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1 Introduction 2 Place Value Representation Exotic Arithmetic, 2019 Fusing Dots, Antidots, and Black Holes 1 Introduction This paper discusses several methods of representing numbers, and several ways to understand these methods of representation.1 We begin with what is called decimal representation, the ordinary method we use to represent integers and fractions. Because the method of representation is an important starting point in learning the arithmetic of integers and decimals, we shall explore alternative methods of representation, that is, representation using bases other than our usual base 10 ten. This is roughly akin to the idea that one does not really understand one's own language until we learn a second language. Instead of trying to develop representation in an arbitrary base b, we select a specific base for the sake of clarity. This is base 5 representation. Also, we'll spend some time learning Martian numeration, base 6. Later we will discuss other representation including those for which the base b is not a positive integer. We also explore the system of enumeration when b is a rational number but not an integer, and then when b is a negative integer. Finally, we'll also see that it is even possible for b to be irrational. 2 Place Value Representation The place value interpretation of 4273 is 4000 + 200 + 70 + 3, which is a sum of multiples of powers of 10. The relevant powers of 10 are 103 = 1000; 102 = 100; 101 = 10; and 100 = 1. Each one has a coefficient or multiplier, 4; 2; 7; and 3, respectively. Thus 4 · 103; 2 · 102; 7 · 101; and 3 · 100 are multiples of powers of 10 and therefore 4273 is a sum of multiples of powers of 10. Roger Howe has given the name place value numbers to these multiples of powers of 10. I've been calling them atoms. So place value notation means decimal notation in this case. Each of the addends in the expanded form of a number will be called an atom. Thus for example 4 · 103 is an atom. Once we learn how to do arithmetic with single place numbers, we can use that knowledge along with the distribution property of multiplication over addition, to do arithmetic with decimal numbers in general. This represents a key virtue of place value: it enables arithmetic computation. The fact that the basic arithmetic operations can be efficiently performed by effectively teachable algorithms was the reason that the place value system, which was only introduced into Europe in the late middle ages (around 1200), sup- planted the well entrenched system of Roman numerals. Here is an example. 1Unauthorized reproduction/photocopying prohibited by law' c 1 Exotic Arithmetic, 2019 Fusing Dots, Antidots, and Black Holes Find the product 23 · 41. First recognize each of these numbers as a place value number, 23 = 20 + 3 and 41 = 40 + 1. Then 23 · 41 = (20 + 3) · (40 + 1) =1 (20 + 3)40 + (20 + 3)1 =2 20 · 40 + 3 · 40 + 20 · 1 + 3 · 1 =3 2 · 10 · 4 · 10 + 3 · 4 · 10 + 2 · 10 · 1 + 3 · 1 =4 8 · 102 + 12 · 10 + 2 · 10 + 3 · 100 =5 8 · 102 + (10 + 2) · 10 + 2 · 10 + 3 · 100 =6 9 · 102 + 1 · 102 + 2 · 10 + 3 · 100 =7 9 · 102 + 4 · 10 + 3 · 100 =8 943; where, we have used the distribution property of multiplication over addition in 1, 2 and 6; commutativity of multiplication and addition in 3 and 6; and place value notation in 6,7, and 8. Of course we have also used the digit multiplication table 4 and the digit addition table in 7. Another objective here is to establish methods of translating between decimal representations and base b representations. In other words, we are given a number expresses as a sum of multiples of powers of 10 and wish to rewrite the number as a sums of multiples of powers of b, where b is an integer bigger than 1. For convenience, let us assume for sections 2, 3, and 4 that b = 5. The same procedures work no matter what the value of b is, but fixing the value of b here makes discussion much easier. Of course there is also the problem of translating from a base b representation into a decimal representation, and this process is called interpretation. The notation 21135 is interpreted as a sum of multiples of powers of 5, just as the decimal number 4273 was above. The subscript 5 must be attached unless we are using base 10, because 10 is the default value of the 3 2 1 0 base. Thus 21135 = 2 · 5 + 1 · 5 + 1 · 5 + 3 · 5 = 250 + 25 + 5 + 3 = 283. The process of finding the decimal (ie, base 10) value of a number from its base 5 representation is called interpreting. Thus we interpreted 21135 as 283. The reverse process, that of finding the base 5 representation of an integer expressed in decimal notation is harder and more interesting. There are two methods, (a) repeated subtraction and (b) repeated division. Each method has some advantages over the other. 