CHAPTER Number Theory FIGURE FIGURE FIGURE Plus hours Plus hours Plus hours + = + = + = FIGURE. Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter described a mathematical system as () a set of elements, along with () one or more operations for combining those elements, and () one or more relations for comparing those elements. We now consider the -hour clock system, which is based on an ordinary clock face, except that is replaced with and only a single hand, say the hour hand, is used. See Figure. The clock face yields the finite set {,,,,,,,,,,, }. As an operation for this clock system, addition is defined as follows: add by moving the hour hand in a clockwise direction. For example, to add and on a clock, first move the hour hand to, as in Figure. Then, to add, move the hour hand more hours in a clockwise direction. The hand stops at, so. This result agrees with traditional addition. However, the sum of two numbers from the -hour clock system is not always what might be expected, as the following example shows. EXAMPLE Find each sum in -hour clock arithmetic. (a) Move the hour hand to, as in Figure. Then advance the hand clockwise through more hours. It stops at, so. (b) Proceed as shown in Figure. Check that. Since there are infinitely many whole numbers, it is not possible to write a complete table of addition facts for that set. Such a table, to show the sum of every possible pair of whole numbers, would have an infinite number of rows and columns, making it impossible to construct. On the other hand, the -hour clock system uses only the whole numbers,,,,,,,,,,, and. A table of all possible sums for this system requires only rows and columns. The -hour clock addition table is shown in Table. Since the -hour system is built upon a finite set, it is called a finite mathematical system. TABLE -Hour Clock Addition
. Clock Arithmetic and Modular Systems EXAMPLE Use the -hour clock addition table to find each sum. Plagued by serious maritime mishaps linked to navigational difficulties, several European governments offered prizes for an effective method of determining longitude. The largest prize was, pounds (equivalent to several million dollars in today s currency) offered by the British Parliament in the Longitude Act of. While famed scientists, academics, and politicians pursued an answer in the stars, John Harrison, a clock maker, set about to build a clock that could maintain accuracy at sea. This turned out to be the key, and Harrison s Chronometer eventually earned him the prize. For a fascinating account of this drama and of Harrison s struggle to collect his prize money from the government, see the book The Illustrated Longitude by Dava Sobel and William J. H. Andrewes. (a) Find on the left of the addition table and across the top. The intersection of the row headed and the column headed gives the number. Thus,. (b) Also from the table,. So far, our -hour clock system consists of the set {,,,,,,,,,,, }, together with the operation of clock addition. Next we will check whether this sytem has the closure, commutative, associative, identiy, and inverse properties, as described in Section.. Table shows that the sum of two numbers on a clock face is always a number on the clock face. That is, if a and b are any clock numbers in the set of the system, then a b is also in the set of the system. The system has the closure property. (The set of the system is closed under clock addition.) Notice also that in this system, and both yield. Also and both yield. The order in which elements are added does not seem to matter. In fact, you can see in Table that the part of the table above the colored diagonal line is a mirror image of the part below the diagonal. This shows that, for any clock numbers a and b, a b b a. The system has the commutative property. The next question is: When three elements are combined in a given order, say a b c, does it matter whether the first and second or the second and third are associated initially? In other words, is it true that, for any a, b, and c in the system, a b c a b c? EXAMPLE Is -hour clock addition associative? It would take lots of work to prove that the required relationship always holds. But a few examples should either disprove it (by revealing a counterexample a case where it fails to hold), or should make it at least plausible. Using the clock numbers,, and, we see that. Thus,. Try another example:. So. Other examples also will work. The -hour clock system has the associative property. Our next question is whether the clock face contains some element (number) which, when combined with any element (in either order), produces that same element. Such an element (call it e) would satisfy a e a and e a a for any element a of the system. Notice in Table that,, and so on. The number is the required identity element. The system has the identity property. Generally, if a finite system has an identity element e, it can be located easily in the operation table. Check the body of Table for a column that is identical to the column at the left side of the table. Since the column under meets this requirement, a a holds for all elements a in the system. Thus is possibly the identity.
CHAPTER Number Theory Minus hours = FIGURE Now locate at the left of the table. Since the corresponding row is identical to the row at the top of the table, a a also holds for all elements a, which is the other requirement of an identity element. Hence is indeed the identity. Subtraction can be performed on a -hour clock. Subtraction may be interpreted on the clock face by a movement in the counterclockwise direction. For example, to perform the subtraction, begin at and move hours counterclockwise, ending at, as shown in Figure. Therefore, in this system,. In our usual system, subtraction may be checked by addition and this is also the case in clock arithmetic. To check that, simply add. The result on the clock face is, verifying the accuracy of this subtraction. The additive inverse, a, of an element a in clock arithmetic is the element that satisfies the following statement: a a and a a. The next example examines this idea. EXAMPLE Determine the additive inverse of in -hour clock arithmetic. The additive inverse for the clock number is a number x such that x. Going from to on the clock face requires more hours, so. This means that is the additive inverse of. The method used in Example may be used to verify that every element of the system has an additive inverse (also in the system). So the system has the inverse property. A simpler way to verify the inverse property, once you have the table, is to make sure the identity element appears exactly once in each row, and that the pair of elements that produces it also produces it in the opposite order. (This last condition is automatically true if the commutative property holds for the system.) For example, note in Table that row contains one, under the, so, and that row contains, under the, so also. Therefore, and are inverses. The following chart lists the elements and their additive inverses. Notice that one element,, is its own inverse for addition. Clock value a Additive inverse a Using the additive inverse symbol, we can say that in clock arithmetic,,,, and so on. We have now seen that the -hour clock system, with addition, has all five of the potential properties of a single-operation mathematical system, as discussed in Section.. Other operations can also be introduced to the clock arithmetic system.
