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Why do this problem
This problem gives a taster of abstract representations framed
around familiar mathematical concepts. It is useful to prepare the
way for students to start thinking about abstract mathematics
such as group theory, as concepts such as Identity, Inverse,
Equivalence and Closure will emerge during the task. The latter
parts of the task are good fun to have ongoing over the course of a
week or term.
This problem would be a difficult challenge for keen students
to consider individually, perhaps as an extended homework or
In lesson time it is suited to a group or class discussion
where the problem is gradually solved together -- focus initially
only on exhibit A and the related questions, saving exhibits B and
C for high-fliers or open problems. All students will know the
mathematics for Exhibits A and C, but perhaps skip B (complex
numbers) if students are not in their final year.
Start with Exhibit A and write up the rules. Say something
like "I've got a set containing pairs of integers, such as (1, 22)
and (-34, 8). I have a rule which allows me to combine pairs to
give another pair in my set.
For example $(1, 2) + (2, 3) = (7, 6)$ and $(7, 2) + (3, 5) =
Oh, and we also have rules such as $(14, 21) \equiv (2,
This is a good place to discuss abstract rules and to
introduce the concept of equivalence if the class has not met this
Allow the class to discuss what might be going on before
writing down the general rule
$(a, b) + (c, d) = (ad+bc, db)$
Suggest that small groups attempt to work out what is
happening by testing out various numerical examples. When a
group feels that they have worked out that the structure is
'addition of fractions' they will then need to convince the rest of
the class of their reasoning.
You can then allow the class to move onto consideration of the
other Exhibits if a longer task is desired.
Have you tried exploring with small numbers, both positive and
negative to get a feel for the structure?
Using our rules, can two pairs be combined to give $(0,
What happens if you combine two of the same pair
Using our rules, which pairs can be combined to give $(N,
Exhibits B and C are likely to offer sufficient extension. If
more exploration is desired, students can attempt to alter the
combination rules and see if any structures emerge.
You could also ask students to devise exhibits of other
structures, such as vectors and matrices.
You can greatly simplify this tasks by asking 'This is similar
to addition of fractions! Can anyone see why?' or 'This is addition
of complex numbers' or 'These are the rules of arithmetic'.