"Donut" Numbers: The Square Root

Let p be a prime number such that the equation

z2 = -1
does 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.

[Numeroscope Applet: displays the donut numbers with variable prime modulus congruent to 3 modulo 4. Can serve as a calculator for this number system, or display additive and multiplicative series, as well as function ranges.]

Lab Activities

Mode Oper. r c Activity
all r + 0 What are the possible sets of multiples and their sizes?

all z × 1 This shows sets of all r'th powers for varying r.

one z × p 1 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?

Discussion Topics

1. How many donut numbers are squares, cubes, fourth powers, ...? Did you have to change your answer, or reasoning, from the same problem in the case of clock numbers?

2. Notice that all the original clock numbers (along the "x-axis") are squares of donut numbers. However, not all donut numbers are squares of donut numbers. Is it possible to somehow make sense of the square root of any donut number?

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