7 Numeration Systems

At this point you may be thinking “Why, do I have to learn about all these other ways of writing numerals? I won’t be teaching my students Egyptian hieroglyphics in math class.” The short answer is that you are too comfortable and have too much experience with the Hindu-Arabic system. You do not have to think about the fact that 1,009 and 109 are two very different numbers. You just know it. This makes teaching it more difficult. As we look at two new systems in this section, keep track of where you get confused and what questions you have. These are likely similar issues your future students will have with the Hindu-Arabic system. As you work through your own questions, you are becoming better prepared to answer their questions!

Chinese Numerals

Chinese numerals are still used today. Symbols for some Chinese numerals are shown below.

Number	Representation in Chinese
One
Two
Three
Four
Five
Six
Seven
Eight
Nine
Ten
[latex]\;\;\;\;\;\;\vdots[/latex]	[latex]\;\;\;\vdots[/latex]
Hundred
[latex]\;\;\;\;\;\;\vdots[/latex]	[latex]\;\;\;\vdots[/latex]
Thousand

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 [latex]3 \times 1000 + 5 \times 10 + 8[/latex] 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 [latex]8 \times 1[/latex], the 1 is not written down. This Chinese numeral (3,058) is shown below on the left. The Chinese numeral for 872 is shown below on the right.

Representation of 3,058	Representation of 872
三	八
千	百
五	七
十	十
八	二

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
六	八	九	二	四	七
千	十	千	千	千	千
四	七	五	六	一
百		百	百	十
		三	五	一
		十
		一

 

Exercise 1

Write each Hindu-Arabic numeral as a Chinese numeral.

a. 5,093

Solution

五
千
九
十
三

 

 

b. 610

Solution

六
百
一
十

 

 

c. 427

Solution

四
百
二
十
七

 

 

d. 8,008

Solution

八
千
八

 

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. IMPORTANT NOTE: This is true even if there is only a “1” in the place value, see 2d above. You MUST remember to write the symbol for “one” above the symbol for “ten.” Although the multiplicative principle allows you to write down fewer symbols than a simple additive system (for most numerals, at least), a further simplification would allow us to skip writing the second numeral of each pair. We could only do this if we used the position of the symbol to indicate the size of that group (10, 100, 1000, etc.) This type of system is called a positional numeration system. The Hindu-Arabic system is one example of a positional numeration system. In order to keep track of a position where no digit is used, a symbol for zero is necessary.

Mayan Numerals

The final system we’ll learn about in this section is similar to the Chinese system in that the symbols for the numerals are written from top to bottom, but it uses a positional system. Mayan Numerals were developed by the Mayan priests of southern Mexico and Central America around 300B.C. It is believed to be the earliest positional numeration system incorporating a zero and using it for a placeholder.

Number	Representation in Mayan	Number	Representation in Mayan
Zero		Ten
One		Eleven
Two		Twelve
Three		Thirteen
Four		Fourteen
Five		Fifteen
Six		Sixteen
Seven		Seventeen
Eight		Eighteen
Nine		Nineteen

 

Exercise 3

Consider the pattern of the Mayan numerals.

a. What value do you think a horizontal bar represents in each symbol?

Solution

It represents five.

 

b. What value do you think a dot represents in each symbol?

Solution

It represents one.

 

c. Describe how the system works for the numerals one to nineteen.

Solution

It’s like the tally system, where you first list out groups of five then list singles, except you use a single bar to represent five and a dot to represent one.

 

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 Mayans used a vertical positional system.

• The bottom level represented how many units (1).
• The second level up represented how many 20’s (20[latex]\times[/latex]1).
• The third level up represented how many 360’s (18[latex]\times[/latex]20[latex]\times[/latex]1)
• The fourth level up represented how many 7,200’s (20[latex]\times[/latex]18[latex]\times[/latex]20[latex]\times[/latex]1)
• The fifth level up represented how many 144,000’s (20[latex]\times[/latex]20[latex]\times[/latex]18[latex]\times[/latex]20[latex]\times[/latex]1)
• Etc.

Except when moving from the second to the 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 has to do with the Mayan calendar having 360 days. We will explore this later. Looking at the chart to the left, which shows the first five place values, may help you understand the system.

To try and make sense of all of this, we will look at some Mayan 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 Mayan system is positional until you count past nineteen.

We have yet to see a numeral system that incorporated a symbol for zero. As previously mentioned, there needs to be a symbol for zero to incorporate the use of place value. As seen in the table above, the Mayans used a shell to represent zero.

Exercise 5

State the Hindu-Arabic equivalent of each Mayan numeral. You may want to use the chart to keep track of your place-values.

a.

𝋫

𝋠

𝋮

b.

𝋥

𝋧

𝋦

c.

𝋩

𝋣

𝋠

d.

𝋤

𝋪

𝋢

𝋮

e.

𝋡

𝋠
𝋥

𝋳

You may be wondering why the Mayans 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 considered “useless” and they didn’t worry about them. Their system made it easy to count time. They didn’t count the “useless days.” For instance, consider the following:

In order to properly teach the Hindu-Arabic system to your future students, it is important to understand the differences between it and other numeration systems. The final exercises of this section will be to compare the Hindu-Arabic, Chinese and Mayan numerals.