Skip to main content Accessibility help
×
Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-25T16:11:28.642Z Has data issue: false hasContentIssue false
This chapter is part of a book that is no longer available to purchase from Cambridge Core

1 - Representing Numbers by Graphical Elements

from Part I - Visualizing Mathematics by Creating Pictures

Claudi Alsina
Affiliation:
Universitat Politècnica de Catalunya
Roger Nelsen
Affiliation:
Lewis & Clark College
Get access

Summary

In many problems concerning the natural numbers (1, 2,…), insight can be gained by representing the numbers by sets of objects. Since the particular choice of object is unimportant, we will usually use dots, squares, spheres, cubes, and other common easily drawn objects.

When one is faced with the task of verifying a statement concerning natural numbers (for example, showing that the sum of the first n odd numbers is n2), a common approach is to use mathematical induction. However, such an analytical or algebraic approach rarely sheds light on why the statement is true. A geometric approach, wherein one can visualize the number relationship as a relationship between sets of objects, can often provide some understanding.

In this chapter we will illustrate two simple counting principles, both of which involve the representation of natural numbers by sets of objects. The principles are:

  1. if you count the objects in a set two different ways, you will get the same result; and

  2. if two sets are in one-to-one correspondence, then they have the same number of elements.

The first principle has been called the Fubini principle [Stein, 1979], after the theorem in multivariate calculus concerning exchanging the order of integration in iterated integrals. We call the second the Cantor principle, after Georg Cantor (1845–1918), who used it extensively in his investigations into the cardinality of infinite sets. We now illustrate the two principles. [Note: The two principles are actually equivalent.]

Type
Chapter
Information
Math Made Visual
Creating Images for Understanding Mathematics
, pp. 3 - 6
Publisher: Mathematical Association of America
Print publication year: 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×