Modular Arithmetic. is just the set of remainders we can get when we divide integers by n

Transcription

1 Modular Arithmetic We are accustomed to performing arithmetic on infinite sets of numbers. But sometimes we need to perform arithmetic on a finite set, and we need it to make sense and be consistent (as far as possible) with normal arithmetic. In this unit we will discuss versions of addition, multiplication, subtraction and division for finite sets of numbers. We will focus on the sets defined by where is just the set of remainders we can get when we divide integers by n One of the important features we want to build into our mathematical operations for finite sets is closure: the property that when we apply an operation to two elements of the set, the result is also an element of the set. Note that we have encountered closure before... when we apply the composition operation to two permutations, the result is another permutation. Since we are working now with, it seems reasonable to use mod n as part of our definitions of addition, multiplication, subtraction and division (because if we end each calculation with mod n we are guaranteed that our results will be in, so we will have closure). In these notes I will use % to represent mod but you can write mod on tests if you prefer it. Since we will be doing a lot of mod operations in this unit, it makes sense to explore the properties of this. Much of this is review none of the new material is difficult. When we write the expression (where and are both integers) we mean the unique integer such that for some integer and The easy way to see that must exist and must be unique is to visualize on a number line where all the multiples of are marked. We can think of as the offset from the closest multiple of that is. It should be clear that for each value of, there is exactly one in the range that works as this offset.

2 When we write the statement we mean Some people get confused by because it looks like the mod n is only being applied to one side. I have to agree... it would make more sense to write to show that the applies to the, and not just to... but nobody is going to listen to me!!! So is a true statement because and All of the examples we have looked have involved positive integers. What happens if we try to include negative integers? For example, what is? It turns out that our definition of works perfectly well if and We can see that and there is no other value of in the range that lets us write... so What if we let be negative? What is? What is? There is no universally accepted definition for these situations. Some mathematicians suggest we should just use the absolute value of, regardless of whether it is negative or positive (so and are handled the same way) but others suggest that when is negative, the remainders should be in the range Fortunately this burning controversy will not be an issue for us... we will always use values of that are > 1

3 The following lemma will be essential to many of the computational tools we will use in this segment of the course: Lemma: Let be an integer and let and be any integers. Then Proof: Let this means is the unique integer such that where is an integer and We want to show by definition We can put that to use in several ways! (Now I really am going to switch over from mod to % ) Claim: Let, and be integers, with. Then Proof: Let and and Claim: Let, and be integers, with. Then I ll let you work out the proof of that. The significance of these results is that if we are doing a large calculation that ends with a %n operation, we can add in more %n operations along the way without changing the result (as long as we are careful).

4 Here are two examples of how we can apply these equalities, similar to the one I did in class. Example 1: What is the value of? Example 2: What is the value of? This can be particularly useful if we attempting to code these calculations and the numbers we are adding and/or multiplying are so big that we are in danger of exceeding the MaxInt for whatever system we are using. This is the end of our discussion of basic principles of modular operations. We now return you to your regularly scheduled notes.

5 We will use the symbols division on to represent addition, multiplication, subtraction and and are very easy to define, and we start with them: Let a and b be elements of. Then, and Example: Let n = 7, a = 3, b = 6. Let n = 8, a = 3, b = 6 It s useful to look at the full and tables for a couple of small values of n. Here is the table for b

6 Here is the table for b a A brief examination of these two tables reveals some interesting patterns. For example, both are symmetric about their main diagonal (that is, the top right triangle is the mirror image of the bottom left triangle ). This is because and are both commutative, which means that and. Let s prove that is commutative. The proof is based on the fact that ordinary addition is commutative (ie ): The proof that is commutative is just as easy. We can also see that in the table, each row is a permutation of. It s not hard to see why this happens: clearly the first row is just a copy of the axis row since we are just adding 0 to each element of. Then in each subsequent row the value increases by 1, so the value increases by 1, so the remainder when we divide by n goes up by 1... until it drops back down to 0. We can see that the same thing would happen for on any, so every table is going to look a lot like the one we just did.

7 The table is a little more complicated but there are certainly patterns there too. The first row is all 0 s of course because each entry is just the remainder of dividing 0*b by n... which is always 0. Each of the other rows is a permutation of. Is that always going to be true? Also, is there any way to predict what the permutations will be? Let s look at the table for b a Well there goes our idea that all the rows (except for the 0 row) would be permutations of : the row for 2 just contains So what property does 5 have that 4 does not? We will see that the crucial property is that 5 is prime. Now notice that two of the rows of the table are permutations of : the rows for 1 and 3. So what property do 1 and 3 have that 2 does not? We will see that the crucial property is that 1 and 3 are both relatively prime with 4 (that is, the only factor they share with 4 is 1). We will explore these relationships and properties in detail over the next couple of classes, so this is just a bit of dramatic foreshadowing. There are some other simple properties of and that we can establish. 0 is the identity element for : We say that 0 is the additive identity for 1 is the identity element for : We say that 1 is the multiplicative identity for

8 We didn t do this in class, but it is also easy to show that and are associative: I ll prove it for... at some point you should satisfy yourself that it is true for also Now let s consider modular subtraction, for which we use the symbol (Incidentally, the LaTeX code for is \ominus which requires me to make the painful pun... modular subtraction looks ominous.) It would make sense to define as x y = (x y) % n and in fact that is exactly where we are going to end up. But we are going to take a slightly round-about route because that will help us when we define (modular division) In normal arithmetic, when we write a b = x, we understand that this is equivalent to writing a = b + x. So when we want to figure out the value of x in the equation a b = x, it makes sense to say x is the element of such that a = b x

9 Let s look at the table for again b From our definition of we can compute as but we can also get this directly from the table for Letting, we can turn this around, as discussed above, to get We can find x by looking at the 3 row of the table and scanning across until we see 2. (How do we know we will find it? Because each row of the table is a permutation of - so 2 must be in the row somewhere. The number at the top of this column gives us the x that adds to 3 to give 2... we see it is 4, which exactly the same result as we got from the formula. In fact, we can show that this method of using the table to compute will always give the same result as the formula. Which method is better? They are really both just doing the same thing. If n is small and we have the table already, perhaps there is a case for using the table. But if n is large, constructing the table (or even just the relevant row of it) might take a long time we re probably better off using the formula So what was the point of all of this? Why not simply define on? and move

10 The point was that we can define completely in terms of (repeating this : ) Our next task will be to define modular division... and we will do it using.