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GRD Journals- Global Research and Development Journal for Engineering | Volume 5 | Issue 2 | January 2020 ISSN: 2455-5703 Evolution of Greek Enumeration & Numerical Notations thereto in furtherance of Arithmetical

Operations

Dr. Sankar Prasad Mukherjee Joydev Bera Research Guide Research Scholar Department of Mathematics Department of Mathematics Seacom Skills University, Kolkata, West Bengal, India Seacom Skills University, Kolkata, West Bengal, India

Abstract

The Greek number system was immensely interesting in respect to their approach and methodology, which were uniquely based upon their alphabets. Twenty four letters of Greek alphabets were in use with additional three strange antique letters , , and along with the symbol M (Myriad representing numbers more than ten thousands).Greek Mathematicians, one of the very important contributions was in giving some ideas of fractions. In case of describing bigger numbers like lakhs and crores, they employed unique approach called Tetrads and Octads. With such kind of numerology, they were able to perform all the arithmetical operations, addition, subtraction, multiplication and division etc. but did not spend much effort attempting to justify these operations. Autors‟ belief is to present an amount of assessment collected from various sources about the contributions of Greeks in the way of advancement of Mathematical evolution. Keywords- Attic System, Herodianic Sign, Alphabetic Numerals, Tetrads, Octads, Arithmetical Operation, Apollonius, Continued Multiplication, Square Root

I. INTRODUCTION We can stand as base with several texts and literatures written in Greek language, which were developed approximately during the period of 6th century BC to 5th century AD. It is appealing and encouraging sign of the times that more and more our acknowledgement was attracted to pay due appreciation with a clear understanding of the gift of the Greeks to mankind. The very important contributory aspect of Greek Mathematics is the major shift from its earlier fragmented concepts to a smooth interwinding growth as a subject along with the inventions, theories, proofs and their implementations/applications. This is of paramount importance and major shift towards overall progress and dividends in Mathematics.

A. Alphabetic Numbers and Large Numbers Greek numeral system was very interesting in respect to its approach and methodology having a bit similarity to other systems viz, Egyptian; Babylonian etc. Numerical symbols in earliest Greece were called “Herodianic Signs”. These signs occur frequently in Athenian inscriptions and these systems numerical notations generally called Attic system. The Ionic Greek replaced the Greek Attic system of enumeration into Ionian numerals (Alphabetic numerals or alike string numeral as used in present day computer arithmetic), which was not positional. . Around 450 B.C, Ionic Greek adopted alphabetic notations and the alphabet denoted the numbers 1-9, 10-90 and 100-900 in same way as Egyptian. The following table shows the alphabetic symbols and respective values- Table 1: Greek Alphabetic Symbols Units Tenths Hundredths Symbol value Symbol value Symbol value  1  10  100  2  20  200  3  30  300  4  40  400  5  50  500 6  60  600

 7  70  700  8  80  800  9 90 900

The first nine letters of Greek alphabet as shown in table no 4.1 were associated with the first nine integers as found in modern numerals, the next nine letters represented the first nine integral multiple of 10 and the last nine letters were used for the first nine integral multiples of 100. So we can infer that they were well versed in array system of ascending form; and at the same time base of 10 i.e, decimal system. In Greek numeral systems, larger numbers were also available, which are appeared below-

As indicated earlier, we find here base of 10. They also wrote the numbers from right to left i.e, position / place value way. They placed unit number first in right, then place tenth in one left, then hundredth and so on. Once idea of place value is available, one can represent large numbers and they have ample knowledge about that Arithmetic.

Example: - 32538 =  Here they used the orthodox way of writing the large number; they used this way of writing tens of thousands was by means of letter M with the number of myriads above it.

Example: - 91755876 = 

B. Tetrad’s and Octads We find a reference in “Pappus, Book II : the only difference” which contains Apollonius arithmetical works, that Apollonius called this „tetrads‟, (set of four digits)  , , ,  etc., „simple myriads‟, „double‟, „triple‟ etc, which means 10000, 100002, 100003 and so on. The abbreviation for these successive powers in Pappus are , ,  etc. Nicolas Rhabdas denotes the successive powers of 10000. Who says a pair of dots above the ordinary numerals denoted the number of myriads, two pairs of dots above the numerals denoted the double myriad and three pair of dots denoted triple myriad and so on. Thus

= 9000000, = 2 (10000)2, = 40 (10000)3 and so on. The Sand-Reckoner of Archimedes was a computational accomplishment of another kind. Archimedes‟s Psammites or Sand-reckoner was another special system to express very large number. It contained a new system of notation for expressing numbers in excess of one hundred million. This goes by Octads: 100002 = 100000000 = 108 All numbers from 1 to 108 from first order. The last number 108 from first order is taken as the unit of second order, which consists of all numbers from 108 to 1016, similarly 1016 is taken as unit of third order, which consists of all numbers from 1016 to 1024 and so on.

