Numerical script of the Aryabhata

The Aryabhata Code , also number system of the Aryabhata , is a representation of numbers by syllables that the Indian astronomer and mathematician Aryabhata defined in order to write his mathematical formulas in Sanskrit verses. The code uses the 33 consonants and 9 vowels of the Indian alphabet and can represent the whole numbers from 1 to 10 ¹⁸ . Since several syllable sequences can be assigned to some numbers, the code is not unique .

prehistory

Very little is known about the history of the code. Probably from the time of the Alexandrian campaigns around 330 BC. BC, but also through India's close ties to the Persian Empire of the Sassanid dynasty 226–641 AD, Indian scholars acquired knowledge of Greek and thus Babylonian astronomy, as well as their sexagesimal system of values . Indian astronomers are already familiar with the first representations of significance around 500. The numbers are not yet written in digits, but in word numbers or symbols and held in verse. Interestingly, they had existed since around 250 BC. The Brahmi digits , i.e. characters that were used exclusively to represent numbers and which were to become the forerunners of our current number symbols, were apparently only used in everyday life until the 7th century, but not for religious or scientific writings in Sanskrit used.

Definition and characteristics

Aryabhata used around 510 in the first chapter of Gitikapada of his book Aryabhatiya his own number system based on 100 by designating the numbers with syllables.

Coding table from Aryabhata

The vowels and so on indicate the powers of 100, those of Aryabhata so called Varga consonants ( Varga = "square") from to have the values 1 to 25 (for example ), and the remaining 8 Avarga consonants from to have the Values 30, 40,…, 100. The syllables with indicate the ones and tens, the syllables with the hundreds and thousands and so on, for example ${\ displaystyle a, i, u}$ ${\ displaystyle ka}$ ${\ displaystyle ma}$ ${\ displaystyle ca = 6, ga = 3, {\ mbox {ṇa}} = 5, cha = 7}$ ${\ displaystyle ya}$ ${\ displaystyle ha}$ ${\ displaystyle a}$ ${\ displaystyle i}$

{\ displaystyle khya = khaya = 2 + 30 = 32}

,

and

{\ displaystyle khyu = khuyu = (2 + 30) \ times 10 ^ {4} = 320,000}

.

Aryabhata's number system is not a position system, a number is written as the sum of its powers of hundreds. In addition, the system is not unique, for example is or . Aryabhata was well aware of these ambiguities, he even mentions the latter expressly, presumably because he wanted to enable more degrees of freedom for the representation of numbers in metric meter. ${\ displaystyle ha = ki = 100}$ ${\ displaystyle {\ dot {n}} ma = ya = 30}$

Remarkably, a zero is not even necessary in this system, missing sum terms are simply left out. Nevertheless, Aryabhata uses the word for "emptiness", "place" or "zero" in explaining his system , which is often taken as an indication that he already knew the decimal system with zero . However, it was not until a century after Aryabhata that his student Bhaskara I used our current decimal system with the first nine Brahmi digits and the zero for the first time in his commentary on Aryabhatiya from the year 629 . Aryabhata's code remained without further mathematical applications and is therefore rather a marginal phenomenon in terms of mathematical history, but its influence on our current number system was probably significant. ${\ displaystyle kha}$

Reasons to develop the code

One can only speculate about Aryabhata's reasons for developing his number system. The numbers formed by the letters are not easy to understand at first and also had to be translated into common numerals by the ancient Indians. The system is also hardly suitable for efficient calculation, and some of the sound combinations are pronounced tongue twisters. The following advantages of the code are mentioned in the literature:

Due to the different display options, the numerals can be incorporated into the metric verse relatively easily, which is hardly possible with fixed numerals.
Large numbers are represented by relatively short sequences of letters. For example, Aryabhata represents the number of sidereal solar orbits of a yuga as

khyughṛ = (2 + 30) ⋅ 10 ⁴ + 4 ⋅ 10 ⁶ = 4,320,000 .

With the aryabatic number system it is relatively easy to carry out approximation methods for taking the square root and the cube root ; they are essentially based on the binomial formulas

{\ displaystyle {\ sqrt {a ^ {2} + 2ab + b ^ {2}}} = a + b, \ qquad {\ sqrt [{3}] {a ^ {3} + 3a ^ {2} b + 3ab ^ {2} + b ^ {3}}} = a + b.}

literature

Kurt Elfering: The Mathematics of Aryabhata I. Text, translation from Sanskrit and commentary . Wilhelm Fink Verlag, Munich 1975, ISBN 3-7705-1326-6 .
Georges Ifrah: Universal History of Numbers . Campus Verlag, Frankfurt a. M./New York 1986, ISBN 3-593-34192-1 .
Bartel Leendert van der Waerden : Awakening Science. Egyptian, Babylonian and Greek mathematics . 2nd Edition. Birkhäuser-Verlag, Basel / Stuttgart 1966, ISBN 3-764-30399-9 .

Individual evidence

↑ Georges Ifrah: Universal History of Numbers . Campus Verlag, Frankfurt a. M./New York 1986, ISBN 3-593-34192-1 , p. 449.
↑ Bartel Leendert van der Waerden : Awakening Science. Egyptian, Babylonian and Greek mathematics . 2nd Edition. Birkhäuser-Verlag, Basel / Stuttgart 1966, ISBN 3-764-30399-9 , p. 90.
^ Kurt Elfering: The Mathematics of Aryabhata I. Text, translation from Sanskrit and commentary. Wilhelm Fink Verlag, Munich 1975, ISBN 3-7705-1326-6 , p. 34.
^ Kurt Elfering: The Mathematics of Aryabhata I. Text, translation from Sanskrit and commentary. Wilhelm Fink Verlag, Munich 1975, ISBN 3-7705-1326-6 , pp. 52ff, 62ff.

[1] Georges Ifrah: Universal History of Numbers . Campus Verlag, Frankfurt a. M./New York 1986, ISBN 3-593-34192-1 , p. 449.

[2] Bartel Leendert van der Waerden : Awakening Science. Egyptian, Babylonian and Greek mathematics . 2nd Edition. Birkhäuser-Verlag, Basel / Stuttgart 1966, ISBN 3-764-30399-9 , p. 90.

[3] Kurt Elfering: The Mathematics of Aryabhata I. Text, translation from Sanskrit and commentary. Wilhelm Fink Verlag, Munich 1975, ISBN 3-7705-1326-6 , p. 34.

[4] Kurt Elfering: The Mathematics of Aryabhata I. Text, translation from Sanskrit and commentary. Wilhelm Fink Verlag, Munich 1975, ISBN 3-7705-1326-6 , pp. 52ff, 62ff.