First year
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I no longer teach first years (since 2002), and so this section is rather ancient, though I will continue to add to it from time to time...

A note of April 2023. I have not added to this page for over twenty years, and now I view it for the first time since I resigned my Headship of St. Patrick's College Mathematics Department.

One of the most dismaying things about trying to maintain a website is to discover broken web links, and immediately below I find that my two references to publications of the London Mathematical Society are amongst those:

In October 1998 I gave my incoming first year students (about 60 of them) a test. I based it rather loosely on a similar test from a recent report of the London Mathematical Society; specifically I based it (for the purposes of comparison) on Appendix A, New undergraduates of the October 1995 LMS published report, Tackling the Mathematics Problem .

I made my test a little more demanding than the above, and you may download it here. I have deleted most of the space between successive questions; in the printed form I allowed more space for the students to do their work in.

I also prepared a lengthy document to comment on their solutions, and to expand a little on the work related to the test questions. You may download that document here.

In my initial contact with first year students, one of the first things I try to get them thinking about is numbers, and generally commence with a simple question like:

what is the value of the square root of 2?

I believe that a great deal of serious mathematical work can flow from that simple sounding question (of course one should not discuss a question like that out of context; one must give students some inkling that that question has deep historical roots, going back to Pythagoras).

[I like to tell my students an anecdote from my student days in London: once, while attending a postgraduate course on Anayltic Sets - given by C.A. Rogers at University College - a group of us was sitting with Rogers, having a chat, and someone asked him which mathematical question - given a choice - he would most like to have resolved. After only a few seconds he replied: to know the decimal expansion of the square-root of 2 ... ]

Here are some related notes which I have prepared, which I only give out some time after class discussions.

A document concerning decimal values, and ... .
You may download it here (309 KB), or in zip format here (58 KB).

A document on rational and irrational numbers.
You may download it here (489 KB), or in zip format here (72 KB).

An alternative way of treating irrationality proofs for beginners.
You may download it here (260KB), or in zip format here (75KB).

A related document dealing with what I call L- and R-approximations.
Want to know what that's all about?
You may download it here (504 KB), or in zip format here (72 KB).

Another related document on irreducible quadratics.
You may download it here (308 KB), or in zip format here (48 KB).


I will add more here as time allows.

Also see the Maple section of my site.

Contact details 

After August 31st 2007 please use the following Gmail address: jbcosgrave at