1	Numeration Thomas J. Bergin © Computer History Museum American University
2	Symbols Symbols are a means of communicating facts and ideas: I have three cows and two sheep I will see you tomorrow Clay tablets in Sumer were used for pictographic writing @ 3300 BC Egyptians use hieroglyphic signs on papyrus
3	Clay Tablet
4	Symbols: English: 4 7 A a Z z International: Ê Š Æ Ü ç ê ñ Mathematics: + - / * ^ ƒ € Special characters: a @ ® ™ © § K Greek: F G P S q w D
5	Numeration Virtually all numeration starts as tallies, using single strokes to represent each additional unit: / for one, // for two, etc. Evidence of tallies has been found on bone fragments from as early as 15,000 BC. A tally system can exist before a language develops words for numbers. Reference: Bunch and Hellemans, The Timetables of Technology, Simon & Schuster, 1993
6	Tally Stick
7	Tokens Early societies developed tokens to represent quantities. By 4000 BC, tokens existed for “ten sheep” ( say: .) and for “one sheep”(say: =) Given the following tokens: . . . = = = How many sheep are represented? There were different tokens for different commodities! Three horses would be represented as “ttt” and not “===,” which is three sheep!
8	Concept of Number Around 4000 BC, traders in Uruk were discovering that the same number could be used to mean ten sheep, ten bags of grain, or ten talents of copper. About 3000 BC, Egyptian tallies show items grouped at ten; these tallies were regrouped at a hundred, and regrouped again at one thousand.
9	Hieroglyphic numbers Source: 195.8.72.23/numbers.htm ©Mark Millmore used with permission
10	Two examples Source: 195.8.72.23/numbers.htm ©Mark Millmore used with permission
11	12,425 Birds Source: 195.8.72.23/numbers.htm ©Mark Millmore used with permission
12	Egyptians Hieroglyphics (pictographic symbols) 1 = Stick / 10 = Arch П 100 = Coiled Rope Ã 1000 = Lotus Flower ë 10,000 = Finger (pointing to sky) æ 100,000 = Tadpole (from the Nile) 1,000,000 = Man (arms reaching to heaven)
13	Hieroglyphics (addition) П П П П / / / / / П П П П П П / П П П / / / / / / / / 45 61 38
14	Hieroglyphics (addition) П П П П / / / / / П П П П П П / П П П / / / / / / / / ----------------------------------- П П П П П П П П П П П П П / / / / / / / / / / / / / / ---------------------------------- Ã П П П П / / / / 45 61 38 --------------------- 144 --------------------- 144
15	Multiplication by Doubling (23 X 13) Number multiplier П П / / / 1 П П П П / / / / / / 2 П П П П П П П 4 П П / / Ã П П П П П П П 8 П / / / / Number multiplier 23 1 46 2 92 4 184 8
16	Multiplication by Doubling (23 X 13) Number multiplier П П / / / 1 a П П П П / / / / / / 2 П П П П П П П 4 a П П / / Ã П П П П П П П 8 a П / / / / Number multiplier 23 1 a 46 2 92 4 a 184 8 a ================= 299 13
17	Check and verify!!! 23 times 13 6 9 2 3_ 2 9 9
18	Greeks and Romans The Greeks adapted their alphabet for numerals; others followed their example. Roman numerals are also alphabetical, but they did not originate as such. Early artifacts show that the X for ten, originated from the way in which scribes drew a slanted line through the number for four: ///// + / became X; one half of X was V, and the habit of putting a circle around the tenth X to indicate one hundred became C
19	Roman Numerals I XI XXX XXXX L II XII III XIII IV XIV V XV VI XVI VII XVII VIII XVIII IX XIX X XX
20	Roman Numerals L = 50 C = 100 D = 500 M = 1000 V (bar)= 5000 “vee bar” XV (bar) = 15,000 L (bar) = 50,000 C (bar) = 100,000 M (bar)= 1,000,000
21	Addition using Roman Numerals 2318 MM CCC X V III +821 DCCC XX I______ 3139 MM D CCC XXX V IIII CCC -------------------------------- MM IX D collecting D D C XXX terms -------------------------------- MMM C XXX IX
22	Subtraction using Roman Numerals 2486 MM CCCC L XXX V I -1343 M CCC XXXX III 1143 expand: MM CCCCXXXXX IIIIII \| XXX \| minus: M CCC XXXX III \| \|-------------------à M C XXXX III
23	Multiplication using Roman Numerals I____V___ _X___L____ C____D___ M V \| V XXV L CCL D MMD V-bar X \| X L C D M V-bar VV-bar 2 8 XXVIII times 1 2 5 6 XXVIII times 1 = XXVIII 2 8__ XXVIII times 1 = XXVIII 3 3 6 XXVIII times 10 = CCLXXX CC L XXXXXXX VV IIIIII Collect terms: CCC XXX VI
24	Hindu-Arabic notation The Indians used horizontal tallies (/) for one, two and three, and special symbols for four through nine. Around 600 CE, the Indians started using place values, i.e., instead of writing the equivalent of 100 + 80 + 7, they wrote 187 Only nine digits were used along with a symbol for zero, probably derived from astronomer’s marking empty places. A famous inscription dated 870 CE contains the first zero that has survived.
