History of Mathematics by Bryan M. Torres

History about Mathematics

1. HISTORY OF MATHEMATICS Class Schedule: TTH | 2:30 – 4:00 PM Consultation: F | 4:00 – 5 :00 PM BRYAN M. TORRES, MMath Assistant Professor I mobile phone: [phone] e-mail address: [email protected] www.facebook.com/sipnayan23

2. The Development of Mathematics Ancient Period Origin of Mathematics: Egypt and Babylonia

3. Objectives Discuss the development of mathematics in the ancient period Show the evolution of numeration systems in ancient times Recognize the symbols and notations used Perform the mathematical operations used in this period

4. Primitive Counting A Sense of Number The term tally comes from the French verb tailler , “to cut,” like the English word tailor; the root is seen in the Latin taliare , meaning “to cut.” It is also interesting to note that the English word write can be traced to the Anglo-Saxon writan , “to scratch,” or “to notch.

5. Primitive Counting Notches as Tally Marks Bone artifacts bearing incised markings seem to indicate that the people of the Old Stone Age had devised a system of tallying by groups as early as 30,000 B.C. The most impressive example is a shinbone from a young wolf, found in Czechoslovakia in 1937; about 7 inches long, the bone is engraved with 55 deeply cut notches, more or less equal in length, arranged in groups of five. (Similar recording notations are still used, with the strokes bundled in fives, like |||| .

6. Primitive Counting Notches as Tally Marks wolfbone used for tallying

7. Primitive Counting The Peruvian Quipus: Knot as Numbers In the New World, the number string is best illustrated by the knotted cords, called quipus, of the Incas of Peru. They were originally a South American Indian tribe, or a collection of kindred tribes, living in the central Andean mountainous highlands.

8. Primitive Counting The Peruvian Quipus: Knot as Numbers

9. Primitive Counting The Peruvian Quipus: Knot as Numbers To appreciate the quipu fully, we should notice the numerical values represented by the tied knots. Just three types of knots were used: a figure-eight knot standing for 1, a long knot denoting one of the values 2 through 9, depending on the number of twists in the knot, and a single knot also indicating 1. The figure-eight knot and long knot appear only in the lowest (units) position on a cord, while clusters of single knots can appear in the other spaced positions.

10. Primitive Counting The Peruvian Quipus: Knot as Numbers Also, the reappearance of either a figure-eight or long knot would point out that another number is being recorded on the same cord. Recalling that ascending positions carry place value for successive powers of ten, let us suppose that a particular cord contains the following, in order: a long knot with four twists, two single knots, an empty space, seven clustered single knots, and one single knot.

11. Primitive Counting The Peruvian Quipus: Knot as Numbers in order: a long knot with four twists, two single knots, an empty space, seven clustered single knots, and one single knot. For the Inca, this array would represent the number  

12. Primitive Counting Mayan The Mayan calendar year was composed of 365 days divided into 18 months of 20 days each, with a residual period of 5 days. This led to the adoption of a counting system based on 20 (a vigesimal system). Numbers were expressed symbolically in two forms.

13. Primitive Counting Mayan The priestly class employed elaborate glyphs of grotesque faces of deities to indicate the numbers 1 through 19. These were used for dates carved in stone, commemorating notable events. The common people recorded the same numbers with combinations of bars and dots, where a short horizontal bar represented 5 and a dot 1.

14. Primitive Counting Mayan

15. Primitive Counting Mayan The symbols representing numbers larger than 19 were arranged in a vertical column with those in each position, moving upward, multiplied by successive powers of 20; that is, by 1, 20, 400, 8000, 160,000, and so on. A shell placed in a position would indicate the absence of bars and dots there. In particular, the number 20 was expressed by a shell at the bottom of the column and a single dot in the second position.

16. Primitive Counting Mayan For example of a number recorded in this system, let us write the symbols horizontally rather than vertically, with the smallest value on the left: For us, this expression denotes the number 62808  

17. Primitive Counting Mayan Because the Mayan numeration system was developed primarily for calendar reckoning, there was a minor variation when carrying out such calculations. The symbol in the third position of the column was multiplied by rather than by , the idea being that 360 was a better approximation to the length of the year than was 400.  

18. Primitive Counting Mayan The place value of each position therefore increased by 20 times the one before; that is, the multiples are 1, 20, 360, 7200, 144,000, and so on. Under this adjustment, the value of the collection of symbols mentioned earlier would be  

19. Primitive Counting Quiz Write 102386 in the following notations 1. Peruvian 2. Mayan 2 different place value principle

20. Hieroglyphic Representation of Numbers As soon as the unification of Egypt under a single leader became an accomplished fact, a powerful and extensive administrative system began to evolve. The census had to be taken, taxes imposed, an army maintained, and so forth, all of which required reckoning with relatively large numbers

21. Hieroglyphic Representation of Numbers

22. Hieroglyphic Representation of Numbers The spectacular emergence of the Egyptian government and administration under the pharaohs of the first two dynasties could not have taken place without a method of writing, and we find such a method both in the elaborate “sacred signs,” or hieroglyphics, and in the rapid cursive hand of the accounting scribe. The hieroglyphic system of writing is a picture script, in which each character represents a concrete object, the significance of which may still be recognizable in many cases.