C. Fraction An ordinary proper Fraction was expressed in various ways. Fractions was denoted by first writing the numerator marked with an accent and then followed by two accented numbers representing the denominator and written as twice. Thus ’ ” ”

and . In case of Fractions having unity for the numerator, the ’ was omitted and denominator

was written only once. Thus ” .

There are a few differences in Diophantus numeral system when compared to their system. In this system, the numerator of any Fraction is written in the line, with the denominator above it, without any accents or marker. A line is put between the denominator above and the numerator below, which describes in Tannery edition of Diophantus. Thus we can say,  . A few more examples from Diophantus be given below- 

Evolution of Greek Enumeration & Numerical Notations thereto in furtherance of Arithmetical Operations (GRDJE/ Volume 5 / Issue 2 / 001)

(1)

(2)

They expressed the Fractions as the sum of submultiples, when numbers are partly integral and partly fractional. Thus from the

previous table ; etc. Also expressed the large numbers by writing the numerator first and the denominator and separated by ’, which means the Fractions are expressed as the numerator divided by the denominator: thus 2507874 / 353147. To calculating Astronomical problems, the Greeks used system of Sexagesimal fractions which appears in the Syntaxis of Ptolemy (100 AD – 170 AD). Around 2nd century AD, in his Syntexis Ptolemy divided the circumference of a circle into 360 parts with four right angles subtended by it at the centre. Each part is called degrees. And also divided each degre () into 60 parts sixtieth or minutes , each of these again into 60 seconds and so on. Thus there was a convenient fractional system exists for arithmetical calculations. So the Fractions we represent by , so many of these we write , and so on. Here the ordinary numerals with units  written first, then the numbers of sixtieths or minutes with one accent and thereafter the seconds with two accents and so on, so forth. Thus   = 3 ( denoted the abbreviation of the unit );   ’ ” = 34 52’ 20”. Similarly, we expressed   ’ ” = 8’ 41” Where p denotes the segment of the diameter.

D. Arithmetical Operations The arithmetic operations are complex in that so many symbols are used. For addition and subtraction, the Greek would keep the several powers of 10 separate in a manner practically corresponding to our system of numerals, the hundreds, the thousands etc being written in separate vertical rows. Example of addition represented bellow- Example of Subtraction be represented bellow-

Greek multiplication method was somewhat different from Egyptian method. This method dependent on the direct use of multiplication table. When they multiply two numbers, they wrote the numbers in two columns. First is multiplier and second is multiplicand. Then they wrote half of the number in first column of the preceeding row and double the number in second column of the preceeding row and so on, until they have 1 in first column. Then cuts the all numbers in the second column which are opposite even numbers in first column and odd all the numbers left in second column, which is the required result. Example :- 203 is to multiplied by 81 81 203 40 406 20 812 10 1624 5 3248 2 6496 1 12992

16,443 = 81 x 203

Evolution of Greek Enumeration & Numerical Notations thereto in furtherance of Arithmetical Operations (GRDJE/ Volume 5 / Issue 2 / 001)

The great commentator Eutocius of 6th century AD, gives a great many multiplications appear at the left and a numerical translation of the symbols and operations at the right are from his commenries on the third proportion of Archimedes on the measure of the circle.

The multiplicand is considered first. Then multiplier is written below where kept before the multiplier i.e,

All the digits of multiplicand are multiplied by the highest power of 10 in the multiplier. Similarly the process continues to required steps developing on the digits and the steps total will be the result. Example :- To find the square of  (or 543)

Example showing multiplication in three steps where multiplicand and multiplier are respectively three digit numbers. The total of the above three steps is the result.

As  represents 500, so  multiplied by  will give 250,000, which expressed as ; as  represents 40, so  multiplied by  will give 20,000, which expressed as ; as  represents 3, so  multiplied by , will give 1,500, which is expressed as .

Therefore (i) the first partial product is or 271,500.

