"Clock" Numbers: Addition and Multiples

Introduction

Every day we perform arithmetic with hours -- and are quite comfortable with the fact that two hours after 11:00, the time has "advanced" to 1:00. Our goal in this course is to study and visualize the mathematics and applications of such "clock numbers". Ordinary clock hours repeat every 12 units, but we can use any positive modulus n instead. The resulting system of n numbers is called Z/nZ.

These numbers have many other practical uses, whenever counting "wraps around" (e.g., odometer rollover, year 2000 problem, etc.). Later, we will see some more sophisticated applications, to cryptography.

[Numeroscope Applet: displays the clock numbers with variable modulus. 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
one z		1		In this mode, the numeroscope simply computes the sum or product of two numbers you choose (c and z). Try a few, and check each one by hand to see that it matches your understanding of how the operations should work. Note: Just click on a number to set the z variable. Hold down the shift key and click, to set the c variable.
sweep r	+		0	Note how additive series progress at uniform speed around the circle, adding the same amount at each step. Compare with how the same kind of series works in the ordinary, unwrapped integers.
sweep r	×		1	Switching to multiplicative series, can you see any intuitive pattern this time? What does this kind of multiplicative series look in the ordinary integers?
all r	+		0	In this mode, the numeroscope displays the set of all multiples of a number z (which you choose). As you vary z, what are the possible sets of multiples? How do the results depend on the modulus n? Record your observations for group discussion (paying particular attention to whether 1 is a multiple of z).

Discussion Topics

1. When in the clock number system modulo 3, we write "3 = 0". This unusual-looking equation demands an answer to the question, what do we mean by "3"? Intuitively, we have to determine what "3" is by counting; mathematically this says that "3" is the number 1+1+1. But then what do we mean by calling a particular clock number "1"? (For hints, look at the background on the laws of algebra.)

2. As a variation on the previous question, what is the relationship between "multiples" and "multiplication"? That is, given an ordinary integer n, we can form the n'th multiple of z by adding z to itself n times; on the other hand, we can consider n as a clock number and then take the product nz in the clock number system. Are these always the same? Why?

3. Compare your observations on the way that sets of multiples vary with z and n. Specifically, consider the consequences of having 1 as a multiple of z. Also, what is the effect of n being prime or not prime?

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