. Clock Arithmetic and Modular Systems Having discussed additive inverses, we can define subtraction formally. Notice that the definition is the same as for ordinary subtraction of whole numbers. Subtraction on a Clock If a and b are elements in clock arithmetic, then the difference, is defined as a b a (b). a b, Chess Clock A double clock is used to time chess, backgammon, and Scrabble games. Push one button, and that clock stops the other begins simultaneously. When a player s allotted time for the game has expired, that player will lose if he or she has not made the required number of moves. Mathematics and chess both involve structured relationships and demand logical thinking. A chess player may depend more on psychological acumen and knowledge of the opponent than on the mathematical component of the game. Emanuel Lasker achieved mastery in both fields. He was best known as a World Chess Champion for years, until. Lasker also was famous in mathematical circles for his work concerning the theory of primary ideals, algebraic analogies of prime numbers. An important result, the Lasker-Noether theorem, bears his name along with that of Emmy Noether. Noether extended Lasker s work. Her father had been Lasker s Ph.D. advisor. EXAMPLE Find each of the following differences. (a) Use the definition of subtraction. The additive inverse of is, from the table of inverses. This result agrees with traditional arithmetic. Check by adding and ; the sum is. (b) Clock numbers can also be multiplied. For example, find the product by adding a total of times:. EXAMPLE Find each product, using clock arithmetic. (a) (b) (c) (d) Some properties of the system of -hour clock numbers with the operation of multiplication will be investigated in Exercises. Modular Systems We now expand the ideas of clock arithmetic to modular systems in general. Recall that -hour clock arithmetic was set up so that answers were always whole numbers less than. For example,. The traditional sum,, reflects the fact that moving the clock hand forward hours from, and then forward another hours, amounts to moving it forward hours total. But since the final position of the clock is at, we see that and are, in a sense, equivalent. More formally we say that and are congruent modulo (or congruent mod ), which is written mod (The sign indicates congruence.)
CHAPTER Number Theory By observing clock hand movements, you can also see that, for example, mod, mod, and so on. In each case, the congruence is true because the difference of the two congruent numbers is a multiple of :, This suggests the following definition.,. Congruence Modulo m The integers a and b are congruent modulo m (where m is a natural number greater than called the modulus) if and only if the difference a b is divisible by m. Symbolically, this congruence is written a b (mod m). Since being divisible by m is the same as being a multiple of m, we can say that a b mod m if and only if a b km for some integer k. EXAMPLE Decide whether each statement is true or false. (a) mod The difference is divisible by, so mod is true. (b) mod This statement is false, since, which is not divisible by. (c) mod This statement is true, since is divisible by. (It doesn t matter if we find or.) There is another method of determining if two numbers, a and b, are congruent modulo m. Criterion for Congruence a b mod m if and only if the same remainder is obtained when a and b are divided by m. For example, we know that mod because, which is divisible by. Now, if is divided by, the quotient is and the remainder is. Also, if is divided by, the quotient is and the remainder is. According to the criterion above, mod since both remainders are the same. Addition, subtraction, and multiplication can be performed in any modular system just as with clock numbers. Since final answers should be whole numbers less than the modulus, we can first find an answer using ordinary arithmetic. Then,
. Clock Arithmetic and Modular Systems as long as the answer is nonnegative, simply divide it by the modulus and keep the remainder. This produces the smallest nonnegative integer that is congruent (modulo m) to the ordinary answer. EXAMPLE Find each of the following sums, differences, and products. (a) (b) (c) (d) (e) mod First add and to get. Then divide by. The remainder is, so mod and mod. mod. Divide by, obtaining as a remainder: mod. mod. When is divided by, a remainder of is found: mod. mod Since, and leaves a remainder of when divided by, mod. mod mod. Problem Solving Modular systems can often be applied to questions involving cyclical changes. For example, our method of dividing time into weeks causes the days to repeatedly cycle through the same pattern of seven. Suppose today is Sunday and we want to know what day of the week it will be days from now. Since we don t care how many weeks will pass between now and then, we can discard the largest whole number of weeks in days and keep the remainder. (We are finding the smallest nonnegative integer that is congruent to modulo.) Dividing by leaves remainder, so the desired day of the week is days past Sunday, or Wednesday. EXAMPLE If today is Thursday, November, and next year is a leap year, what day of the week will it be one year from today? A modulo system applies here, but we need to know the number of days between today and one year from today. Today s date, November, is unimportant except that it shows we are later in the year than the end of February and therefore the next year (starting today) will contain days. (This would not be so if today were, say, January.) Now dividing by produces with remainder. Two days past Thursday is our answer. That is, one year from today will be a Saturday.
