CSE 8351 Computer Arithmetic Fall 2005 Instructor: Peter-Michael Seidel

CSE 8351 Computer ArithmeticSeidel - Fall Administrative Issues Class times: TTh 5:00-6:20 Office hours: Seidel: W 2-3, Th 2-3 Course Webpage: Class material –Handouts, slides and references will be provided on course webpage References will be provided on WWW Computer Arithmetic Page (to be setup) Grade distribution –Project40% –Paper Summaries/Presentation 40% –Examination 20% 8000 level - Research focus in class –Quality Research is based on a combination of Knowledge (facts) / Skills (methodology) / Motivation (effort)

CSE 8351 Computer ArithmeticSeidel - Fall Historical Perspectives Need Input to Compute –Numbers –How to represent them? –In early days very few people could write or had access to paper (writing was expensive) –Communication of numbers: with the purpose of trading –Numbers represented with hands, fingers and body parts

CSE 8351 Computer ArithmeticSeidel - Fall Development of numbers –First number symbols found are about 6000 years old (in caves in stone) –3300 years ago (Egypt): report on cows won after a battle –200 years ago (Micronesia) : Some Indian tribes can only count to 3 everything larger is called 4. –Counting started with fingers, stones, lines, cherry stones –Number range with fingers is very limited, is it ? –Counting possible, computing difficult –Combining of different symbols giving them different weights

CSE 8351 Computer ArithmeticSeidel - Fall Development of numbers Roman numbers: I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = = MMCCCLXXXVIII ( =2388) Even counting is difficult

CSE 8351 Computer ArithmeticSeidel - Fall Greek Enumeration and Basic Number Formation Use Greek Letters: with the three additional symbols: digamma koppasampi Numbers < 1000 composed:

CSE 8351 Computer ArithmeticSeidel - Fall More on Greek Numbers Larger Greek Numbers (apostrophe signals units of 1000) Fractions: Calculations: (multiplication based on distributive law)

CSE 8351 Computer ArithmeticSeidel - Fall Types of numbers systems Additional Number Systems (Greece – Syria – China - Roman) Number value is generated as addition of digits (independent of position), e.g. –Unary numbers: |, ||, |||, ||||, … Positional Number Systems (India - Arabic): value of a digit depends not only on symbol, but also on position in the representation (that defines the weight of that digit) Which advantages do additional Number Systems have over Positional number systems for computation?

CSE 8351 Computer ArithmeticSeidel - Fall Development of Numbers Numbers similar to ours developed around 600 a.D. Computing device still the fingers, but: 40 finger positions allowed number representations up to 200,000 also allows to compute Digital has been defined back then (latin Word: digitus (finger)). Promoted by Leonardo Fibonnacci ( ) in his book on the Abacus (1202) allows not only to count, but also to compute 16 th century: print digit letters

CSE 8351 Computer ArithmeticSeidel - Fall Computing devices Very early computing device: tally (write numbers in wood) Allow more computation with the Abacus: Allows to add and subtract. Multiplication and division requires additional tables

CSE 8351 Computer ArithmeticSeidel - Fall Importance of Number Representations A German merchant of the fifteenth century asked an eminent professor where he should send his son for a good business education. The professor responded that German universities would be sufficient to teach the boy addition and subtraction but he would have to go to Italy to learn multiplication and division. Before you smile indulgently, try multiplying or even just adding the Roman numerals CCLXIV, MDCCCIX, DCL, and MLXXXI without first translating them. John Allen Paulos, Beyond Numeracy

CSE 8351 Computer ArithmeticSeidel - Fall Simple Computing devices Multiplication and Division more complicated than addition subtraction => Motivation to reduce multiplication to addition Use Logarithms, reduce Multiplications to Additions Use of logarithm tables Corresponding Tool: Sliding rule Still in use in high schools 25 years ago, before calculators were used …

CSE 8351 Computer ArithmeticSeidel - Fall Abacus Structure

CSE 8351 Computer ArithmeticSeidel - Fall Abacus Operations Introduction and Counting: Addition: Subtraction: Book (includes also Multiplication & Division, Square Roots, Cube Roots ):Multiplication & DivisionSquare RootsCube Roots Applet: