Keeping CountWriting Whole Numbers

K. McGivney

MAT400

Introduction

• Civilizations that developed writing also had mathematical knowledge.

• The early history of math is traced by identifying records that indicate the number systems used within a society. Were the numbers used for accounting? for solving problems of commerce? for “academic” problems?

• Other evidence of math is based on accomplishments that (we believe) require mathematical sophistication – for example, the Great Pyramids of Gizeh.

Main topics

• Counting and numbers

• Some ancient systems– Egyptian, Babalonyian, Mayan, Greek,

Chinese

• Roman numerals

• Hindu-Arabic numbers

Counting

• Some evidence from the early and diverse records of human societies

• Tally marks found on bones in Zaire (around 6000 BCE)

• Quipu knots from Incas in Peru (1400 CE) –non-writing recording device

Egyptian number system• (as early as 3000 BCE)

– Hieroglyphic system.– Used papyrus as *paper*– Base 10 “grouping” system– Additive – not positional.– Rhind papyrus (1650 BCE)– How would you write 3,244? 21,237?

Egyptian number system

• Egyptian (as early as 3000 BCE)– How would you write 3,244?– How would you write 21,237?

Babylonian number system

• Babylonian (as early as 2000 BCE) – originated in Mesopotamia (now part of Iraq)– Cuneiform on clay tablets, used two symbols– Sexagesimal system. Base 10 for the “digits” up to 59,

and base 60 for large numbers. (Today: trig, clock). Multiply successive groups of symbols by increasing powers of 60 -- similar to our system – we multiply successive digits by increasing powers of 10.

– Place value system with no 0; that is, it used the position of the symbols to determine the value of a symbol combination.

Babylonian number system

Representing Numbers 60 and Beyond

• Numbers between 60 and 3599 were represented by two groups of symbols; second is placed to the left of the first and separated by a space or comma.– The value of the entire quantity is found by adding the

values of the symbols within each group and then multiplying the value of the left group by 60 and adding in to the value of the group on the right.

• For numbers 3600 and beyond, use more combinations of the two basic shapes and place the groups further to the left. Each group was multiplied by successive powers of 60.

Examples of Babylonian Number System

• Convert the following numbers to Base 10 numbers:                 

Problems with the Babylonian System

• Spacing between symbol groups.

Mayan number systems

• Around 300 BCE – Central America– Similar to the Babylonian system, but without the

spacing problems.– Two basic symbols: Dots and lines for 1’s and 5’s– Written vertically– Sort of base 20 (vigesimal) with a strange use of 18– Place value system with a 0– Examples

Greek and Roman Systems

• More primitive than the Babylonians.

• Mayan culture was not known to the Europeans until several centuries later so the system had no influence on the development of number systems in Western culture.

Greek (alphabetic) number system (450BCE)

• One of two Greek systems• Ciphered numeral system• Greek letters stand for numbers• Non-positional (additive) decimal system

Chinese-Japanese number system

• Multiplicative grouping system

• Symbols for digits and symbols for value

• Essentially like the way we write number names: four hundred and three, or one thousand twenty five

• See Wikipedia entries for more

Roman numerals

• As late as 500 CE• I = 1, V = 5, X = 10, L = 50,

C = 100, D = 500, M = 1000• Simple grouping system (later) with

subtractive principle• No symbol for 0• Used in eighteenth century academic

papers, and still used today in limited form

Roman numerals

1. Write the following as modern numerals:1. MDCCCXXVIII

2. CDXCV

2. Without translating to modern numerals, find XV11 + XX = ___

Hindu-Arabic numerals

• Developed in India around 600 CE• Transmitted by Islamic expansion into India

around 700 CE• “Invented” 0• Spread to the West initially by Latin translations

of Arabic texts as early as 1100 CE• Western trade with the Middle East at the end of

Europe’s “Dark Ages” helped spread the system• Fibonacci’s Liber Abaci (1202 CE) begins with a

page explaining these numerals

Summary

• The concept of “number” developed independently in every culture, much like language.

• The various systems are similar in many ways. Some are positional, some have zero, and some are still used today.

• Commerce and communication led to widespread use of the Hindu-Arabic system based on its elegance and computational ease.

References

• Berlinghoff and Gouvea• MacTutor Math History Archive• Jamie Hubbard’s Mayan Numerals web page (8/31/04) at

http://mathcentral.uregina.ca/RR/database/RR.09.00/hubbard1/MayanNumerals.html

• Victor J. Katz, A History of Mathematics, Pearson/ Addison Wesley, 2004

• Howard Eves, An Introduction to the History of Mathematics, Saunders College Publishing, 1991.

• Wikipedia entry on Number Names (8/31/04) at http://en.wikipedia.org/wiki/Number_names

• http://www.michielb.nl/maya/math.html

Approximate Timeline for the Development of Numbers

• 3000 BCE Egyptian numerals• 2000 BCE Babylonian (Iran/Iraq)• 1000 BCE Chinese-Japanese• 600 BCE–500 CE Roman Empire• 300 BCE Mayan (Central America)• 600 CE Hindu-Arabic numerals• 500 CE–1100 CE Dark Ages in Europe• 1100 CE Arabic texts translated• 1202 CE Fibonacci publishes

Liber Abaci

Test questions

(Note: These should not be part of your PowerPoint presentation.)

1. Which of the following numbers is the largest? Which is the smallest? Which is illegal?

A. XV

B. XL

C. IC

D. LI

E. IL

Test questions

2. Which book begins with a page explaining Hindu-Arabic numerals?A. The Elements, by Euclid, around 300BCE

B. Liber Abaci, by Leonardo Pisano, around 1200 CE.

C. Principia Mathematica, by Isaac Newton, around 1700 CE.

Test questions

3. Give two numbers (in our modern notation) that when translated to Babylonian system might be confused with one another.

Test questions

4. Give an example of how each of the following number systems is still used today. Babylonian

Roman

Hindu-Arabic

Test questions

5. How many different symbols must be memorized to write all of the numbers less than 1000 in each of the following systems?1. Hindu-Arabic

2. Babylonian cuneiform

3. Egyptian hieroglyphics