Greek numerals

The Greek numbers are represented by letters and digits. Since ancient times, three different display types can be distinguished.

The oldest, the akrophone principle, put in the first letter of the number words, while the other two each emanating from the order of the letters in the alphabet, which you either go to the Milesian principle in decades numerical values ​​or by the Thesis principle directly from their position derived numerical values ​​were assigned in the alphabet. The latter was, however, as it is limited to only 24 values ​​at the mathematicians no use.

The alphabetical number system is an idea of the Greeks. Other alphabets were later adapted according to their model. However, the principle of the system itself is in use already in the hieratic and demotic Egyptian spelling of numbers. The alphabetical number system was in ancient Greece, the most important by far. It was the standard number system of all Greek mathematicians from antiquity to modern times, that is, until the takeover of Indian numerals in modern European mathematics.

Since the Hellenistic period, the Greek alphabetical numbering system was also used in numerology oriented geriatric practices in the area of ​​number exegesis, the Onomatomantik and magic.

It is in the Greek is still widely used for the posting of ordinal numbers ( for example, for rulers name, the use of Roman numerals in the German equivalent), while for cardinal numbers today usually also in Greek - if not value emphasizes on a traditional notation is placed - the decimal Indo- Arabic numerals are common. Each modern Greeks are the old numbers still familiar, such as the count of classes is based on the old system: A child who attends the fifth grade, that is in the Epsilon class.

According to the Greek model, almost all alphabets, such as the Hebrew alphabet, the Arabic alphabet, the Cyrillic alphabet, and most other alphabets have been adapted as a number alphabets. However, as in West Rome, by the end of the Middle Ages, the Roman numerals were considered standard numbering system, the Latin alphabet was up in the modern era never adapted. Instead, one related in the Eastern Roman Empire during the Middle Ages, the Greek alphabetic numerals.

The acrophonic numbers

Since about the beginning of the fifth century BC is attested use of the so-called acrophonic numbers. The first letter of the word number can be used to override the corresponding numerical value. The result is a system very similar to - developed at about the same - Roman numerals.

It was: Ι ( single prime ) = 1, Π (from pente ) = 5, Δ ( by decade ) = 10, Η (of Hecaton ) = 100, Χ (of chilioi ) = 1000 and Μ (of myrioi ) = 10 000

In addition to five same pi as pente, there are also signs of 50, 500, 5000 and 50,000. Here, 10, 100, 1000 and 10000 written into the Pi, which corresponds to a multiplication by five and avoids a fivefold juxtaposition of the same number.

Note: For practical reasons, Π was in the table as | | shown.

1982 = X | Η | HHHH. | Δ | = ΔΔΔII MCM. LXXXII.

The index numbers

The linear, defective system according to the Thesis principle

As a " thesis " system is referred to in the research, building on a Formulierweise Artemidors a linear number alphabet, in which the numerical values ​​of the letters without any other increase principle or other arithmetic operations directly even from a position (thesis ) of the respective letters in the order of the alphabet result. As an indication that such a thesis system based on the principle Alpha to Omega = 1 to 24 ( ie excluding the three numeric characters Digamma, even stigma Qoppa and Sampi ) could have played a role in the Greek culture of antiquity, one has in particular the tradition of Homer's Iliad and Odyssey valued, their respective 24 songs will be numbered and cited at least in the tradition of the Alexandrian grammarians after the letters of the alphabet, and also Zählbuchstaben on the frieze panels of the Pergamon Altar and for values ​​less than ten ( so far without clear indication of a thesis system) on Lostafeln or to designate the parts of the city of Alexandria.

Through the dream book Artimedors is also testified that in the geriatric interpretation of personal names sometimes the Thesis values ​​of the letters were applied when the usual for such purposes Milesian values ​​would have been inappropriately high.

The complete number system after the Milesian principle

Since the middle of the 3rd millennium BC, the so-called hieratic figures are attested. These were the contractions of the even older, analog representation of Egyptian hieroglyphics numbers. In its original form, for example, the number 397 as three Kringelchen (three hundred ) plus nine Bögchen (ninety ) and seven strokes has been shown. Each site was initially separately, but was a sign, a digit, contracted later. These figures were used, for example, in the Rhind Papyrus.

