Sheep in Wolf's Clothing
Can you work out what simple structures have been dressed up in these advanced mathematical representations?
Problem
In the following exhibits we give an advanced or alternative way of thinking about mathematics concepts which are likely to be known in a more familiar form.
Explore these structures and experiment by substituting particular values such as $0, \pm 1$. Can you work out what they represent?
Exhibit A
All pairs of integers such that:
$$(a, b) + (c, d) = (ad+bc, bd)\quad\quad (Na, Nb) \equiv (a, b) \mbox{ for all } N\neq 0$$
Can you find two pairs which add up to give $(0, N)$ or $(0, M)$ for various values of $N$, $M$?
A set of ordered pairs of real numbers which can be added and multiplied such that
$(x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 +y_2)$
$(x_1, y_1)\times (x_2, y_2) = (x_1x_2 -y_1y_2, x_1y_2+y_1x_2)$
Exhibit C
A set defined recursively such that
$+_k(1) = +_1(k)$
$+_k(+_1(n)) = +_1(+_k(n))$
$\times_k(1) = k$
$\times_k(+_1(n)) = +_k(\times_k(n))$
In these rules, $k$ and $n$ are allowed to be any natural numbers
Once you have figured out what these structures represent ask yourself this: Are these good representations? What benefits can you see to such a representation? How might familiar properties from the structures be represented in these ways?
Student Solutions
Exhibit A
The condition $$(Na, Nb) \equiv (a, b) \mbox{ for all } N\neq 0$$ ought to give it away: $$(a,b) \iff \frac{a}{b}$$ This statement simply says that if the numerator and denominator of a fraction share a common factor, they can be cancelled down.
Exhibit B
If we represent a complex number a+bi by the ordered pair (a,b), we get the required properties:
$$(a+bi) + (c+di) = (a+c) + (b+d)i \iff (a,b) + (c,d) = (a+c, b+d)$$
$$(a+bi) \times (c+di) = (ac-bd) + (ad+bc)i \iff (a,b) \times (c,d) = (ac-bd, ad+bc)$$
Exhibit C
These formally define addition and multiplication over the natural numbers. Can you see how the familiar properties we're used to follow from them?
The first implies $k+1 = 1+k$, i.e. addition is commutative.
The second implies $k+(1+n) = 1+(k+n)$, i.e. addition is associative.
The third implies $k\times 1 = k$, i.e. 1 is the multiplicative identity.
The fourth implies $k\times(1+n) = k+(k\times n)$, that mulitplication is distributative over addition.
This is a rigorous treatment of a very familiar concept. For more information on this subject, you could start by reading this Wikipedia article.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
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.
Possible approach
Key questions
Possible extension
Possible support