A Story of Numbers · Lesson 6 of 8
Place Value Representation — Mesopotamian and Mayan Systems
“Understand positional notation, base 60, placeholders and the Mayan place-value system.”
• Explain how a positional system uses position to assign landmark value. • Read a base-60 place-value representation. • Explain why blank positions can create ambiguity. • Describe the role of a placeholder symbol. • Interpret the main features of the Mayan number system.
Place Value Representation
A base system still needs a way to write its landmark groups compactly. Place value provides the key idea: instead of drawing a different symbol for every landmark, reuse symbols and let their position tell us which landmark they belong to.
I. The Mesopotamian Number System
The later Mesopotamian system used base 60. Its landmark values were powers of 60: 1, 60, 3600, 216000, ... . Our modern time units still preserve a familiar trace of this grouping: 60 seconds make a minute and 60 minutes make an hour.
In a positional representation, the rightmost group corresponds to 1s, the group to its left to 60s, then 3600s, and so on. The same kinds of symbols can therefore be reused in different positions.
Problem
Interpret 2 groups of 3600, 5 groups of 60 and 30 ones.
- 1.2×3600 = 7200.
- 2.5×60 = 300.
- 3.30×1 = 30.
- 4.7200 + 300 + 30 = 7530.
Problem
Break 640 into base-60 landmark groups.
- 1.The largest useful landmark below 640 is 60.
- 2.640 = 10×60 + 40.
- 3.So the representation needs 10 in the 60s position and 40 in the 1s position.
A number system with a base in which the position of a symbol determines the landmark value associated with that symbol.
Early Mesopotamian notation had a serious difficulty. If a certain power of 60 did not occur, a blank space was left. Without perfectly consistent spacing, readers could not always tell how many positions were blank.
Later Mesopotamian notation used a placeholder symbol for an empty internal position. This is similar to the role played by 0 in a modern place-value system. The placeholder makes the position explicit even when that place contributes no quantity.
If an empty position is not recorded clearly, the same visible symbols may be interpreted as belonging to different powers of the base.
II. The Mayan Number System
The Mayan system independently developed place-value notation. It used a dot for 1, a bar for 5, and a shell-like symbol for an empty place. Its symbols were arranged vertically.
Its landmark sequence was unusual: 1, 20, 360, ... . The third landmark is 360 rather than 20² = 400, so this system is not a pure base-20 system.
Problem
Interpret one 360, zero 20s and one 1.
- 1.1×360 = 360.
- 2.0×20 = 0.
- 3.1×1 = 1.
- 4.360 + 0 + 1 = 361.
Problem
Interpret two 360s, zero 20s and one 1.
- 1.2×360 = 720.
- 2.0×20 = 0.
- 3.1×1 = 1.
- 4.Total = 721.
Quiz
What determines the value of a symbol in a positional system?
What is 60²?
Why did blank positions cause trouble in Mesopotamian notation?
What is the role of a placeholder?
Why is the Mayan system described here not a pure base-20 system?
Practice Problems
- Break 3605 into powers-of-60 landmark groups.
- Represent 200 as groups of 60 and 1.
- Explain how the same visible symbols could be ambiguous if a blank place is not clearly marked.
- Use the Mayan landmark values 360, 20 and 1 to decompose 77, 100 and 721.
- Compare the Mesopotamian and Mayan systems: identify one place-value similarity and one important difference.
Key Takeaways
• Place value reuses symbols by letting position determine the landmark value. • The Mesopotamian system used powers of 60 as positional landmarks. • Blank positions can make a numeral ambiguous. • A placeholder records an empty position without changing the other place values. • The Mayan system also used place value and a placeholder symbol. • The Mayan landmark sequence included 1, 20 and 360, so it was not a pure base-20 system.