Attic numerals

Short description: Symbolic number notation used by the ancient Greeks

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Detail of stela showing tributes paid by allies of Athens in the League of Delos. The amounts are in Attic numerals, using the drachma sign "𐅂" instead of the generic unit sign "Ι". Some amounts are "𐅄" = 50, "ΗΗΗ" = 300, "𐅅ΗΗΗ" = 800, "ΔΔΔ𐅂𐅂𐅂" = 33, "Χ" = 1000, and "Χ𐅅𐅄Δ𐅂𐅂"? = 1562?. Plaque above the main entrance to the orphanage, which later became a prison, on the Greek island of Aegina. The ancient Greek inscription translates as “The Governor erected this orphanage in the year 1828”. The year is shown as Χ𐅅ΗΗΗΔΔΠΙΙΙ.

The Attic numerals are a symbolic number notation used by the Ancient Greece . They were also known as Herodianic numerals because they were first described in a 2nd-century manuscript by Herodian; or as acrophonic numerals (from acrophony) because the basic symbols derive from the first letters of the (ancient) Greek words that the symbols represented.

The Attic numerals were a decimal (base 10) system, like the older Egyptian and the later Etruscan, Roman, and Hindu-Arabic systems. Namely, the number to be represented was broken down into simple multiples (1 to 9) of powers of ten — units, tens, hundred, thousands, etc.. Then these parts were written down in sequence, in order of decreasing value. As in the basic Roman system, each part was written down using a combination of two symbols, representing one and five times that power of ten.

Attic numerals were adopted possibly starting in the 7th century BCE and although presently called Attic, they or variations thereof were universally used by the Greeks. No other numeral system is known to have been used on Attic inscriptions before the Common Era.[1][2] Their replacement by the classic Greek numerals started in other parts of the Greek World around the 3rd century BCE. They are believed to have served as model for the Etruscan number system, although the two were nearly contemporary and the symbols are not obviously related. [citation needed]

## Contents

* 1 The system
* 1.1 Symbols
* 1.1.1 Special symbols
* 1.1.2 The symbol for 100
* 1.2 Simple multiples of powers of ten
* 1.3 General numbers
* 2 Unicode
* 3 See also
* 4 Notes and references

## The system

### Symbols

The Attic numerals used the following main symbols, with the given values:[1][2]

Value  | Symbol  | Talents  | Staters  | Notes  | Etruscan  | Roman

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1 | Ι  |  |  | Tally mark?  | 𐌠  | I
5 | Π  | 𐅈  | 𐅏  | Old Greek: ΠΕΝΤΕ [pɛntɛ] Modern: πέντε | 𐌡  | V
10 | Δ  | 𐅉  | 𐅐  | Old Greek: ΔΕΚΑ [deka] Modern: δέκα | 𐌢  | X
50 |  | 𐅊  | 𐅑  | "Δ" in "Π": 10 × 5 = 50  | 𐌣  | L
100 | Η  | 𐅋  | 𐅒  | Old Greek: ΗΕΚΑΤΟΝ [hɛkaton] Modern: εκατό | 𐌟  | C
500 |  | 𐅌  | 𐅓  | "Η" in "Π": 100 × 5 = 500  | ?  | D
1000 | Χ  | 𐅍  | 𐅔  | Old Greek: ΧΙΛΙΟΙ [kʰilioi] Modern: χίλιοι | ?  | M
5000 |  | 𐅎  |  | "Χ" in "Π": 1000 × 5 = 5000  | ?  | V
10000 | Μ  |  | 𐅕  | Old Greek: ΜΥΡΙΟΙ [myrioi] Modern: μύριοι | ?  | X
50000 |  |  | 𐅖  | "Μ" in "Π": 10000 × 5 = 50000  | ?  | L

The symbols representing 50, 500, 5000, and 50000 were composites of an old form of the capital letter pi (with a short right leg) and a tiny version of the applicable power of ten. For example, 𐅆 was five times one thousand.

#### Special symbols

The fractions "one half" and "one quarter" were written "𐅁" and "𐅀", respectively.

The symbols were slightly modified when used to encode amounts in talents (with a small capital tau, "Τ") or in staters (with a small capital sigma, "Σ"). Specific numeral symbols were used to represent one drachma ("𐅂") and ten minas "𐅗".

#### The symbol for 100

The use of "Η" (capital eta) for 100 reflects the early date of this numbering system. In the Greek language of the time, the word for a hundred would be pronounced [hɛkaton] (with a "rough aspirated" sound /h/) and written "ΗΕΚΑΤΟΝ", because "Η" represented the sound /h/ in the Attic alphabet. In later, "classical" Greek, with the adoption of the Ionic alphabet throughout the majority of Greece, the letter eta had come to represent the long e sound while the rough aspiration was no longer marked.[3][4] It was not until Aristophanes of Byzantium introduced the various accent markings during the Hellenistic period that the spiritus asper began to represent /h/, resulting in the spelling ἑκατόν.[5]

### Simple multiples of powers of ten

Multiples 1 to 9 of each power of ten were written by combining the two corresponding "1" and "5" digits, namely:

