Number theory in byzantium according to codex vindobonensis

Number Theory in Byzantium according to Codex Vindobonensis phil. Graecus 65 of the 15th cent (fols 11r126r). The numbers and the numerical positioning system

Number Theory in Byzantium according to Codex Vindobonensis phil. Graecus 65 of the 15th cent. The numbers and the positional numeral system

Dr. aria Chalkounternational Congress MICOM 2015Athens 22-26 of September

Codex Vindobonensis Phil. Gr. 65, is a 15th century Byzantine manuscript kept in the National Library of Austria in Vienna.In this article we describe elements from Number theory, as these are presented in the Codex 65This Ms contains a comprehensive programme for teaching mathematics, addressed to an audience consisting of students probably of all the grades of what is todays primary and secondary education, but also state functionaries, merchants, craftsmen of various specialities such as silversmiths and goldsmiths, builders, etc. Dr. Maria Chalkou http:// www.drchalkou.simplesite.com

Codex Vindobonensis phil. Gr. 65t is the first Greek Ms in which there appears the problem of the construction of a square inscribed in an equilateral triangle, so that one of its sides touches one side of the triangle. he theoretical part of the construction is not recorded in the Ms; what is only asked is to calculate the side of the square in relation to the side of the triangleThe Ms is of considerable importance since the scientific research carried out indicated that it is in essence the Mathematical Encyclopaedia of the Byzantines, and possibly the first Mathematical Encyclopaedia

Dr. Maria Chalkou http:// www.drchalkou.simplesite.com





The initial study of the codex was published in 2004 by NKUA as a Doctoral Thesis by the Department of Mathematics The improved and completed edition of the codex was published in 2006 by the Byzantine Research Centre of AUT. This part of the codex, which had remained unpublished until then, since studies of the content of such Mss are rare, was called

Tractatus Mathematicus Vindobonensis GraecusDr. Maria Chalkou http:// www.drchalkou.simplesite.com

The units of Tractatus Mathematicus Vindobonensis Graecus

UNIT 1. Operations between real numbers.UNIT 2. Fractions, ratios, proportions.UNIT 3. Progressions.UNIT 4. Problems on first-degree equations. Solution through practical arithmetic.UNIT 5. Interest on loans or debts.UNIT 6. Problems of sharing in proportional parts.UNIT 7. Problems on manufacture of gold and silver.UNIT 8. Roots of real numbers.UNIT 9. Solution of equations.UNIT 10. Systems of equations.UNIT 11. Plane geometry.UNIT 12. Areas of plane figures.UNIT 13. Solid geometry.


Table of correspondence between Greek and Arabic numbers

(alfa) 1 (giota) 10 (ro) 100 1000 (vita) 2 (kapa) 20 (sigma) 200 2000 (gama) 3 (lamda) 30 (taf) 300 3000 (delta) 4 (mi) 40 (ipsilon) 400 4000 (epsilon) 5 (ni) 50 (fi) 500 (stigma) 6 (ksi) 60 (chi) 600 (zita) 7 (omikron) 70 (psi) 700 (ita) 8 (pi) 80 (omega) 800 (thita) 9 (koppa) 90 (sabi) 900

0 In the Transcription, and anywhere else in the Edition of the Codex 65 it was necessary to be done, the symbol u was used instead of to declare 0. Dr. Maria Chalkou http:// www.drchalkou.simplesite.com

UNIT 1The symbols used for the numbers are the letters of the Greek alphabet, but the calculations are done with the then new decimal Arabic numeration. The author appears not to have adjusted to the new method.Yet it must be stressed that using letters and not numbers did not influence the result since the positional numeral system was used, that is the position of the letter determined its arithmetical value.


Examples of writing numbers

denotes number 123. So does . denotes number 995. So does . denotes number 5360. So does u.

We observe that the numbers in the examples are symbolized by the author in the Greek system of writing in two ways. The second way can be used because each letter takes its value according to its position in the number.


Execution of the operationsFirst example (eight) by (21) 8 21 168We observe that the operations do not differ from those we would execute today, that is we would multiply (one) by (eight), and exactly below and we would write the result, which is again . Next we would multiply (two) by (eight) and we would write (16), in which case the final result of the operation will be (168). In the above example we can observe that the numbers , and are placed on the right-hand column and denote the units. Numbers and are placed on the second column and denote the tens, and the number on the third column and denotes one hundred.The important element here is that the result denotes the number 168 according to the operations. This number could also be symbolized as since denotes one hundred, the number 60, and the number eight. That is we see that when (six) is in the second place from the right, it denotes six tens, and when is in the third place from the right, it denotes one hundred. Therefore each letter takes an arithmetical value according to its position in the number.


