Extreme Numbers


Read here about extreme numbers, both very large and very small. Scale research requires such numbers.

Extremely Large Numbers


Sand Reckoner

In the Sand Reckoner Archimedes (287 BCE- ?) shows Gelon, King of Syracuse, there would be 1021 x 1042 = 1063 grains of sand in a universe compacted wholly of sand. But Archimedes doesn't stop there. He then leaves the sand-filled universe behind, diminished itself to a grain of sand, as he piles up order on order, even unto the 108 order, which frighteningly enough contains all the numbers from [(100,000,000).sup.99,999,999 to ([100,000,000).sup.100,000,000]. He then creates periods, imagines a process up to a myriad-myriad units of the myriad-myriadth order of the myriad-myriadth period, and derives what we would call 1080,000,000,000,000,000. See the stories document.

Hindu Epochs


The Indians had a passion for high numbers, which is intimately related to their religious thought. For example, in texts belonging to the Vedic literature, we find individual Sanskrit names for each of the powers of 10 up to a trillion and even 1062. (Even today, the words 'lakh' and 'crore', referring to 100,000 and 10,000,000, respectively, are in common use among English-speaking Indians.) One of these Vedic texts, the Yajur Veda, even discusses the concept of numeric infinity (purna "fullness"), stating that if you subtract purna from purna, you are still left with purna.
The Lalitavistara Sutra (a Mahayana Buddhist work) recounts a contest including writing, arithmetic, wrestling and archery, in which the Buddha was pitted against the great mathematician Arjuna and showed off his numerical skills by citing the names of the powers of ten up to 1 'tallakshana', which equals 1053, but then going on to explain that this is just one of a series of counting systems that can be expanded geometrically. The last number at which he arrived after going through nine successive counting systems was 10421, that is, a 1 followed by 421 zeros.
There is also an analogous system of Sanskrit terms for fractional numbers, capable of dealing with both very large and very small numbers.
Larger number in Buddhism works up to Bukeshuo bukeshuo zhuan or 1037218383881977644441306597687849648128, which appeared as Bodhisattva's maths in the Avataṃsaka Sūtra., though chapter 30 (the Asamkyeyas) in Thomas Cleary's translation of it we find the definition of the number "untold" as exactly 1010*2122, expanded in the 2nd verses to 1045*2121 and continuing a similar expansion indeterminately.
A few large numbers used in India by about 5th century BCE (From Georges Ifrah: A Universal History of Numbers, pp 422–423):
Hindu Name
Value
Exponent
lakṣá
10^5
5
kōṭi
10^7
7
ayuta
10^9
9
niyuta
10^13
13
pakoti
10^14
14
vivara
10^15
15
kshobhya
10^17
17
vivaha
10^19
19
kotippakoti
10^21
21
bahula
10^23
23
nagabala
10^25
25
nahuta
10^28
28
titlambha
10^29
29
vyavasthanapajnapati
10^31
31
hetuhila
10^33
33
ninnahuta
10^35
35
hetvindriya
10^37
37
samaptalambha
10^39
39
gananagati
10^41
41
akkhobini
10^42
42
niravadya
10^43
43
mudrabala
10^45
45
sarvabala
10^47
47
bindu
10^49
49
sarvajna
10^51
51
vibhutangama
10^53
53
abbuda
10^56
56
nirabbuda
10^63
63
ahaha
10^70
70
ababa
10^77
77
atata
10^84
84
soganghika
10^91
91
uppala
10^98
98
kumuda
10^105
105
pundarika
10^112
112
paduma
10^119
119
kathana
10^126
126
mahakathana
10^133
133
asaṃkhyeya
10^140
140
dhvajagranishamani
10^421
421
bodhisattva
10^37218383881977644441306597687849648128
37218383881977644441306597687849648128
lalitavistarautra
10^200infinites
200infinites
matsya
10^600infinites
600infinites
kurma
10^2000infinites
2000infinites
varaha
10^3600infinites
3600infinites
narasimha
10^4800infinites
4800infinites
vamana
10^5800infinites
5800infinites
parashurama
10^6000infinites
6000infinites
rama
10^6800infinites
6800infinites
khrishnaraja
10^7000infinites
7000infinites
kaiki
10^8000infinites
8000infinites
balarama
10^9800infinites
9800infinites
dasavatara
10^10000infinites
10000infinites
bhagavatapurana
10^18000infinites
18000infinites
avatamsakasutra
10^30000infinites
30000infinites
mahadeva
10^50000infinites
50000infinites
prajapati
10^60000infinites
60000infinites
jyotiba
10^80000infinites
80000infinites

time_Hindu_Epochs_from_Wikipedia.png

Googol


Modern day names of this type include the googol — 1 with 100 zeroes after it — and the googolplex (10 to the googol power). There are other names for very large numbers, such as primo-vigesimo-centillion for 10366, and the mellifluous milli-millillion for 103,000,003.

Extremely Small Numbers


Infinitesimal Difference Between 0.99999… And 1


Students resist equating 0.9999… to 1, and with good reason. Katz and Katz 2009 show that “So long as the number system has not been specified, the students’ hunch that .999 . . . can fall infinitesimally short of 1, can be justified in a mathematically rigorous fashion.”

Katz and Katz define the conditions under which it is real:
(1) the reals are not, as the rationals are not, the maximal number system;
(2) there exist larger number systems, containing infinitesimals;
(3) in such larger systems, the interval [0, 1] contains many numbers infinitely close to 1;
(4) in a particular larger system called the hyperreal numbers, there is a generalized notion of decimal expansion for such numbers, starting in each case with an unbounded number of digits “9”;
(5) all such numbers therefore have an arguable claim to the notation “.999 . . .” which is patently ambiguous (the meaning of the ellipsis “. . .” requires disambiguation);
(6) all but one of them are strictly smaller than 1;
(7) the convention adopted by most professional mathematicians is to interpret the symbol “.999 . . .” as referring to the largest such number, namely 1 itself;
(8) thus, the students’ intuition that .999 . . . falls just short of 1 can be justified in a mathematically rigorous fashion;
(9) the said extended number system is mostly relevant in infinitesimal calculus (also known as differential and integral calculus);
(10) if you would like to learn more about the hyperreals, go to your teacher so he can give you further references.