Units  | Ι | II | III | IIII | Π | ΠI | ΠII | ΠIII | ΠIIII

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1  | 2  | 3  | 4  | 5  | 6  | 7  | 8  | 9
Tens  | Δ | ΔΔ | ΔΔΔ | ΔΔΔΔ | 𐅄 | 𐅄Δ | 𐅄ΔΔ | 𐅄ΔΔΔ | 𐅄ΔΔΔΔ
10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90
Hundreds  | Η | ΗΗ | ΗΗΗ | ΗΗΗΗ | 𐅅 | 𐅅Η | 𐅅ΗΗ | 𐅅ΗΗΗ | 𐅅ΗΗΗΗ
100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 | 900
Thousands  | Χ | ΧΧ | ΧΧΧ | ΧΧΧΧ | 𐅆 | 𐅆Χ | 𐅆ΧΧ | 𐅆ΧΧΧ | 𐅆ΧΧΧΧ
1000 | 2000 | 3000 | 4000 | 5000 | 6000 | 7000 | 8000 | 9000
Tens of thousands  | Μ | ΜΜ | ΜΜΜ | ΜΜΜΜ | 𐅇 | 𐅇Μ | 𐅇ΜΜ | 𐅇ΜΜΜ | 𐅇ΜΜΜΜ
10000 | 20000 | 30000 | 40000 | 50000 | 60000 | 70000 | 80000 | 90000

Unlike the more familiar Roman numeral system, the Attic system used only the so-called "additive" notation. Thus, the numbers 4 and 9 were written ΙΙΙΙ and ΠΙΙΙΙ, not ΙΠ and ΙΔ.

### General numbers

In general, the number to be represented was broken down into simple multiples (1 to 9) of powers of ten — units, tens, hundred, thousands, etc.. Then these parts would be written down in sequence, from largest to smallest value. For example:

* 49 = 40 + 9 = ΔΔΔΔ + ΠΙΙΙΙ = ΔΔΔΔΠΙΙΙΙ
* 2001 = 2000 + 1 = ΧΧ + I = ΧΧΙ
* 1982 = 1000 + 900 + 80 + 2 = Χ + 𐅅ΗΗΗΗ + 𐅄ΔΔΔ + ΙΙ = Χ𐅅ΗΗΗΗ𐅄ΔΔΔΙΙ
* 62708 = 60000 + 2000 + 700 + 8 = 𐅇Μ + ΧΧ + 𐅅ΗΗ + ΠΙΙΙ = 𐅇ΜΧΧ𐅅ΗΗΠΙΙΙ.

## Unicode

Main article: Ancient Greek Numbers (Unicode block)

Numeral systems

Hindu–Arabic numeral system

* Western Arabic
* Eastern Arabic

* Bengali
* Devanagari
* Gurmukhi
* Sinhala
* Tamil

* Balinese
* Burmese
* Dzongkha
* Gujarati
* Javanese
* Khmer
* Lao
* Mongolian
* Thai

East Asian

* Chinese
* Suzhou
* Hokkien
* Japanese
* Korean
* Vietnamese
* Tangut

* Counting rods

Alphabetic

* Abjad
* Armenian
* Āryabhaṭa
* Cyrillic
* Ge'ez
* Georgian
* Greek
* Hebrew
* Roman

Former

* Aegean
* Attic
* Babylonian
* Brahmi
* Chuvash
* Egyptian
* Etruscan
* Glagolitic
* Inuit
* Kharosthi
* Mayan
* Muisca
* Quipu

* Prehistoric

Positional systems by base

* 2
* 3
* 4
* 5
* 6
* 8
* 10
* 12
* 16
* 20
* 60

Non-standard positional numeral systems

* Bijective numeration (1)
* Signed-digit representation (Balanced ternary)
* factorial
* negative
* Complex-base system (2i)
* Non-integer representation (φ)
* mixed
* Asymmetric numeral systems

List of numeral systems

* v
* t
* e

## See also

* Etruscan numerals - Words, phrases and symbols for numbers of the Etruscan language
* Greek numerals - System of writing numbers using Greek letters
* History of ancient numeral systems
* List of numeral system topics
* List of numeral systems

## Notes and references

1. ↑ 1.0 1.1 "The Greek numerical alphabet". The Journal of Philology (Cambridge) XXII: 278–9. 1883. https://books.google.com/books?id=gZZNAQAAMAAJ&pg=PA278.
2. ↑ 2.0 2.1 "Reading and writing numbers: Greek numerals". History of Mathematics. 2. New York: Dover Publications. 1958. pp. 49–51. ISBN 9780486204307. https://books.google.com/books?id=uTytJGnTf1kC&pg=PA49.
3. ↑ The Study of Greek Inscriptions (Second ed.). Cambridge: Cambridge University Press. 1981. p. 18. ISBN 0-521-23188-4.
4. ↑ "§14". Greek Grammar (Revised ed.). Cambridge, MA: Harvard University Press. 2002. p. 10. ISBN 0-674-36250-0.
5. ↑ As found in Katharevousa, polytonic Greek in general and Modern Greek in particular before the 1982 official adoption of the monotonic system; however, neither the /h/ phoneme nor the spiritus asper, or "rough breathing mark", are present in monotonic Standard Modern Greek, resulting in the standard modern spelling εκατό.

* v
* t
* e

Ancient Greek mathematics

Mathematicians
(timeline)|

* Anaxagoras
* Anthemius
* Archytas
* Aristaeus the Elder
* Aristarchus
* Apollonius
* Archimedes
* Autolycus
* Bion
* Bryson
* Callippus
* Carpus
* Chrysippus
* Cleomedes
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* Democritus
* Dicaearchus
* Diocles
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* Geminus
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* Hipparchus
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* Hypsicles
* Isidore of Miletus
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* Theodosius
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* Theon of Smyrna
* Thymaridas
* Xenocrates
* Zeno of Elea
* Zeno of Sidon
* Zenodorus

Treatises|

* Almagest
* Archimedes Palimpsest
* Arithmetica
* Conics (Apollonius)
* Elements (Euclid)
* On the Sizes and Distances (Aristarchus)
* On Sizes and Distances (Hipparchus)
* On the Moving Sphere (Autolycus)
* The Sand Reckoner

Problems|

* Problem of Apollonius
* Squaring the circle
* Doubling the cube
* Angle trisection

Centers|

* Cyrene
* Library of Alexandria
* Platonic Academy

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