User blog:Denis Maksudov/Googology 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. It was very interesting for me how Archimedes could write numbers up to $$10^{8 \times 10^{16}}$$ using only  Greek numerals, which you can see in table below

Greek Numerals
Example: ͵θτπεʹ=9385

1-μονὰς, 10-δέκα, 100-ἑκατόν, 1000-χιλιάς, 10000-μύριον (myriad)

Notation of Archimedes
Greek bold text, which you can see below, was copied from The Sand Reckoner, translation in my words

ἀριθμοὶ ἐς τὰς μυρίας μυριάδας πρώτοι καλουμένοι

natural numbers up to myriad of myriads $$(10^{8})$$ are numbers of first order

πρώτων ἀριθμῶν αἱ μυρίαι μυριάδες μονὰς καλείσθω δευτέρων ἀριθμῶν

the myriad of myriads of first order is equal to the unit of second order $$(10^{8})$$

μυρίαι μυριάδες τῶν δευτέρων ἀριθμῶν μονὰς καλείστω τρίτων ἀριθμῶν

the myriad of myriads of second order is equal to the unit  of  third order $$(10^{8}\times 10^{8}=10^{16})$$

τρίτων ἀριθμῶν μυρίαι μυριάδες μονὰς καλείσθω τετάρτων ἀριθμῶν

the myriad of myriads of third order is equal to the unit  of  fourth order $$(10^{16}\times 10^{8}=10^{24})$$

τετάρτων ἀριθμῶν μυρίαι μυριάδες μονὰς καλείσθω πέμπτων ἀριθμῶν

the myriad of myriads of fourth order is equal to the unit  of  fifth order $$(10^{24}\times 10^{8}=10^{32})$$

and so on up to

μυριακισμυριοστῶν ἀριθμῶν μυρίας μυριάδας

myriad of myriads of 100 000 000-th order $$(10^{8\times 10^8})$$

Examples

ιʹ μονάδες τῶν δευτέρων ἀριθμῶν = ten of units of second order $$=10\times 10^{8\times(2-1)}= 10^{9}$$

μυρίαι μυριάδες τῶν δευτέρων ἀριθμῶν = myriad of myriads of second order=αʹ μονάδες τῶν τρίτων ἀριθμῶν = one unit of third order $$=10^{8\times(3-1)}= 10^{16}$$

ιʹ μυριάδες τῶν τρίτων ἀριθμῶν = ten myriads of third order $$=10\times10^4\times10^{8\times(3-1)}= 10^{21}$$

ιʹ μονάδες τῶν πέμπτων ἀριθμῶν = ten of units of fifth order $$=10\times 10^{8\times(5-1)}= 10^{33}$$

ιʹ μυριάδες τῶν ἕκτων ἀριθμῶν = ten myriads of sixth order $$=10\times10^4\times10^{8\times(6-1)}=10^{45}$$

͵α μονάδες τῶν ἑβδόμων ἀριθμῶν = thousand of units of seventh order $$=10^3\times10^{8\times(7-1)}=10^{51}$$

= χιλίαι μονάδες τῶν ἑβδόμων ἀριθμῶν

͵α μυριάδες τῶν ὀγδόων ἀριθμῶν = thousand of myriads of eighth order= $$=10^3\times10^4\times10^{8\times(8-1)}=10^{63}$$

͵θτπεʹ μονάδες τῶν μυριακισμυριοστῶν ἀριθμῶν = $$9385 \times 10^{799 999 992}$$

Extension
Let all mentioned numbers are the numbers of first period (ἀριθμοὶ πρώτας περιόδου )

ὁ δὲ ἔσχατος ἀριθμὸς τᾶς πρώτας περιόδου μονὰς καλείσθω δευτέρας περιόδου πρώτων ἀριθμῶν

And last number of first period $$(10^{8\times 10^8})$$ is unit of first order of second period

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

the myriad of myriads of first order of second period is equal to the unit of second order of second period $$=10^{8\times 10^8 +8}$$

and so on up to

μυριακισμυριοστᾶς περιόδου μυριακισμυριοστῶν ἀριθμῶν μυρίας μυριάδας

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

And this is the limit of the notation