Let p be a prime number such that the equation
z2 = -1does not have any solutions in the "clock" numbers modulo p.
Then we can create a new number system by adjoining an imaginary square root of -1 (we will call it i) to the original system of clock numbers. By adding and multiplying this new number i together with the original clock numbers, we get numbers of the form x + yi (where x and y are clock numbers). This set of p2 numbers is in fact closed under all the operations of a field.
To picture this number system, we can think of x and y as coordinates, only we have to remember that they wrap around after p steps. Because of this double wrap-around, we could actually picture these numbers as being on the surface of a donut.
What are the possible sets of multiples and their sizes?
This shows sets of all r'th powers for varying r.
This shows the p'th power of the number you click. What is the effect? If you know about complex numbers, can you find an analogy to this operation in that system?