maybe the American
Googols
were an invention after all?!
Anyhow, it happened that the literary Buddha dried out and the mahâyâna (Great Vehicle) buddhists developed a big thirst for stupendous baroque visions of universal Buddhas. These godlike creatures inspired lengthy arrays of numbers to count their manifold virtues and huge hosts of followers, in a library packed with scriptures of which the most renowned are the Lotus Sutra and the Flower Ornament Sutra.
# Counting on the Lotus Sutra
The Buddha answered the bodhisattva Infinite Thought:
"Good son! If there be countless {10^?} hundred thousand {10^5} myriad {10^4} kotis {10^7}* of living beings suffering from pain and distress who hear of this bodhisattva Regarder of the Cries of the World, and with all their mind call upon his name, the bodhisattva Regarder of the Cries of the World will instantly regard their cries, and all of them will be delivered."– the Lotus Sutra, chapter 24 or 25
The AllSidedness of the bodhisattva Regarder of the Cries of the World
_{ * <Note>} Sanskrit original reads sattvakotînayutasatasahasrâni {10^21} in total
If there really is such a higher being as bodhisattva Avalokiteshvara, who looks down upon us and is capable of assisting at least 10^16 living beings in an instant, then this enlightening being (whom the Chinese call Guan Yin, a motherly figure like the Virgin Mary), or the God powered rescue mechanism he/she employs can only exist in a larger universe – the buddhaland of which ours is a small but still accessible part.
The question is, "How much larger?"
We think the size of a divine lifeline must be exponential
compared to the being saved.
If the scales don't match,
the abstract spark of a soul may be lifted from its bodily enclosure,
but all physical information will be lost,
because it's too small to contain in a higher order transitory environment.
So if a small creature contains 2^k {k=18}
bits of personal information (such as the Name),
we surmise its helper God will at most be 2^(k1)
times bigger than the creature's universe – not so very Big.
The bodhisattva Avalokita, while moving in the deep course of perfect understanding, shed light on the five skandhas and found them equally empty. After this penetration he overcame all pain.
"Listen Shariputra! Form is emptiness, emptiness is form, form is inseparable from emptiness, emptiness does not differ from form. The same is true with perceptions, ideas, feelings and consciousness. Hear Shariputra! All dharmas are marked with emptiness. They are neither produced nor destroyed."– the Heart Sutra
(Listen! It's back to zero again ;)