2 Exotic Arithmetic, 2019 Fusing Dots, Antidots, and Black Holes 3 Repeated Subtraction To see how to use repeated subtraction, first make a list of all the integer powers of 5 that are not bigger than the number we are given. In the case of 283, we need the powers 50 = 1; 51 = 5; 52 = 25; and 53 = 125. Next repeatedly subtract the largest power of 5 that is less than or equal to the current number (which changes during the process). So we have 283 = 125 + 158. At this point our current number becomes 158 and we repeat the process. Then 283 = 125 + 158 = 125 + 125 + 33, and our current number is 33. Repeating the process on 33 gives 33 = 25 + 8 and incorporating that in the above gives 283 = 2 · 125 + 25 + 8 = 2 · 125 + 1 · 25 + 8. Continuing this with 8 leads to 283 = 2 · 53 + 1 · 52 + 1 · 51 + 3 · 50, which is a sum of multiples of powers of 5, just what we want. Thus 283 = 21135, just as we saw above. Repeated subtraction has two advantages over the repeated division method. First, it is closely related to the definition, hence it leads to a better conceptualization. Second, it can be used in other situations when repeated division cannot, as in the case of Fibonacci representation. 4 Repeated Division The repeated division method requires that we repeatedly divide the given integer by base 5 and record the remainder at each stage. First we divide 283 by 5 to get 283 ÷ 5 = 56:6. We can interpret this as 283 = 5 · 56 + 3, so the quotient is 56 and the remainder is 3. Notice that the remainder can never exceed 5 since in such a case the quotient would have been larger. Next divide the quotient by 5 and record the new quotient and the remainder. Thus 56 = 5 · 11 + 1. Repeat the process with the new quotient 11 = 5 · 2 + 1 and finally, 2 = 5 · 0 + 2. Next write the remainders in reverse order, 2; 1; 1; and 3 to get 21135 as the base 5 representation of 283. You'll see why the order must be reversed in the following example. Example 1. Repeated Division To see why 283 = 21135, we can repeatedly replace each quotient with its value obtained during the division process. Thus 3 Exotic Arithmetic, 2019 Fusing Dots, Antidots, and Black Holes 283 = 5 · 56 + 3 = 5(5 · 11 + 1) + 3 = 5(5(5 · 2 + 1) + 1) + 3 = 5(5 · 5 · 2 + 5 · 1 + 1) + 3 = 5 · 5 · 5 · 2 + 5 · 5 · 1 + 5 · 1 + 3 = 2 · 53 + 1 · 52 + 1 · 51 + 3 · 50 = 21135 The advantage of repeated division is that it is computationally more effi- cient. Also, the method of justification can be applied in other situations (synthetic division and Euclidean algorithm). When we get to the section on fusing dots, you'll see in yet another way why it makes sense to record the remainders upon division by b. 5 Repeated Multiplication and Repeated Sub- traction In sections 3 and 4 we saw two methods (algorithms) for writing a given integer in a base different from 10. Before we consider representing fractions, let's review the place value ideas in decimal notation. For example, 5:234 is, as in the first part, a sum of multiples of powers of 10. This time, the powers are (except one) negative exponents: 5:234 = 5 · 100 + 2 · 10−1 + 3 · 10−2 + 4 · 10−3: Using this interpretation as a guide, we can interpret 0:1245 similarly, as a sum of multiples of (negative) powers of 5. Thus −1 −2 −3 0:1245 = 1 · 5 + 2 · 5 + 4 · 5 1 2 4 25 + 10 + 4 = + + = 5 25 125 125 39 = 125 As in the discussion of integers, there are two methods for dealing with numbers in the range 0 < x < 1. They are called (a) repeated subtraction 4 Exotic Arithmetic, 2019 Fusing Dots, Antidots, and Black Holes and (b) repeated multiplication. As before, each has advantages over the other. Example 2. Repeated Subtraction To use the method of repeated subtraction on 39=125, first list the powers of 5 with negative integer expo- nents: 5−1 = 1=5; 5−2 = 1=25; 5−3 = 1=125;:::: Find the largest of these powers of 5 and subtract it from the original number. Thus 39=125 − 1=5 = 14=125. Therefore, 39=125 = 1=5 + 14=125. Now repeat the process on the number 14=125. Note that 1=25 = 5=125.
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