Greek myriad system

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The Greek myriad system is a series of two distinct systems, both developed in the late 3rd century BC, that aim to extend the Greek numerals beyond a myriad, or 10,000. The smaller of the two allows for the denotation of any number up to 10,000² = 100,000,000, while the larger of the two allows for any number up to 10,000²⁸ = 10¹¹². Archimedes also describes a variation of this method and defines numbers up to \(10^{8 \times 10^{16}}\) in his work The Sand Reckoner (ca. 250 BC).

Greek numerals

The Greek numerals from 1 to 900 use a series of 27 symbols based on the 24 letters of the Greek alphabet along with the now disused digamma, koppa and sampi. To distinguish these characters from other Greek writing, a small mark (similar to a diacritic) is placed to the right of the numbers.

List of Greek numerals up to 1000

Greek name	Symbol	Value
Alpha	Αʹ	1
Beta	Βʹ	2
gamma	Γʹ	3
Delta	Δʹ	4
Epsilon	Εʹ	5
Digamma	Ϛʹ	6
Zeta	Ζʹ	7
Eta	Ηʹ	8
Theta	Θʹ	9
Iota	Ιʹ	10
Kappa	Κʹ	20
Lambda	Λʹ	30
Mu	Μʹ	40
Nu	Νʹ	50
Xi	Ξʹ	60
Omicron	Οʹ	70
Pi	Πʹ	80
Koppa	Ϙʹ	90
Rho	Ρʹ	100
Sigma	Σʹ	200
Tau	Τʹ	300
Upsilon	Υʹ	400
Phi	Φʹ	500
Chi	Χʹ	600
Psi	Ψʹ	700
Omega	Ωʹ	800
Sampi	Ϡʹ	900

Numerals from 1000 to 9000 used the symbols for 1 to 9 with the mark to the top left or bottom left instead of to the right or on top. Alternatively, these numbers can be denoted using the lowercase versions of the symbols for 1 to 9. This system allowed for numbers up to the myriad (10,000), commonly denoted as M.

Multiplication system (ca. 225 BC)

Multiplication system diagram up to 99,990,000

It is unknown who specifically developed this system, but around the late 3rd century BC the Greeks developed a system for writing multiples of the myriad. In this system, a number would be multiplied by a myriad if written above an M, which when combined with numbers smaller than a myriad allowed for the denotation of any number up to a myriad myriads, or 100,000,000 (represented as an M over and M). Although this system could technically be continued by placing a number above an M above another M, this quickly becomes unwieldy and wasn't used by the ancient Greeks as they had little use for numbers above 100,000,000.

Exponential system (ca. 200 BC)

Pappus of Alexandria (ca. 290-350 AD) states that Apollonius of Perga (ca. 260-190 BC) laid out a much larger system for numbers beyond a myriad.

Numbers were divided into “classes” based on powers of the myriad: The first class, called the elementary class, contains all the numbers up to 9,999, that is to say all numbers less than the myriad. The second class, called the class of primary myriads, contains the multiples of the myriad by all numbers up to 9,999 (So the numbers 10,000, 20,000, 30,000, and so on up to 99,990,000, or 9,999 x 10,000). This class is denoted by an a lowercase alpha (α) above the letter M (to denote 10,000¹), followed by a number up to 9,999 to represent the multiple. The third class is the secondary myriads, which contains the multiples of a myriad myriads (so the numbers 100,000,000, 200,000,000, 300,000,000, and so on up to 999,900,000,000). This class is denoted by an a lowercase beta (β) above the letter M (to denote 10,000²), followed by a number up to 9,999 to represent the multiple. After the secondary myriads is the tertiary myriads, also known as the gamma myriads, then the delta myriads, then the epsilon myriads, and so on.

Pappus of Alexandria gave the example of the number 5,462,360,064,000,000, expressed as 5,462 tertiary myriads, 3,600 secondary myriads, and 6,400 primary myriads. Theoretically up to 27 powers of the myriad (using the 27 symbols used for the numerals) are possible, allowing for numbers as high as 10¹¹², although it seems numbers this large were never used in practice.

