We look at
Check that the horizontal coordinates of c+z and cz depend only on the horizontal coordinates of c and z. Check the same for vertical coordinates.
Check by hand in examples that the rule for how the horizontal and vertical coordinates behave under + and × agrees with those operations in |
Notice that because 0×(anything) = 0, if you set c to zero, multiplying by anything gets you nowhere.
Now pick a c which is not zero, but is outside of the shaded region. what happens when you multiply it by different numbers? Which numbers can you get as products with c?
Check that in each instance the numbers at a at (1,0) and b at (0,1) have a special property: their squares are themselves, while their product is zero.
By hand, verify that based on these properties,
Notice how addition simply proceeds in straight lines, except for the "wraparound" effect.
Which additive series (of multiples of z) fill up the whole number system? What are all the different possible results of filling in all multiples? How does this correspond to the previous question?
Notice how the effect of multiplication does not have any geometrically obvious interpretation -- at least in this view.
What are all the different possible results of filling in all powers of a number?
How do the results compare with those for clock numbers modulo a prime?
Specifically, try the following combinations, and compare:
p 11 13 13 q 7 5 7