Around this time of the year - end of one year and start of another - the topic of time, its nature and how it seems to pass by, often comes up in private reflection and public discussions. On the last day of 2012, I came across an article in the New York Times, The Life of Pi and Other Infinities, and an accompanying podcast (An Endless Subject and a Way to End Bedbugs? - free on iTunes) about infinity that I found interesting and plan to share with my students when school resumes after the holiday break (skip the part on the podcast about bedbugs). The article is not very long but does manage to mention the efforts of Aristotle, Newton, Leibniz, Cantor and Einstein to make use of the concept of infinity to make important developments in mathematics and science.

I often find myself remarking to students that infinity is not a number but a concept ... and that you need to be very careful when encountering it in mathematics (one of the points made in the NY Times article).  For example, if a student writes then I tell them that it is more proper to indicate that the limit does not exist or increases without bound. Writing implies that the limit is equal to a numerical value and that it's infinity. Many textbooks get away with stating that certain expressions "equal" infinity but I go out of my way to tell students that this is not quite right. In fact, there is more than one kind (or magnitude) of infinity - which is probably the primary point of the NY Times article.

When discussing different sets of numbers (e.g. integers, rational, complex) is a natural time to have students grasp the idea of different infinities.  Students understand that the set of real numbers and the set of integers are both uncountable - i.e. there are an infinite number of elements in both sets. However, they also know that the set of integers is a subset of the set of real numbers. Thus, it makes intuitive sense that the 'infinity' attached to the set of real numbers is 'larger' than the 'infinity' attached to the set of integers.

As a part of this discussion with students, I also like to share with them a very brief (little over a minute) youtube video about the paradox of the infinite hotel first devised by the infludential German mathematician David Hilbert (1862-1943).

I will certainly devote future blog posts on ways to engage with students about infinity - and try to show how the idea of infinity is not only interesting and tricky but an absolute critical part of many important mathematical methods and developments.