## Egyptian numerals

The Egyptian numerals were invented in ancient Egypt about 3000 BC.

Egyptian numerals system uses additive principle, for example:

The glyphs for addition and subtraction are and , respectively.

## Greek numerals

### Attic numerals

The Attic numerals appeared in the 7th century BC.

 Value Symbol Greek numeral 1 Ι μονὰς(monas) 5 Π πέντε(pente) 10 Δ δέκα (deka) 100 Η ἑκατόν (hɛkaton) 1,000 Χ χιλιάς (kilias) 10,000 Μ μύριον (myrion)

### Ionic numerals

In the 4th century BC the attic numerals were replaced by the ionic numerals.

 1 αʹ 10 ιʹ 100 ρʹ 1,000 ͵α 2 βʹ 20 κʹ 200 σʹ 2,000 ͵β 3 γʹ 30 λʹ 300 τʹ 3,000 ͵γ 4 δʹ 40 μʹ 400 υʹ 4,000 ͵δ 5 εʹ 50 νʹ 500 φʹ 5,000 ͵ε 6 ϛʹ 60 ξʹ 600 χʹ 6,000 ͵ϛ 7 ζʹ 70 οʹ 700 ψʹ 7,000 ͵ζ 8 ηʹ 80 πʹ 800 ωʹ 8,000 ͵η 9 θʹ 90 ϟʹ 900 ϡʹ 9,000 ͵θ

Greek numerals system uses additive principle, for example: ͵θτπεʹ = 9000+300+80+5 = 9,385

Ten thousand (one myriad) ancient greeks wrote as "M" and quantity of myriads they wrote before "M". For example:

͵θτπεM͵θτπεʹ = 93,859,385

Thus, the largest number which ancient greeks could write was ,θϡϟθM,θϡϟθ' = 99,999,999.

## Notation of Archimedes

Earliest googological notation in human history was invented by Archimedes (c. 287 – c. 212 BC) with aim to calculate how much grains of sand the Universe can contain (in supposition, that all universe filled by sand).

### Definition

All natural numbers up to myriada of myriads (100,000,000) are first numbers (ἀριθμοὶ ἐς τὰς μυρίας μυριάδας πρώτοι καλουμένοι). The unit of (n+1)ths numbers is myriada of myriads of n-ths numbers. And so on up to 100,000,000-ths numbers.

### Examples

• ιʹ μονάδες τῶν δευτέρων ἀριθμῶν = ten of units of second numbers $$=10\times 10^{8\times(2-1)}= 10^{9}$$
• μυρίαι μυριάδες τῶν δευτέρων ἀριθμῶν = myriad of myriads of second numbers=αʹ μονάδα τῶν τρίτων ἀριθμῶν = one unit of third order $$=10^{8\times(3-1)}= 10^{16}$$
• ιʹ μυριάδες τῶν τρίτων ἀριθμῶν = ten myriads of third numbers $$=10\times10^4\times10^{8\times(3-1)}= 10^{21}$$
• ιʹ μονάδες τῶν πέμπτων ἀριθμῶν = ten of units of fifth numbers $$=10\times 10^{8\times(5-1)}= 10^{33}$$
• ιʹ μυριάδες τῶν ἕκτων ἀριθμῶν = ten myriads of sixth numbers $$=10\times10^4\times10^{8\times(6-1)}=10^{45}$$
• ͵α μονάδες τῶν ἑβδόμων ἀριθμῶν = thousand of units of seventh numbers $$=10^3\times10^{8\times(7-1)}=10^{51}$$ = χιλίαι μονάδες τῶν ἑβδόμων ἀριθμῶν
• ͵α μυριάδες τῶν ὀγδόων ἀριθμῶν = thousand of myriads of eighth numbers= $$=10^3\times10^4\times10^{8\times(8-1)}=10^{63}$$
• ͵θτπεʹ μονάδες τῶν μυριακισμυριοστῶν ἀριθμῶν = 9,385 units of 100,000,000-ths numbers = $$9,385 \times 10^{799,999,992}$$

In general, each natural number up to $$10^{800,000,000}$$ can be uniquely written in this form:

$$N=\sum_{i=1}^k$$(Number of myriads of i-ths numbers+Number of units of i-ths numbers)

### Extension

All natural numbers up to myriada of myriads of 100,000,000ths numbers are numbers of first period (ἀριθμοὶ πρώτας περιόδου). The unit of first numbers of (n+1)th period is myriada of myriads of 100,000,000ths numbers of the n-th period.

μυρίαι μυριάδες τᾶς δευτέρας περιόδου πρώτων ἀριθμῶν μονὰς καλείσθω τᾶς δευτέρας περιόδου δευτέρων ἀριθμῶν

= the myriad of myriads of first numbers of 2nd period is equal to the unit of 2-ths numbers of 2nd period $$=10^{8\times 10^8 +8}$$

$$10^{16\times 10^8}$$ is unit of first numbers of 3rd period

$$10^{24\times 10^8}$$ is unit of first numbers of 4th period

$$10^{32\times 10^8}$$ is unit of first numbers of 5th period

and so on, up to:

μυριακισμυριοστᾶς περιόδου μυριακισμυριοστῶν ἀριθμῶν μυρίας μυριάδας

= myriad of myriads of 100,000,000th numbers of 100,000,000th period $$=10^{8\times 10^{16}}$$

## Roman numerals

 Symbol I V X L C D M Value 1 5 10 50 100 500 1,000

Roman numerals system uses additive principle, for example:

MMMCCCXXXIII=MMM+CCC+XXX+III=3000+300+30+3=3,333.

