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Sheep in Wolf's Clothing

Age 16 to 18 Challenge Level:

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 $N$, $M$?


Next explore the properties of these structures:
Exhibit B

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?