The demotic writing simplified the digits again. But middle of the 4th century BC, the Greeks came up with the idea, the first three of each consisting of nine digits hieratic - demotic series of numbers, to be replaced by the letters of their own alphabet. Since then one speaks of the "alphabetical number system ". It divides the alphabet into three groups of nine characters for the representation of the One, the tens and hundreds.

To this takes the total number of 3 × 9 = to have 27 characters available were for the purpose of the numerical scale three old letters that were already eliminated in the classical Greek alphabet, reintegrated.

• 6 = Digamma - It corresponds to the Latin small letter F. As the word end variant of the sigmoid colon ( ς ) together with a Tau ( τ ) as a ligature ( ς ), which is interpreted as a stigma. In today's printing works the letters sigma -tau ( στ ) is used mostly.
• 90 = Koppa - This is the old Qoph, so the Latin Q. Originally written in the form ϙ, and later in the form of writing: ϟ.
• 900 = Sampi - Sampi or Tsampi corresponds to the Phoenician Sade (San ) and the Hebrew Tzade; graphically in Greek: ϡ.

While F and Q are on their original place in the alphabet, was the old San or Tzade, which is actually between P and Q, as Tsampi, set to the last place.

With these 27 characters and they lay assigned numerical values ​​could be already the numbers between 1 and 999 write by additive combination of units, tens and hundreds, so 8 = η ( eta ), 88 = πη ( pi eta = 80 8), 318 = τιη ( Tau Iota Eta = 300 10 8). A sign of the zero did not exist and was not necessary for the purposes of the case number by about 200 = σ (Sigma ), 202 = σβ ( Sigma Beta 2 = 200 ), 220 = σκ (Sigma Kappa = 200 20) wrote.

To distinguish the numbers in the typeface of words, the former were usually overwritten in the manuscripts with a dash, for example, = 310, while this in the age of printing an apostrophe ( Δεξιά Κεραία ) has naturalized before the first digit in the Unicode U 0374th If the identity number as clear, however, sometimes it is omitted.

The hieratic figures 1000-9999 can be represented as alphabetic numerals. For this, the first number letter was multiplied by adding a diacritic with thousands. Hand one usually uses a character which stands in the form of a small open left hook at the top left before the number. In letterpress printing, for the subscript apostrophe ͵ ( Αριστερή Κεραία ) has prevailed in Unicode U 0375.

Expansion of the number space

The standard extension to one hundred million

To date, the use of the word million number is unusual in Greece. One speaks instead of " hundred myriads " ( εκατομμύριο, ekatommýrio ), a myriad corresponds to the tens of thousands. The number 99 million 999 thousand 999 were made in ancient times 9999 myriads 9999. This is traditionally the highest number of the Greek number system.

This number can be Θ ϡ ϟ θ M Θ ϡ ϟ θ be written easily and clearly; "M" being the myriad mean. ( A possible M = 40 000 is YES in the system not available. ) Likewise, for example, writes Aristarchus of Samos 71 million 755 thousand 875 as 7175 myriads 5875: ͵ ZPOE M ͵ EΩOE. While Diophantus of Alexandria in 4372, for example, myriads 8097, the icon for the myriads even reduced to a point: ͵ ΔTOB. ͵ H ϟ Z.

In order to characterize the Myriadenzahlen clear preference was given to it in ancient times to write the units of the myriads over the akrophone Μ icon.

Today, we used the number of words δισεκατομμύριο ( disekatommýrio ) = 1,000,000,000, τρισεκατομμύριο ( trisekatommýrio ) = 1,000,000,000,000 and so on.

The potency of extension to 10 high 36

Another, but very rarely used system for the representation of large numbers can be found in Apollonius of Perge, the myriads of first, second, third, etc., until ninth order differed according to the testimony of Pappus of Alexandria in ascending potency by Μ the with the character α to θ = 1 to 9 overwrote that were here therefore no longer valued as a multiplier, but as the exponent of a power. As a representation of the number 5,462,360,064,000,000, this resulted in a case such as the following:

Importance of alphabetic numbers

In theory, the number systems apply to both the hieratic, and the alphabetic numbers as addition systems of the third kind

Except in the western Roman territories, where you always clung to the Roman numerals, dominated this progressive system - in their adaptations to the respective alphabets - a very long time the science and management of Persia, Armenia, Georgia, Arabia, Ethiopia, the Byzantine Empire and the old Russia. Only the Hindu numerals solved the system, after four thousand years of dominance, gradually. In the Arab world at the end of the first millennium after Christ, or until the middle of the second millennium.