 Investigation: Number concepts in cultures

Lesson

Introduction

Much of the mathematics we use today has come from many different cultures and has looked very different over time. We might assume that all cultures have the same number system or ways of counting, but there are diverse ways of organizing numbers that show interesting number concepts.

Objective

Research and explore a number concept to tell a story about how it came about and how it is used in a specific culture. Explain how the number concept is relevant in a current context.

Materials

• Device with internet access or library access

Activity 1

As a class, create a list of number concepts that you have seen in the past. This could be in math class or in your own experiences. Here is a list to get you started:

• How numbers are named
• The concept of zero
• The concept of infinity
• The concept of square roots
• Tallies
• Pythagorean triples
• Prime numbers
• Remainders
Activity 2

In a group of two or three do the following:

1. Choose a number concept that you will research. This could be a concept from the list from Activity 1, or a number concept that you came up with as a group. If you came up with it as a group, ask your teacher to check it before going to the next step.
2. Create a 3 - 5 minute presentation telling the story of the number concept of your choice. Things that could be included are:
• In present day what this concept looks like in different cultures
• Across one or more cultures, how the concept was initially developed compared to what it looks like in present day - who was it named after versus who first used it
• A timeline
• Any underlying assumptions that we might make about the concept
• How the concept can be used in different contexts - present and past
• Images of what the number concept looks like when written down
• Sources for your information - if not from personal experience
Group work tips
• Decide on the group's approach.
• Keep everyone focused.
• Help everyone work together.
• Make sure everyone understands.
• Push the group to explore more ideas.
• Ask questions to test solutions.

Here is an example on the number concept of number bases and naming.

Example on number systems

In many cultures, in the present day we use the digits $0,1,2,3,4,5,6,7,8,9$0,1,2,3,4,5,6,7,8,9, when writing numbers. These can be called the Arabic numerals and come out of the Hindu-Arabic numeral system which was developed in India around 500 CE and can still be seen in Sanskrit. These are designed for a base ten number system. This means that when we each digit can take on one of these ten digits and that the digit to the left of the units digit tells us the number of "tens" and so on. Many cultures have used base ten as long as they have been counting because we have ten fingers. In computer science we use binary which has base $2$2 (digits $0$0 and $1$1) or hexadecimal which has base $16$16 (digits $0$0 to $9$9 and letters A to F). Being able to think in different bases can be beneficial, just like knowing different languages.

Before society had these ten digits to represent numbers, there were many systems including Roman numerals which were based on addition and subtraction of the digits shown in the table below.

Symbol I V X L C D M
Value $1$1 $5$5 $10$10 $50$50 $100$100 $500$500 $1000$1000

A smaller symbol before another symbol tells us to subtract it and a larger symbol before another symbol means to add it. For example, IV is $4$4, while VI is $6$6. This system led to some very long expressions for numbers and the maximum number was $3999$3999 which would be MMMCMXCIX. However, this idea of a number being composed of operations like addition and subtraction still is used. For example in French, the number $97$97 is quatre-vingt-dix-sept which translates to four-twenty-ten-seven and comes from the fact that $97=4\times20+10+7$97=4×20+10+7.

Not all cultures or languages use the Arabic numeral system. Especially those that do not use base ten. One example is the Kaktovik or Iñupiaq numerals which is a base $20$20 number system that was created to be able to write out the numbers used in the Iñupiaq language - which revolves around $20$20s, because we have $10$10 fingers and $10$10 toes. The written notation is very recent, in the mid 1990s, and was developed in Alaska by Iñupiat students! The names of the numbers are also helpful as they are in terms of operations. For example, notice how $7$7 is the two markings for $2$2 and $5$5 put together and that the name is also $2$2 and $5$5put together. Digit Spoken
0 There is no word for zero
1 atausiq
2 malġuk
3 piŋasut
4 sisamat
5 tallimat
6 itchaksrat
7 tallimat malġuk
8 tallimat piŋasut
9 quliŋuġutaiḷaq
10 qulit
19 iñuiññaŋŋutaiḷaq
20 iñuiññaq

We are curious to learn more about how to do arithmetic and how the names work for the Iñupiaq numbers.

Sources:

• The History of Arithmetic, Louis Charles Karpinski, 200 pp, Rand McNally & Company, 1925.
• Bartley, William Clark (2002). "Counting on tradition: Iñupiaq numbers in the school setting". In Hankes, Judith Elaine; Fast, Gerald R. (eds.). Perspectives on Indigenous People of North America. Changing the Faces of Mathematics. Reston, Virginia: National Council of Teachers of Mathematics. pp. 225–236. ISBN 978-0873535069

Outcomes

9.B1.1

Research a number concept to tell a story about its development and use in a specific culture, and describe its relevance in a current context.