Sample of an Ancient Number System and Place Value Lesson

I also teach a course for elementary educators titled Integrated Mathematics. This course is taught over two semesters.  I teach the portion of the course that addresses number theory. We do a lesson over place value and ancient number systems in this course, and this a worksheet I use to introduce some of the concepts associated with these topics. I also teach this lesson to my Fundamentals of Mathematics course and they do fine with it. Of course the changing bases is too challenging for my MAT 030 students, so it is a lighter version of this for them.

MAT 155 Integrated Mathematics         Name______________

Ancient Number Systems/Place Value/Rounding

The Egyptians

The Ancient Egyptians would write number quantities using symbols or pictures. In the spaces to the right, draw the symbol that represents each quantity.

Egyptian Number Pictographs
1 =		Staff ____
10 =		heel bone ____
100 =		coil of rope ____
1000 =		lotus flower ____
10,000 =		pointing finger ____
100,000 =		Tadpole ____
1,000,000 =		astonished man ____

Write the number 1324 in Egyptian Number Pictographs ___________________________

Write the number 820 in Egyptian Number Pictographs ____________________________

Write the number 231 in Egyptian Number Pictographs ____________________________

What would the Egyptian Number look like if we were to combine 820 and 231 using Egyptian Number Pictographs?

____________________________________________________

The Greeks

The original Greek alphabet consisted of 27 letters and was written from the left to the right. These 27 letters make up the main 27 symbols used in their numbering system. Later special symbols, which were used only for mathematics vau, koppa, and sampi, became extinct. The New Greek alphabet nowadays uses only 24 letters.

Write the number 820 using the Greek Number System ____________________________

Write 231 using the Greek Number System _____________________________________

Try to combine the numbers 820 and 231 using these Greek symbols. How can this be done?

The Mayans

The number systems discussed so far still lacked an important part of the number system we use today. They lacked a place-value system. The symbols used in the Egyptian and Greek number systems did not need to be in any particular order. If a person knew what the symbols meant it would not matter what order they were in.

Some ancient cultures who attempted to use a place-value system were the ancient Babylonians and the Mayan cultures. Both of these cultures used place-value systems but they each used a different “base” than we do.

In the table below are represented some Mayan numbers. The left column gives the base with place value for each position of the Mayan number and its multiplier. Remember the numbers are read from bottom to top. Below each Mayan number is its decimal equivalent. A flat line represents the value 5, and a dot represents the value 1.

203	8,000
202	400
201	20
200	1
Base	Multiplier	20	40	445	508	953	30,414

It has been suggested that counters may have been used, such as grain or pebbles, to represent the units and a short stick or bean pod to represent the fives. Through this system the bars and dots could be easily added together as opposed to such number systems as the Romans but, unfortunately, nothing of this form of notation has remained except the number system that relates to the Mayan calendar.

The 360 day calendar also came from the Mayan's who actually used base 18 when dealing with the calendar. Each month contained 20 days with 18 months to a year. This left five days at the end of the year which was a month in itself that was filled with danger and bad luck.

Exercise: Try to write the number 456 using Mayan number symbols.

The Babylonians

The ancient Babylonian number system worked the closest to the way ours works, but instead of a base-10 system like ours, the ancient Babylonians used a base-60 number system. The Mayans used a base-20 system. The ancient Babylonians would understand the number:

37 25 41

What would this Babylonian number mean to us today?

….0x(60⁴) + 0x(60³) + 37x(60²) + 25x(60¹) + 41x(60⁰)

So, (37 x 3600) + (25 x 60) + (41x1) = 133200 + 1500 + 41 = 134,741.

The ancient Babylonians only had two symbols that were used to represent all numbers possible. It was the idea of having place value that allowed this system to work.

The symbol for a single unit is a staff with a wedge at the top. _______

The symbol for a collection of 10 units is a wedge shape on its side. _______

So the number 32 would look like…________

They would often “bunch” these unit symbols together instead of writing the symbols side-by-side.

So the number 58 might look like…____________

What would the value 125 look like in ancient Babylonian number symbols?

The Romans

The Romans also used symbols to represent numerical values. Their system placed an importance on symbols and their position, but not place-value like the Babylonians or the Mayans.

The symbols would read left-to-right with the rule of addition between each symbol.

They allowed for up to “one” symbol of lesser value be written to the left of a larger value that meant subtract the lesser from the larger.

For example, LXIII means ___________

Where XLIII means ____________

Write the value 859 in Roman numerals in the space below.

How does our number system really work?

The number system we use today has only been in use the way we understand it for the last 500 years or so. It has gone through several changes over time while developing. Several cultures contributed to its evolution giving us its present state.

Hindu-Arabic numerals are now used in most of the countries of the world. It took more than 1,500 years for the numerals to develop their modern shape. People who write in the Arabic alphabet still use an older form of Hindu-Arabic numerals called East Arabic numerals.

Exercise: Write the number 8,743 in each system discussed using the appropriate symbols. Be sure to label each response accordingly.

For which ancient number systems did you find this exercise more or less challenging? Explain why or why not.

Of the number systems introduced so far, which number system do you feel you could best adapt to if they were the only ones available? Explain your reasoning for your response.

The Modern Number System

In our modern base-10 number system there are 10 “digits” to choose from 0-9. Using our modern number system, label the following slots with the corresponding place-value name for that slot.

3 5 , 5 7 3 , 0 9 3

We use estimation, or “rounding”, to talk about numbers in less detail. For example, the number above is about 36, 000, 000. We can “round” numbers to different place-values depending on how accurate we feel we need to be. Rounding the number above to 36, 000, 000 is an example of rounding to the millions place.

Round the number above to the following place values.

Ten-Millions Place _________________________________________

Millions Place _____________________________________________

Hundred-Thousands Place ___________________________________

Ten-Thousands Place _______________________________________

Thousands Place ___________________________________________

Hundreds Place ____________________________________________

Tens Place ________________________________________________

Ones Place _______________________________________________

The number 35,573,093 is written in what is called “standard form”. The meaning of this number is 30,000,000 + 5,000,000 + 500,000 + 70,000 + 3,000 + 90 + 3. A number written in this form is written in what is called “expanded form”.

Complete the following table.

Standard Form	Expanded Form
456
8745
	6,000 + 40 + 3
43, 020
	50,000 + 400 + 30 + 7

Answer the following.

1. Write a number in standard form that has a “7” in the “tens place” and a “4” in the “hundred-thousands place”.

2. Write a number that has a “5” in the ten-millions place, a “3” in the hundreds place, and a “9” in the ten-thousands place.

3. Write a number that contains 4 digits and has a “1” in the tens place.