How to test knowledge on the real numbers in a written exam?

Question

In German universities, the first-year students typically start their analysis courses with introducing the real numbers. Most commonly, the incompleteness of $\mathbb{Q}$ is discussed using the example of $\sqrt{2}$. Then the real numbers are mostly introduced in an axiomatic way or sometimes constructed.

Unfortunateley, when it comes to the written exams at the end of the term, this topic is usually not covered. It seems to be not easy at all, since we don't just want our students to learn the axioms off by heart. We want them to be aware of the fact, that the reals aren't the rationals and that completeness is not natural at all.

Question: Is there a good way to ask for the knowledge on real numbers in a written exam? Good answers would give examples for questions or describe experiences with this topic.

Edit: The knowledge I want to test includes at least: Knowing that the real numbers have to be introduced and do not exist per se (as unique concept extending the rationals, see nonstandard analysis); knowing the property of completeness; understanding that the every real number has a decimal representation which yet is not always unique; knowing that "infinitly small numbers" don't exist as real numbers; being aware of the fact, that analysis has no use if it is restricted to $\mathbb{Q}$.

Answer 1

I agree with you that this is in general not really questioned. But I also think that is is very difficult in contrast to other topics in such a course. In my opinion this should be more a part of an oral exam than a written exam.

To answer your questions, I've seen the following examples in exams:

• Multiple choice: True or false? "Every bounded subset of $\mathbb{R}$ has an infimum."
• You can ask for consequences of the construction/definition of the real numbers, e.g., questions involving the density of $\mathbb{Q}$ in $\mathbb{R}$. For example, asking if a function like $$f(x) = \begin{cases} x, & x\in\mathbb{Q}, \\ -x, & x\in\mathbb{R}\setminus \mathbb{Q} \end{cases}$$ is continuous.
• Depending on the method you used to introduce the real numbers, you can hide some part of the construction into a questions. For example, if you use equivalance of Cauchy sequences to construct the real numbers, you can define a similar equivalence relation (or just exactly this) and let the students show that it is an equivalence relation.
• You can ask to show that the real numbers are uncountable (Okay, this was maybe also learned by heart, but since this is not a very short proof, you can ask them only of a part of it).

Answer 2

Some possible exam questions:

1) Prove that if $x<1/n$ and $x>-1/n$ for all natural numbers $n$, then $x=0$.

2) Prove or disprove: If $x$ and $y$ are reals with $x<y$, there is a rational $q$ with $x<q<y$.

3) Construct an injective map $f:\mathbb{Q}\times \mathbb{Q} \times \mathbb{Q \rightarrow R}$, and prove that it is injective.