Powers of the myriad

Class name	Minimum number (scientific notation)	Maximum number (scientific notation)	Maximum number (full)
Elementary class	1×100	9.999×103	9,999
Alpha myriads (Primary myriads) (A)	1×104	9.999×107	99,990,000
Beta myriads (Secondary myriads) (B)	1×108	9.999×1011	999,900,000,000
Gamma myriads (Tertiary myriads) (Γ)	1×1012	9.999×1015	9,999,000,000,000,000
Delta myriads (Δ)	1×1016	9.999×1019	99,990,000,000,000,000,000
Epsilon myriads (Ε)	1×1020	9.999×1023	999,900,000,000,000,000,000,000
Digamma myriads (Ϛ)	1×1024	9.999×1027	9,999,000,000,000,000,000,000,000,000
Zeta myriads (Ζ)	1×1028	9.999×1031	99,990,000,000,000,000,000,000,000,000,000
Eta myriads (Η)	1×1032	9.999×1035	999,900,000,000,000,000,000,000,000,000,000,000
Theta myriads (Θ)	1×1036	9.999×1039	9,999,000,000,000,000,000,000,000,000,000,000,000,000
Iota myriads (Ι)	1×1040	9.999×1043	99,990,000,000,000,000,000,000,000,000,000,000,000,000 ,000
Kappa myriads (Κ)	1×1044	9.999×1047	999,900,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000
Lambda myriads (Λ)	1×1048	9.999×1051	9,999,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000
Mu myriads (Μ)	1×1052	9.999×1055	99,990,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000
Nu myriads (Ν)	1×1056	9.999×1059	999,900,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000
Xi myriads (Ξ)	1×1060	9.999×1063	9,999,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000,000
Omicron myriads (Ο)	1×1064	9.999×1067	99,990,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000,000,000
Pi myriads (Π)	1×1068	9.999×1071	999,900,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000,000,000,000
Koppa myriads (Ϙ)	1×1072	9.999×1075	9,999,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000,000,000,000,000,000
Rho myriads (Ρ)	1×1076	9.999×1079	99,990,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000,000,000,000,000,000,000
Sigma myriads (Σ)	1×1080	9.999×1083	999,900,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000,000,000,000,000,000,000 ,000
Tau myriads (Τ)	1×1084	9.999×1087	9,999,000,000.000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000
Upsilon myriads (Υ)	1×1088	9.999×1091	99,990,000,000,000.000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000
Phi myriads (Φ)	1×1092	9.999×1095	999,900,000,000,000,000.000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000
Chi myriads (Χ)	1×1096	9.999×1099	9,999,000,000,000,000,000,000.000,000,000,000,000,000 ,000,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000
Psi myriads (Ψ)	1×10100	9.999×10103	99,990,000,000,000,000,000,000,000.000,000,000,000,000 ,000,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000,000
Omega myriads (Ω)	1×10104	9.999×10107	999,900,000,000,000,000,000,000,000,000.000,000,000,000 ,000,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000,000,000
Sampi myriads (Ϡ)	1×10108	9.999×10111	9,999,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000,000,000,000,000

This limit assumes that the powers of a myriad are limited to one symbol (and thus only 27 powers are possible). If you instead interpret the symbol above the M as its own number you can go much higher. For example, the 10,000th power of the myriad (Or the myriadth myriad), represented by an M over an M, would be equal to 10^40,000.

The Sand Reckoner (ca. 250 BC)

Archimedes used a similar system to define a series of large numbers.

He created a series of “orders” based on powers of 100,000,000 (or a myriad squared). Numbers up to 100,000,000 (10⁸) were part of the first order, numbers up to 10¹⁶ were part of the second order, and so on all the way up to the 100,000,000th order, which would be equal to \((10^8)^{10^8}\) = 10^800,000,000. These orders were called the “orders of the first period”. He then constructed the orders of the second period by taking multiples of the last largest value (giving a largest value of 10^1,600,000,000), and then again for the third period, and so on and so on until he had reached the 100,000,000th period. The largest number named by Archimedes was the last number in this period, equal to \(((10^8)^{10^8})^{10^8}\) = \(10^{8 \times 10^{16}}\), or 10^{80,000,000,000,000,000}. This number is given the name "μυριακισμυριοστᾶς περιόδου μυριακισμυριοστῶν ἀριθμῶν μυρίας μυριάδας (Myriakismyriostás periódou myriakismyriostón arithmón myrías myriádas) ", meaning"Myriad of myriads of the 100,000,000th order of the 100,000,000th period". The number can also be considered to be called "Archimedes' Number", although this just a hypothetical name and not used by official sources.

See Ancient numeral systems - Notation of Archimedes for a more in-depth examination.

See also

• Ancient numeral systems