As a second example we mention the multiplication of (13) by (15). This operation appears in the manuscript as follows: 13 1 15 195The author multiplies (three) by (five) and has (15). He writes and carries . As he says, he multiplies by crosswise, that is he multiplies by , which gives a result , and by , which gives a result , and adds the results and , in which case he has a result . He adds , which he has carried from the first multiplication, to , and he places , which is their sum, in the place of tens. Finally, he multiplies of by of and writes their product, which is again , in the place of hundreds. Thus the result of the multiplication is the number or (195).


These operations by the anonymous author are justified by applying the distributive law of multiplication over addition and as follows: is written as .10+, and as .10+. Therefore the product of by is:.=(.10+). (.10+)=..100+..10+..10+.=..100+..10+..10+.10+=..100+(.+.+).10+=100+(5+3+1).10+5=100+90+5=195


As a third example we mention the multiplication of (15) by (123). u 0 1 5 1 1 2 3 1 1 8 4 5 First he multiplies by and he has . He writes and carries . Next he multiplies by , by , and adds to them. The result is . He writes and carries . He multiplies ouden by , by , by , and adds to what he finds. The result is . He multiplies ouden by , and by . The result is . The product of the multiplication is . According to the distributive law of multiplication over addition we have: 15.123= (10+5)(100+2.10+3)= 1000+2.100+3.10+5.100+10.10+3.5= 1000+8.100+4.10+5= 1845


On the check of the operation of multiplication 15/7= 2+1/7 6/7 1.6= 6 u 90/7= 12+6/7

The author asks for the remainder of the division of 15 by seven, which is the unit. Because the remainder of the division of six by seven is six, he multiplies the unit by six and places the product, which equals six, in a circle. Finally he finds the remainder of the division of 90 by seven, which is six, and compares it with the number he has placed in the circle, that is 6. If the two results are the same, then the multiplication is correct.


His method is based on the following:If we symbolize the multiplicand with M and the multiplier with , then M. M. (mod 7), and ifM=7k+m, and =7l+n with k, l, m, n and m, n < 7, M. (7k+m).(7l+n) (mod 7), M. m.n+(7k.l+k.n+l.m).7 (mod 7), M. m.n (mod 7)This is the mathematical justification of the method with the remainders of the division by seven, which is used by the author of the 15th century.


On the check of the operation of addition In another problem the Byzantine author deals with the addition of final sums which result from various loan accounts. He adds 1695+1393+3454+4565 and finds 11107. Nowadays we do not refer to a check of additions with more than two addends. However, in Codex Vindobonensis phil. gr. 65 there is a check with the same method which the author applies to the operation of multiplication, which is as follows: He divides each addend by seven, in which case from the first division he has the unit (1) as remainder, from the second one a remainder zero (0), from the third one a remainder three (3), and from the fourth one as remainder the unit (1). Next he divides the sum 11107 by 7 and he finds a remainder five (5). He adds the remainders of the four addends 1+0+3+1, and again he finds (5). Then he writes that the addition is correct.


This procedure can be justified as follows:We symbolize the above addends with 1, 2, 3, 4, the respective remainders of their division by seven with u1= 1, u2= 0, u3= 3, u4= 1, and the respective quotients of the same divisions with k, l, m, n . Then we will have: 1+2+3+4 1+2+3+4 (mod 7),In which case 7k+u1+7l+u2+7m+u3+7n+u4 1+2+3+4 (mod 7), that is u1+u2+u3+u4+7(k+l+m+n) 1+2+3+4 (mod 7),or u1+u2+u3+u4 1+2+3+4 (mod 7)


Thus we observe that apparently the problems are dealt with, by using Arithmetic, while in essence the author seems to have a good knowledge of Number Theory, elements of which he uses in his teachingDr. Maria Chalkou http:// www.drchalkou.simplesite.com


The text after the transcription

' , . , , . . : - . . /. / /. / / /. / Dr. Maria Chalkou http:// www.drchalkou.simplesite.com

The Mathematical Comments

. 171. (). = 12 , . . = ( ), ===6. ==/2, . : 6= +(/2), = 144/5, = (144/5). , , /5, . Dr. Maria Chalkou http:// www.drchalkou.simplesite.com


Number theory in byzantium according to codex vindobonensis

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