1. Writing Whole Numbers K. McGivney MAT400
Keeping Count
Writing Whole Numbers
K. McGivney
MAT400

2. 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.

3. Main topics Counting and numbers Some ancient systems Roman numerals
Egyptian, Babalonyian, Mayan, Greek, Chinese
Roman numerals
Hindu-Arabic numbers

4. 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

5. 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?

6. Egyptian number system
Egyptian (as early as 3000 BCE)
How would you write 3,244?
How would you write 21,237?

7. 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 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.

8. Babylonian number system

9. 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.

10. Examples of Babylonian Number System
Convert the following numbers to Base 10 numbers:                 

11. Problems with the Babylonian System
Spacing between symbol groups.

12. 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

13. 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.

14. Greek (alphabetic) number system (450BCE)
One of two Greek systems
Ciphered numeral system
Greek letters stand for numbers
Non-positional (additive) decimal system

15. 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

16. 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

17. Roman numerals Write the following as modern numerals:
MDCCCXXVIII
CDXCV
Without translating to modern numerals, find XV11 + XX = ___

18. 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

19. 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.

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

21. 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

22. Test questions (Note: These should not be part of your PowerPoint presentation.)
Which of the following numbers is the largest? Which is the smallest? Which is illegal? XV XL IC LI IL

23. Test questions
Which book begins with a page explaining Hindu-Arabic numerals?
The Elements, by Euclid, around 300BCE
Liber Abaci, by Leonardo Pisano, around 1200 CE.
Principia Mathematica, by Isaac Newton, around 1700 CE.

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

25. Test questions
Give an example of how each of the following number systems is still used today.
Babylonian
Roman
Hindu-Arabic

26. Test questions
How many different symbols must be memorized to write all of the numbers less than 1000 in each of the following systems?
Hindu-Arabic
Babylonian cuneiform
Egyptian hieroglyphics