Subtractive notation is also used, in the following cases:

I placed before V or X reduces number by one, X placed before L or C reduces number by ten, C placed before D or M reduces number by hundred,

The largest number representible in Roman numerals is:

MMMMCMXCIX = 4,000+(1,000-100)+(100-10)+(10-1) = 4,000+900+90+9 = 4,999.

## Chinese numerals

Chinese numerals are numerals of ancient and modern usage in China. The invention of chinese numerals and math calculations traditionally ascribed to the mythological Yellow Emperor (2698–2598 BC).

### The numbers of everyday usage

 Chinese Characters (Simplified) Value Pinyin (Standard Chinese/Mandarin) 〇 0 líng 一 1 yī 二 2 èr 三 3 sān 四 4 sì 五 5 wǔ 六 6 liù 七 7 qī 八 8 bā 九 9 jiǔ 十 10 shí 百 100 bǎi 千 1,000 qiān 万 10,000 wàn

### Multiplicative principle

Chinese numerals system uses a multiplicative principle, for example:

• 一千一百五十八 = 1,158
• 八 = 8
• 八十八 = 88
• 八百八十八 = 888
• 八千八百八十八 = 8,888
• 八千〇八十 = 8,080

### Notations for large numbers

Notation 1:

• $$a_0=10^4$$
• $$a_{n+1}=a_n\cdot10$$

Notation 2:

• $$a_0=10^4$$
• $$a_{n+1}=a_n\cdot10^4$$

Notation 3:

• $$a_0=10^8$$
• $$a_{n+1}=a_n\cdot10^8$$

Notation 4:

• $$a_0=10^4$$
• $$a_{n+1}=a_n^2$$

### Names and values of the large numbers

 Chinese Characters (Simplified) Pinyin (Standard Chinese/Mandarin) N1 N2 N3 N4 万 wàn $$10^4$$ $$10^4$$ $$10^4$$ $$10^4$$ 亿 yì $$10^5$$ $$10^{8}$$ $$10^{8}$$ $$10^{8}$$ 兆 zhào $$10^6$$ $$10^{12}$$ $$10^{16}$$ $$10^{16}$$ 京 jīng $$10^7$$ $$10^{16}$$ $$10^{24}$$ $$10^{32}$$ 垓 gāi $$10^8$$ $$10^{20}$$ $$10^{32}$$ $$10^{64}$$ 秭 zǐ $$10^9$$ $$10^{24}$$ $$10^{40}$$ $$10^{128}$$ 穰 ráng $$10^{10}$$ $$10^{28}$$ $$10^{48}$$ $$10^{256}$$ 沟 gōu $$10^{11}$$ $$10^{32}$$ $$10^{56}$$ $$10^{512}$$ 涧 jiàn $$10^{12}$$ $$10^{36}$$ $$10^{64}$$ $$10^{1,024}$$ 正 zhèng $$10^{13}$$ $$10^{40}$$ $$10^{72}$$ $$10^{2,048}$$ 载 zǎi $$10^{14}$$ $$10^{44}$$ $$10^{80}$$ $$10^{4,096}$$

## Avatamsaka Sutra's notation

In Avatamsaka Sutra, written in the India before 5 century AD, Buddha gave the name "Nirabhilapya nirabhilapya parivarta (Bùkěshuō bùkěshuō zhuǎn 不可說不可說轉)" for number which is defined as follows:

• $$a_0=10^{7}$$
• $$a_{n+1}=a_n^2$$
• Nirabhilapya nirabhilapya parivarta $$=a_{122}$$, or in other words, Nirabhilapya nirabhilapya parivarta $$= (10^{7})^{2^{122}}$$ or $$10^{7 \times 2^{122}}$$ or $$10^{37,218,383,881,977,644,441,306,597,687,849,648,128}$$, which is approximately $$10^{3.722 \times 10^{37}}$$.

## Cyrillic numeral system

Cyrillic numeral system is a numeral system based on Cyrillic alphabet. This numeral system was used in Russia in 10th-17th centuries (before reforms of Peter the Great). Sometimes sign ҃ (titlo) was drawn over the number to separate it from the text.

Value Symbol Value Symbol Value Symbol Cyrillic numerals 1 А 10 І 100 Р 2 В 20 К 200 С 3 Г 30 Л 300 Т 4 Д 40 М 400 У 5 Е 50 Н 500 Ф 6 Ѕ 60 Ѯ 600 Х 7 З 70 О 700 Ѱ 8 И 80 П 800 Ѿ 9 Ѳ 90 Ч 900 Ц

For example ЦЧѲ = 999.

### Medieval Russian notations for large numbers

There were two notations for large numbers in the medieval Russia: Lesser count and Greater count. In both notations if a letter is preceded by the sign ҂, then the number, which is denoted by this letter, is multiplied by 1000. For example ҂ФЦ=500900 and ҂Ф҂ЧЦ =590900. If a letter is inside one of the following signs

• simple circle;
• circle consisting of dots;
• circle consisting of commas;
• circle consisting of crosses;
• square brackets top and bottom,

then the number is correspondingly multiplied by:

• 10 4, 105, 106, 107, 108 in the Lesser count;
• 106, 1012, 1024, 1048, 1049 (or 1096) in the Greater count.