3.3: Numeration Systems
- Page ID
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In set of practice problems, you will learn about two ancient numeration systems: those of the Chinese and Maya.
Chinese Numerals
Chinese numerals are still used today. Symbols for some Chinese numerals are shown below.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 100 | 1,000 |
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Chinese numerals are formed by writing the symbols vertically and using the multiplicative principle, which simplifies the recording of numerals by eliminating the repetition of symbols. For instance, the Chinese write the numeral for 3,058 by thinking \(3 \times 1000 + 5 \times 10 + 8\) and write down the symbols 3, 1000, 5, 10 and 8 in that order to represent that number. Even though you can think of the 8 as \(8 \times 1\), the 1 is not written down. This Chinese numeral (3,058) is shown to the left. The Chinese numeral for 872 is shown to the right.
Below are some more Chinese numerals. Make sure you understand how to read all of them before trying the exercises.
| 6,400 | 87 | 9,531 | 2,605 | 4,011 | 7,000 |
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Write each Hindu-Arabic numeral as a Chinese numeral.
- 5,093
- 610
- 427
- 8,008
So you won't have to keep going back to look up these symbols, here again are the Chinese numerals you probably need to look at to do the following exercise.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 100 | 1000 |
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Rewrite each Chinese numeral in Hindu-Arabic numeral .
| a. ____ | b. ____ | c. ____ | d. ____ | e. ____ |
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Notice in the Chinese system that numbers over nine have symbols written in pairs. To write 800, you must write the symbol for 8 above the symbol for 100. This is true even if there is only a "1" in the place value, see 2d above. Although the multiplicative principle allows you to write down fewer symbols than a simple additive system for most numerals, a further simplification would allow us to skip writing the second numeral of each pair. This would work if we used the position of the symbol to indicate the size of that group (10, 100, 1000, etc.). In other words, we could use a positional numeration system. In order to keep track of a position where no digit is used, a symbol for zero is necessary. Although the Chinese system doesn't need a symbol for zero, a circle was introduced to represent zero in the 1200s.
Maya Numerals
The Maya numeration system uses a positional system and is similar to the Chinese system in that the symbols for the numerals are written from top to bottom. Maya numerals were developed by the Maya priests of southern Mexico and Central America around 300 BCE. It is believed to be the earliest positional numeration system incorporating a zero and using it for a placeholder.
Some Maya numerals are shown below. Try to figure out the pattern and then fill in the missing numerals.
Explain what symbols are in this system, what they stand for and how the system works for at least the numerals one to nineteen.
From what you have seen of this system so far, it might look like a simple additive system. One might guess that the numeral for 20 would be four line segments and that the numeral for 103 would be twenty line segments and three dots. At first glance, it is very similar to the tally system. However, the Maya used a vertical positional system. The bottom level represented how many units (or ones), the second level up represented how many 20's, the third level up represented how many 360's (20x18), the fourth level up represented how many 7200's (20x18x20), the fifth level up represented how many 144,000's (20x18x20x20), etc. Except between the second and third level, each place value increased by a multiple of 20. It is almost a Base Twenty system except for that strange third level. Why the third level is 18 times the second level is explained later. This chart shows the first four place values.
| 7200s |
| 360s |
| 20s |
| 1s |
To try and make sense of all of this, we will look at some Maya numerals that have more than one position now. The numerals from one to nineteen only utilize the bottom level so it isn't apparent that the Maya system is positional until you count past nineteen.
Here is a two-level Maya numeral. There is a 16 in the bottom level, representing 16 ones, or 16 (\(16 \times 1\)), plus there is a 7 in the second level up, representing 7 groups of twenty, or 140 (\(7 \times 20\)). We add the values of each level, 16 + 140, so the numeral you see represents 156.
You have seen two basic symbols in this system so far: a dot, which represents the number one and a line segment, which represents the number five. As previously mentioned, there needs to be a symbol for zero to incorporate the use of place value. For that, the Maya used a symbol representing a shell.
This is a two-level Maya numeral with a zero in the bottom level (representing zero ones, or 0) plus a 13 in the second level up (representing 13 groups of twenty, or 260 since \(13 \times 20 = 260\)). Adding the values together, we have \(0+260 = 260\).
Write the Hindu-Arabic equivalent of each Maya numeral. Show how you obtained your answers. Note that proper space should be left between each place value. Otherwise, someone might incorrectly conclude the number shown for 5a represents 14, which would be the answer if there was no space.
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Now, we'll go on to some three- and four-level Maya numerals. Remember that the place value for the third level up is 360 and the place value for the fourth level up is 7200. See if you can figure these out on your own first.
Write the Hindu-Arabic equivalent of each Maya numeral. Show how you obtained your answers.
| a. ____ | b. ____ | c. ____ | d. ____ | e. ____ |
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How did you do? Make sure that if you still have any trouble understanding any of these that you go back and work through them again or ask for help.
It's a little trickier to start with a Hindu-Arabic numeral and convert it to Maya, but with a little patience and practice, you'll be doing it quickly and accurately! Before showing a method for doing this, try the following exercise. Hint: It should be easy and no calculator is required. Think about the place values of the various levels in the Maya system.
Write the Maya numeral equivalents for each of the following numbers:
| a. 1 | b. 20 | c. 360 | d. 7200 | e. 144000 |
To convert a number to a Maya numeral, the first thing you'll have to determine is how many levels the numeral will have. Remember the levels: 1, 20, 360, 7200, 144000, 2880000, etc. So any numeral less than 20 has one level, a numeral between 20 and 359 has two levels, a numeral between 360 and 7199 has three levels, a numeral between 7200 and 143999 has four levels, and so on.
