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

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.

## Extreme Numbers

## Table of Contents

## 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):

## 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.