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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?