It might help if you set up a table with the correct number of levels set up already with a space to fill in what symbol you'll be using at each level. For instance, look below at the four most common charts you'll be using.
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Let's start with the number 174. Convince yourself that this will be a two-level numeral. We'll start at the top, which is the 20s place value. We have to ask ourselves how many 20s are in 174? This is a division question: \(174 \div 20 = 8\), remainder 14. We can begin to construct the Maya numeral by starting with the two-level chart and filling an 8 in second level up as shown below.
So far, we have eight groups of twenty, or 160 filled in, which leaves 14 more (the remainder) to accommodate. When you get down to the units place, the remainder is filled in there. So the next step is to fill 14 in the units place. Do that in the vacant place in the chart shown. Before feeling satisfied that everything is correct, check your answer by computing the Maya numeral you have just constructed and see if it is indeed 174. Then, get rid of the chart and write the answer as a Maya numeral as shown to the right.
Let's try another number. We'll convert 6017 to a Maya numeral. This will be a three-level numeral (make sure you understand why!), so we'll start off with a three-level chart and figure out how many 360s are in 6017. We do this division exercise: \(6017 \div 360 = 16\), remainder 257. This tells us to put the symbol for 16 in the third level up (360s place). Now we have to take the remainder, 257, and find out how many 20s there are in it to find out what to put in the second level. We do this division exercise: \(257 \div 20 = 12\), remainder 17. Since only the units place value is left (the bottom level), the remainder of 17 goes in that level. The sequence of filling in the charts is shown below. Make sure you go back and check your work by converting the numeral back to Hindu-Arabic (\(1 \times 17 + 12 \times 20 + 16 \times 360\)) and seeing if it really equals 6017.
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Let's convert 71509 to a Maya number. Fill in the Maya symbols on the chart as we work through the problem.
This is a four-level Maya numeral (make sure you understand why!). Find what number will go in the 7200s place by performing a division calculation. Did you get 9, remainder 6709?
So, 9 is in the fourth level up. Write the Maya numeral for 9 in the fourth level. Now, do the next calculation. If you did this correctly, you should have \(6709 \div 360 = 18\), remainder 229. So, 18 is in the third level up. Write the Maya numeral for 18 there. Do the next computation. You should have \(229 \div 20 = 11\), remainder 9. So, 11 is the second level up and that leaves 9 in the units place. Write the Maya numerals for 11 and 9 in the appropriate locations. The final answer is shown.
Let's try a five-level number. How about 430040?
Fill in the Maya symbols on the chart to the as we work through the problem. We first make sure you understand why 430040 requires five levels to write with Maya numerals. Now, do the first division calculation.
If you have done this correctly and correctly interpreted the result, you should have a 2 at the fifth level up (the 144000s place) with a remainder of 142040 to represent with the first through fourth levels. This is a big remainder but is less than 144000. Now, do the next calculation and interpret the results.
You should have divided 142040 by 7200 to get 19 at the fourth level up and a remainder of 5240. Do the next calculation and interpret the results.
You should have divided 5240 by 360 to get 14 at the third level up and a remainder of 200. What happens next? Think about this before reading on.
There are exactly ten 20s in 200 and no remainder for the units, so we will need a zero at the bottom level. Finally, the numeral we seek is shown. Make sure you check it by computing \(2 \times 144000 + 19 \times 7200 + 14 \times 360 + 10 \times 20 + 0\). Does it really equal 430040? If so, it's right.
Let's convert 1584060 to Maya. Try this on your own. Then read on.
This is another five-level Maya numeral, so our first division of \(1584060 \div 144000\) gives us 11 at the fifth level up with a remainder of 60. There are no 7200s or 360s in 60, so there must be zeroes at those two levels. There are 3 20s in 60 with no remainder, so the units level must also have a zero. The numeral is shown. Please check it. Is it really 1584060?
Think about the highest numeral value that can be on any given level. For instance, the bottom level could go up to 19, or . But what about the second level up? What number would be represented by a two level Maya numeral if there was a 19 at the second level up and a zero at the bottom level? Write the Maya numeral for 380.
What is the highest Maya numeral value that should be on the second level up? What is the highest Maya symbol that could be on any level except the second level up?
Convert each of the following to Maya numerals. Make sure you check your answers!
- 1549
- 4750
- 53981
- 145804
- 1000000
You may be wondering why the Maya chose 360 for the third level up instead of 400, which seems more natural. Their counting system was based on their calendar, which consisted of 18 months of 20 days each, hence 360. The extra five days were treated differently. Their system made it easy to count time since those extra five days were not counted like the rest. For instance, consider these representations of the number of days in each time period:
| 1 year | 2 years | 3 years | 5 months, 9 days | 8 years, 11 months |
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Write the Maya numeral for the number of days (not including the extra 5 in a year) in:
- 7 Maya months, 15 days
- 13 Maya years
- 20 Maya years
Let's compare the Hindu-Arabic, Chinese and Maya numerals.
State how many different symbols a person has to memorize to use each system:
- Hindu-Arabic: _____
- Chinese: _____
- Maya: _____
Write each number as a numeral in Chinese and Maya. Note how many symbols it takes to write the given number in Hindu-Arabic, Chinese and Maya. Show work.
- 15
- 100
- 100
- 9999