CHAPTER Number Theory Problem Solving A modular system (mod m) allows only a fixed set of remainder values,,,,, m. One practical approach to solving modular equations, at least when m is reasonably small, is to simply try all these integers. For each solution found in this way, others can be found by adding multiples of the modulus to it. EXAMPLE Solve x mod. In a mod system, any integer will be congruent to one of the integers,,,,,, or. So, the equation x mod can be solved by trying, in turn, each of these integers as a replacement for x. x : Is it true that mod? No x : Is it true that mod? No x : Is it true that mod? Yes Try x, x, x, and x to see that none of them work. Of the integers from through, only is a solution of the equation x mod. Since is a solution, find other solutions to this mod equation by repeatedly adding :,,, and so on. The set of all positive solutions of x mod is,,,,,,. EXAMPLE Solve the equation x mod. Because the modulus is, try,,,,,,,, and : Is it true that mod? No Is it true that mod? No Continue trying numbers. You should find that none work except x : mod. The set of all positive solutions to the equation x mod is,,,, or,,,,,,. EXAMPLE Solve the equation x mod. Try the numbers,,,,,,, and. You should find that none work. Therefore, the equation x mod has no solutions at all. Write the set of all solutions as the empty set,. This result is reasonable since x will always be even, no matter which whole number is used for x. Since x is even and is odd, the difference x will be odd, and therefore not divisible by. EXAMPLE Solve x mod. Trying the integers,,,,,,, and shows that any replacement will work. The solution set is,,,,.
. Clock Arithmetic and Modular Systems An equation such as x mod in Example that is true for all values of the variable (x, y, and so on) is called an identity. Other examples of identities (in ordinary algebra) include x x x and y y y. Some problems can be solved by writing down two or more modular equations and finding their common solutions. The next example illustrates the process. EXAMPLE Julio wants to arrange his CD collection in equal size stacks, but after trying stacks of, stacks of, and stacks of, he finds that there is always disc left over. Assuming Julio owns more than one CD, what is the least possible number of discs in his collection? The given information leads to three modular equations, x mod, x mod, x mod, whose sets of positive solutions are, respectively,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, and,,,,,,,,,,,. The smallest common solution greater than is, so the least possible number of discs in the collection is. EXAMPLE A dry-wall contractor is ordering materials to finish a -footby--foot room. The wallboard panels come in -foot widths. Show that, after uncut panels are applied, all four walls will require additional partial strips of the same width. The width of any partial strip needed will be the remainder when the wall length is divided by (the panel width). In terms of congruence, we must show that mod. By the criterion given above, we see that this is true since both and give the same remainder (namely ) when divided by. A -foot partial strip will be required for each wall. (In this case four -foot strips can be cut from a single panel, so there will be no waste.) FOR FURTHER THOUGHT A Card Trick Many card tricks that have been around for years are really not tricks at all, but are based on mathematical properties that allow anyone to do them with no special conjuring abilities. One of them is based on mod arithmetic. In this trick, suits play no role. Each card has a numerical value: for ace, for two,, for jack, for queen, and for king. The deck is shuffled and given to a spectator, who is instructed to place the deck of cards face up on a table, and is told to follow the procedure described: A card is laid down with its face up. (We shall call it the starter card.) The starter card will be at the bottom of a pile. In order to form a pile, note the value of the starter card, and then add cards on top of it while counting up to. For example, if the starter card is a six, pile up seven cards on top of it. If it is a jack, add two cards to it, and so on. (continued)
CHAPTER Number Theory When the first pile is completed, it is picked up and placed face down. The next card becomes the starter card for the next pile, and the process is repeated. This continues until all cards are used or until there are not enough cards to complete the last pile. Any cards that are left over are put aside for later use. We shall refer to these as leftovers. The performer then requests that a spectator choose three piles at random. The remaining piles are added to the leftovers. The spectator is then instructed to turn over any two top cards from the piles. The performer is then able to determine the value of the third top card. The secret to the trick is that the performer adds the values of the two top cards that were turned over, and then adds to this sum. The performer then counts off this number of cards from the leftovers. The number of cards remaining in the leftovers is the value of the remaining top card! For Group Discussion. Obtain a deck of playing cards and perform the trick as described above. (As with many activities, you ll find that doing it is simpler than describing it.) Does it work?. Explain why this procedure works. (If you want to see how someone else explained it, using modulo arithmetic, see An Old Card Trick Revisited, by Barry C. Felps, in the December issue of the journal The Mathematics Teacher.)