Context: This is the second half of the tutorial to students Cathy, George and Ben. In the first half they have been doing Linear Algebra which the tutor claims is worth less tutoring time than the more problematic Analysis: Cathy has not delivered a draft with her work to the tutor. When he asks her if she has done any work she replies 'kind of'. The tutor turns to CD2.1ii and points out that George and Ben have given different solutions: George's solution is longer than Ben's. The tutor invites Ben to present his solution on the b/b 'in two lines'.

Ben writes his proof for CD2.1ii on the b/b (fig.2a). The tutor disagrees with the student's claim that a and b are rational numbers. George then intervenes in order to outline his solution: you can always find a rational number between two other rationals and then do this for reals. The tutor agrees that this idea can be the basis for a proof and stresses that part of George's problem in his writing was that he hadn't 'written down the data we are starting from'.

G1: Yeah, because I was trying to prove that there was a rational between any two real numbers... when your real numbers are divided by an irrational... it was a fact that there is a rational between two real numbers... you said it was a fact that there is a rational between two real numbers, I wrote that 'suppose...' and you said it wasn't a supposition, it's a fact... so could I just have written...?

T1: Oh, that's different. You see what you've written is not what you meant.

G2: Oh,... I see yeah...

T2: You see what you've written is precisely what you just said: there exist two real numbers and a rational between them. Sure they do. I can prove that by exhibiting two of them, like zero and 1.

G3: You still want to prove there that this is true for any a and b.

T3: You want to look for them all. You want to quantify...

The tutor then explains the problem with Ben's proof: in CD2.1ii they need to start from two arbitrary real numbers. In part i the numbers for which the statement is proved are rational. So part i, at least at this stage, cannot be used. The tutor then presents a proof by contradiction (fig.2b) of which in the end he appears rather critical: it is, he says, a 'bad way to present it because it is not constructive, but it is quicker'. 'But does that prove that there is an irrational between any real numbers?' asks Cathy. The tutor stresses that a and b are arbitrary numbers. She then wonders about part i of the question: 'it seems to assume that irrational numbers exist'. 'They do', the tutor says, and Ö2 is one of them, as they have 'all accepted during his presentation'. He then concludes that there are other irrational numbers too but it is sufficient to point at the existence of one. Then it is legitimate to assume the existence of rational numbers.

Return to Section 6(ii).

Return to Appendices for Chapter 6.