25	Hindu-Arabic numerals Ancient Hindus: zero place values and decimal system (base 10) Positional Notation: 4, 4 2 8 Arab traders brought the system to Europe where it became known as “Arabic numerals” base: X⁵ X⁴ X³ X² X¹ X⁰ 10 100000 10000 1000 100 10 1 2 32 16 8 4 2 1
26	House of Wisdom Caliph Al-Mamun 800 AD Baghdad Prophet Mohammed: “Seek Learning Though It Be In China” Astronomical tables - Feast of Ramadan Mecca (geography and geometry)
27	Al-Khowarizmi Abu Jafar Mohammed Ibn Musa Al-Khowarizmi born 780 AD Kiva, USSR Hisab Al Jabr Wal-Mugabalah (The Compendious Book on Calculations by Completion And Balancing) Used Hindu numerals and decimal system Spread throughout Europe “business” problems: inheritance of estates modern words: algorithm from “Al-Khowarizmi” and algebra from “Al Jabr” wrote 2 additional books on the Astrolabe
28	Mesopotamians Number system based on sixty: 60 Through the ages this system has been used by astronomers: 60 seconds in a minute 60 minutes in an hour 360 degrees in a circle longitude and latitude
29	Early European Textbooks Hero of Alexandria: multiplication by factoring 13 times 8 = (10 + 3) times (10 + 8) = 100 + 80 + 30 + 24 = 234 Ptolemy (The Almagest) Used Babylonian number system Base 60, our source of 60 minutes, 60 seconds, and 360º angular measurements Mathematical Operations Addition, Subtraction, Multiplication, Division, Duplation (doubling), and Mediation (halving)
30	Multiplication by doubling 297 times 22 5 9 4 5 9 4__ 6 5 3 4 297 1 594 2 a 1188 4 a 2376 8 4752 16 a 6534 22
31	The Instruction of Ptahotep: No limit may be set to art, neither is there any craftsman that is fully master of his craft. Source: 195.8.72.23/numbers.htm ©Mark Millmore used with permission
32	Alphabetic symbols Source: 195.8.72.23/numbers.htm ©Mark Millmore used with permission
33	Source: 195.8.72.23/numbers.htm ©Mark Millmore used with permission
34	Alphabets 1500 BC one of the first alphabets is developed in Ugarit (Syria) by stripping down Mesopotamian cuneiform characters to only 30 signs; elsewhere in the middle east, scribes developed symbol sets that were easier to write than the wedge-shaped letters of cuneiform. 1000 BC Phoenicians develop an alphabet of 22 signs for consonants; although not the first alphabet, it is adapted by both Greeks and Israelites to their own needs.
35	Spread of Alphabets Because the Phoenicians were great traders they spread their version of the alphabet around the Mediterranean: Greeks, Etruscans, and Romans. After printing was invented, a form of the Roman alphabet from Italy became the standard printed alphabet. [Johann Gutenberg invented a system for casting type as a flat surface around 1440 CE] Note how the English language, Windows®, and HTML have permeated present societies around the world.
36	References Bunch and Hellemans, The Timetables of Technology, Simon & Schuster, 1993 A major source of information about hieroglyphics can be found at: 195.8.72.23/numbers.htm ©Mark Millmore
37	Show and Tell Sample hieroglyphic tablets Prayer rug Astrolabe; hourglass