Investigation: Algebraic story

Introduction

The algebra that we use today has come from many diverse cultures, has changed over time, and still looks different in different cultures. However, we often have many of the same concepts, but represented in different ways.

 

Objective

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

 

Materials

• Device with internet access or library access

 

Activity 1

As a class, create a list of algebraic 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:

• The use of variables
• Substitution
• Operations with algebraic expressions
• Systems of equations
• The origins of algebra in different cultures
• Connection between algebra and geometry

 

Activity 2

In a group of two or three do the following: 

1. Choose an algebraic 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 notation has looked like
   • What you want to learn more about
   • Sources for your information - if not from personal experience

 

 

Here is an example on the algebraic concept of solving cubic equations. 

Example on the history of cube roots

This algebraic concept actually started as a geometric concept, like much of algebra. In modern days, consider how we might solve this cubic equation:

x^{3} = 2

We might follow these steps:

x^{3}	=	2	Equation to solve
\sqrt[3]{x^3}	=	\sqrt[3]{2}	Take the cube root of both sides
x	=	\sqrt[3]{2}	Simplify
x	\approx	1.25992 \ldots	Evaluate on calculator

 

Since 2 is not a perfect cube, we don't get an integer answer, but we can use our calculators, unlike mathematicians a long time ago (calculators could be a whole separate story).

However, in ancient Greek times, sometimes called "antiquity", there was not the same abstract concept of algebra that we have and everything was done in terms of geometry, so the number 2 was not represented with the numeral 2, but as a line segment with length 2. Squaring a number was literally drawing a square with side lengths of that number. Cubing a number was drawing a cube with side lengths of that number. So when something couldn't be constructed geometrically, which all rational numbers can be, it was troubling. Here is a story of why solving x^{3} = 2 to get x= \sqrt[3]{2} also known as "Doubling the cube" was such a big deal historically.

Eratosthenes, in his work called Platonicus, says that the Delians were going through a plague. And through an oracle there were told that, in order to get rid of a plague, they should construct an altar double the volume of the existing one. This sounded easy, but when they doubled one side length, the volume was eight times the size of the original. The craftsmen were completely stuck so went to ask Plato, a philosopher, about it. Plato told them that the god did not actually want an altar of double the size, but wanted to shame the Greeks for focusing all of their efforts on geometry instead of mathematics as a whole.

 

There are a couple of other stories that introduce the concept of "doubling the cube", which is now solving x^{3} = 2, which use different characters. 

This story highlights that all of the different fields of mathematics are important and that just focusing on one, like geometry, means we miss out. The connections between mathematical concepts can be beautiful.

 

Sources: 

• O'Connor, J.J., and E.F. Robertson. “Doubling the cube.” MacTutor, School of Mathematics and Statistics University of St Andrews, April 1999, https://mathshistory.st-andrews.ac.uk/HistTopics/Doubling_the_cube/. Accessed 21 July 2021.