23. Hieroglyphic Representation of Numbers In one of the tombs near the Pyramid of Gizeh there have been found hieroglyphic number symbols in which the number one is represented by a single vertical stroke, or a picture of a staff, and a kind of horseshoe, or heelbone sign ꓵ is used as a collective symbol to replace ten separate strokes. In other words, the Egyptian system was a decimal one (from the Latin decem , “ten”), which used counting by powers of 10.

24. Hieroglyphic Representation of Numbers That 10 is so often found among ancient peoples as a base for their number systems is undoubtedly attributable to humans’ ten fingers and to our habit of counting on them. For the same reason, a symbol much like our numeral 1 was almost everywhere used to express the number one.

25. Hieroglyphic Representation of Numbers Special pictographs were used for each new power of 10 up to 10,000,000: 100 by a curved rope, 1000 by a lotus flower, 10,000 by an upright bent finger, 100,000 by a tadpole, 1,000,000 by a person holding up two hands as if in great astonishment, and 10,000,000 by a symbol sometimes conjectured to be a rising sun.

26. Hieroglyphic Representation of Numbers Other numbers could be expressed by using these symbols additively (that is, the number represented by a set of symbols is the sum of the numbers represented by the individual symbols), with each character repeated up to nine times. Usually, the direction of writing was from right to left, with the larger units listed first, then the others in order of importance.

27. Hieroglyphic Representation of Numbers Thus, the scribe would write to indicate our number

28. Hieroglyphic Representation of Numbers For example, all stood for the number 1232. Thus the Egyptian method of writing numbers was not a “positional system”—a system in which one and the same symbol has a different significance depending on its position in the numerical representation.

29. Hieroglyphic Representation of Numbers Addition and subtraction caused little difficulty in the Egyptian number system. For addition, it was necessary only to collect symbols and exchange ten like symbols for the next higher symbol. This is how the Egyptians would have added, say, 345 and 678:

30. Hieroglyphic Representation of Numbers

31. Hieroglyphic Representation of Numbers

32. Hieroglyphic Representation of Numbers

33. Egyptian Hieratic Numeration With the introduction of papyrus, further steps in simplifying writing were almost inevitable. The first steps were made largely by the Egyptian priests who developed a more rapid, less pictorial style that was better adapted to pen and ink. In this so-called “hieratic” (sacred) script, the symbols were written in a cursive, or free-running, hand so that at first sight their forms bore little resemblance to the old hieroglyphs.

34. Egyptian Hieratic Numeration In both of these writing forms, numerical representation was still additive, based on powers of 10; but the repetitive principle of hieroglyphics was replaced by the device of using a single mark to represent a collection of like symbols. This type of notation may be called “ cipherization .” Five, for instance, was assigned the distinctive mark instead of being indicated by a group of five vertical strokes.

35. Egyptian Hieratic Numeration

36. Egyptian Hieratic Numeration The hieratic system used to represent numbers is as shown in the preceeding table. Note that the signs for 1, 10, 100, and 1000 are essentially abbreviations for the pictographs used earlier. In hieroglyphics, the number 37 had appeared a

37. Egyptian Hieratic Numeration but in hieratic script it is replaced by the less cumbersome The larger number of symbols called for in this notation imposed an annoying tax on the memory, but the Egyptian scribes no doubt regarded this as justified by its speed and conciseness.

38. The Greek Alphabetic Numeral System Around the fifth century B.C., the Greeks of Ionia also developed a ciphered numeral system, but with a more extensive set of symbols to be memorized. They ciphered their numbers by means of the 24 letters of the ordinary Greek alphabet, augmented by three obsolete Phoenician letters (the digamma for 6, the koppa for 90, and the sampi for 900).

39. The Greek Alphabetic Numeral System The initial nine letters were associated with the numbers from 1 to 9; the next nine letters represented the first nine integral multiples of 10; the final nine letters were used for the first nine integral multiples of 100. The following table shows how the letters of the alphabet were arranged as numeral

40. The Greek Alphabetic Numeral System

41. The Greek Alphabetic Numeral System Because the Ionic system was still a system of additive type, all numbers between 1 and 999 could be represented by at most three symbols. The principle is shown by  

42. The Greek Alphabetic Numeral System For larger numbers, the following scheme was used. An accent mark placed to the left and below the appropriate unit letter multiplied the corresponding number by 1000; thus represents not 2 but 2000. Tens of thousands were indicated by using a new letter M, from the word myriad (meaning “ten thousand”). The letter M placed either next to or below the symbols for a number from 1 to 9999 caused the number to be multiplied by 10,000, as with  

43. The Greek Alphabetic Numeral System

44. Quiz

45. References Oronce , Orlando A. (2016) RBS GENERAL MATHEMATICS . First Edition. Rex Printing Company, Inc. https://www.wolframalpha.com https://www.geogebra.org