Similarly, we get the second and third partial product. (ii) The second partial product is or 21,720 and (iii) the third partial product is or 1629.

Then add all partial product and get the final result. Thus the square of the number  (or 543) is or 294,849.

Greek methods for division depends on multiplication and subtraction. For example, suppose (169,744) is divided by  (412).

Evolution of Greek Enumeration & Numerical Notations thereto in furtherance of Arithmetical Operations (GRDJE/ Volume 5 / Issue 2 / 001)

Divisor Dividend Quotient

 (412) (169,744)  (400) 1st term 412 x 400 = 164800 Remainder = 4944 ( ) 412 x 10 = 4120  (10) 2nd term Remainder = 824 412 x 2 = 824  (2) 3rd term Hence the whole quotient is  (412).

E. Apollonius Process of Continued Multiplication Apollonius ( late 3rd – early 2nd centuries BC) was a great geometerand astronomer in Greece. His objectives was to discover a easy method of constructed product of numbers of factors. Each factor is represented by single letter in Greek notation. So, instead of multiplication of large numbers, he confined the multiplication of any number of factors as shown below : 1) A number of units as 1,2,3,……….,9 2) A number of even tens i.e, 10, 20, 30, ………. , 90 3) A number of even hundred 100, 200, 300, ……….., 900 It may be noted that it does not deal with factors above 100 i.e, not in thousands. The reason behind it was the Greek numerical alphabet was upto only 900. The attracction of this method is consecutive separate multiplication as (a) the base of several factors and (b) the factors containing the powers of ten. Also given that 30 is the multiple of 10 with base 3 and 800 is the multiple of 100 with base 8. The multiplication of three numbers such as 2, 30, 800; Apollonius multiplication process expressed the product of tens and the hundred and lastly multiples of two products, where the bases of these three numbers are 2, 3 and 8. Finally the product can be expressed as the number of units less than myriad, then number of myriads, number of „double 2 myriad‟, „triple myriad‟ etc; which expressed in the form A0 + A1M + A2 M +………… 4 Where M is a myriad i.e, 10 an A0, A1, A2 ……. represents some number less or equal to 9999.

F. Extracting the Square Root Now, we discuss the process of extracting the square root of a number. Theon‟s ((70 AD- 135 AD), Greek Mathematician) extraction of square root is rather geometrical, yet it is based on the Algebra‟s binomials expansion. In his commentary of Ptolemy almagest explained the process of extracting square root of a number. To extract square root, he took first the root of nearest square number and subtracted from the number, itself. Then he doubled this first root and divided this above subtracted reduced number, and took the remainder term. Finally, took the root of this remainder. Thereafter, added first root and root of remainder terms. First, Theon‟s algorithm was based on Euclid‟s elements-II. 4, which claimed: “If a straight line is cut at random, then the square on the whole equals the sum of the squares on the segments plus twice the rectangle contained by the segments.” That expressed in the form as,

Fig. 1: Square to extract square root

Theon first illustrated this process by finding square root of 144. The steps were as follows- 1) To take highest possible denomination in the square root is 10. 2) Subtract 102 from 144, leaves 44. 3) Double the root I.e, 2 x10 = 20, because there are two rectangle. 4) Divide 44 by 20 to get 2 with a remainder 4, i.e, 2 . 20 + 22 = 44 5) Then, 4 is the final small square; whose root is 2. 6) Thus, √

Evolution of Greek Enumeration & Numerical Notations thereto in furtherance of Arithmetical Operations (GRDJE/ Volume 5 / Issue 2 / 001)

We can say that process of reducing by squares on the basis of Geometrical concept.

II. CONCLUSION It was most amazing that at the initial stage, the Greeks used a very unique approach to tackle numerology and arithmetical operations using alphabetic numbers. They represent numbers by alphabetic symbols (but not positional). With such kind of numerology they were able to perform all the Arithmetical operations. Later on, one of the eminent Mathematicians, Apollonius (late 3rd century BC to early 2nd century BC) developed continued multiplication methodology of big numbers in a simple manner. The other amazing methodology to be mentioned was extracting of square root using Geometry by Theon ((70 AD- 135 AD), another great Greek Mathematician). So we can conclude that the evolutionary history of Mathematics is indebted to various countries, places and great inventors at different point of times. As in this article we see the alphabetic numbers along with amazing approach of arithmetical operations; continued simple multiplication methodology of big numbers and uncommon way of extracting square root using Geometry.

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