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EDUC 343/543

Ancient Numeration systems

Ancient numeration systems can be an interesting topic of study for elementary and middle school students. First they can learn more about the mathematics of our own system by comparing it to the systems of ancient civilizations. In addition they can learn more about those ancient civilizations from understanding better how they wrote numbers.

Objective: Students will be able to demonstrate their understanding of the four different numeration systems by accurately adding and subtracting two numbers, and showing the regrouping process in the appropriate numeration system.
  • e.g. 5825 + 6941 written correctly in the appropriate system, regrouped as needed, and then calculated accurately.
Coil of rope
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The Babylonian cuneiform method of recording quantities, approximately 5000 years old, is among the oldest numeral systems in existence. They developed a base-60 (sexidecimal) system with numbers less than sixty represented in base-ten. They also developed a positional system for writing larger numbers with fewer symbols, But they had no number for zero, so their numerals can be difficult to interpret.

Babylonian image
For a more thorough look at the Babylonian system, see the Babylonian mathematics website from the University of Edinburgh.

Babylonian numerals link has a terrific chart that shows the symbols for the numbers 1-59...and then shows how to write a number as large as 424000 in Babylonian–1,57,46,40 written in Babyonian cuneiform symbols!

How did they multiply in the Babylonian system? They used a table of squares to calculate products using the following relationship: ab=[(a + b)2 - a
2- b2 ]/ 2. More about this at this site from the United Kingdom about Babylonian Mathematics.

The Babylonian sexidecimal system has its influence in our own world today. Angle measurement (360 degrees equals a circle) and time measurements (60 seconds equals one minute) are two examples.

Egyptian image


The Egyptian method for recording quantitities is based on 10 with a symbol for 1, ten, and each successive power of ten. A distinct hieroglypic was used for each power of 10. There was no symbol for zero, therefore a particular symbol was omitted in a numeral when that multiple of ten was not part of the number.

For a more thorough look at the Egyptian system, see the Egyptian mathematics website from the University of Edinburgh.

Egyptian numerals link has a terrific chart that shows the symbols for the Egyptian numeral hieroglyphs...and then shows how to write a number as large as 4622 using Egyptian numeral hieroglyphs.


The Mayan system was a base- 20 system (vigesimal) that used a system of bars and dots in a vertical place value system. A dot stood for one and a bar stood for five.

For more information about this system, see this Mayan math site.

Mayan image

Roman image


The Roman method (no place value, instead an additive system, rules of subtraction, base ten with fives)

For a more in depth look at Roman numerals, see this "Ask Dr. Math" website about Roman numerals. Here Dr. Math summarizes the basics of the Roman system, including a summary of the rules of subtraction for the Roman system. This site is part of the larger "Ask Dr. Math website," , a searchable resource for answering mathematics questions online. This is in turn a part of the larger "Math Forum" website at Drexel University– a valuable resource for mathematics educators to which you may subscribe. There is also a "Roman Numeral Calculator" you might enjoy using with your students (thanks to EDUC 3/543 student Lisa Jernstedt Webster for bringing this to our attention).

Further Exploration
Written by South Korean students, this website about Oriental Mathematics is part of an award winning ThinkQuest. It introduces students to the Babylonian, Egyptian, Mayan, and Roman numeration systems and provides more background information about each. Its explanation of each system, while a bit terse at times, has good graphics that show what the symbols actually looked like and goes into greater depth into the number theory and history behind these fascinating systems.

Last Updated 2/2007
Copyright Dr. Mike Charles © all educational uses encouraged
All images courtesy of Art Images for College Teaching (AICT)
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