chapter 3 - Bentham Science

More documents

Recommendations

Info

Spreadsheet Conditional Formatting Applications of Spreadsheets in Education The Amazing Power of a Simple Tool 65 be included in the school curriculum is long overdue. With just a little effort, this material may be made accessible and, hopefully, appealing to secondary school mathematics students. In this chapter we show that informative investigations may be carried out using spreadsheets[2] and present new spreadsheet-based approaches for computing and illustrating multiplication tables for modular arithmetic. The modular arithmetic spreadsheet described in this chapter allows investigation of a variety of properties and concepts in modular arithmetic, including that of modular inverse, which is essential for understanding of public-key cryptosystems such as RSA. 4.2 Modular Arithmetic Due in significant part to its widespread application to cryptography and coding theory, basic knowledge of this important branch of mathematics is now considered almost essential for IT professionals, and others. However, in the experience of the authors, relatively few IT graduates seem to appreciate that integer arithmetic on a digital computer is none other than modular arithmetic. This is so, as all integer operations are performed modulo some power of 2 – it is the nature of a digital, binary CPU. All integer arithmetic done on a digital computer is with respect to a maximum word size, and therefore is modulo some positive integer. For example, on a contemporary desktop machine, an integer may be represented by 32 bits. This means that additions and multiplications will be modulo 2 32 (for cardinals), or 2 31 (for signed integers). When two integers are added on such a machine, and the sum exceeds the maximum value representable with 32 bits, e.g., 2 32 − 1 for a cardinal, then the processor overflow flag will be set, and the sum will only be correct modulo 2 32 . In many instances, this situation corresponds to a serious run-time error. However, if we are doing arithmetic modulo 2 32 , then we may legitimately ignore the overflow flag, as this is just the result we want. 4.2.1 Rules of the Game What is modular arithmetic and how does it work? What are the basic rules? For the most part, these are very simple. The first concept we need is that of remainder or residue. Suppose we wish to divide 38 by 9. We have 38=(4×9)+2. In this equation, 38 is the dividend, 9 is the divisor, 4 is the quotient, and 2 called the remainder or residue. The word residue means “something left over”. By definition, it is non-negative and less than the divisor, so it could be zero. In other words, residues run from 0 up to one less than the divisor. In some contexts the divisor is called the modulus (plural moduli), and we consider this very soon. Given any dividend and a non-zero divisor, the quotient and remainder are unique. This is guaranteed by the so-called division algorithm. Theorem 4.1 (The division algorithm) Given integers n and d = 0, there exist unique integers q (quotient) and r (residue or remainder) such that n=qd+ r and 0≤r≤d− 1. Definition 4.1 In Theorem 4.1, the residue r is also written as n mod d. Example 4.1 We have 38=(4×9)+2, so 38 mod 9=2. Example 4.2 We have−38=(−5×9)+7, so−38 mod 9=7.
66 Applications of Spreadsheets in Education The Amazing Power of a Simple Tool Miller and Sugden Remark 4.1 When computing a mod b for specific integers a and b, we may think of adding or subtracting as many copies of b from a as required until we finally arrive at a number in the correct range, i.e., between 0 and b−1 inclusive. For example, to compute 89 mod 12, keep removing (subtracting) twelves until you are left with 5. This can be regarded as removing seven copies of 12, or equivalently, dividing 89 by 12 to obtain a quotient of 7 (irrelevant for modular arithmetic) and remainder 5 (the important part). Thus, 89 mod 12=5. To compute(−46) mod 12, do pretty much the same thing, except that we need to keep adding twelves until we are “in range,” i.e., between 0 and 11 inclusive. Adding four twelves to (−46) gives us 2, which is the answer. Thus, (−46) mod 12=2. Exercise 4.1 Verify that the conditions of Theorem 4.1 are satisfied for each of the examples just considered. 4.2.2 Congruence with Respect to a Modulus If two integers x and y have the same residue with respect to a given modulus m, i.e., x mod m = y mod m, then we say that they are congruent with respect to this modulus and write x≡y(mod m). For example, 22 and 40 are congruent with respect to the modulus 6, since they both have remainder 4 after division by 6. We write 22≡40(mod 6). Another way to think of this is that the numbers x and y differ by a multiple of m, i.e., x−y≡0(mod m). Thus, 22≡40(mod 6) as 22 and 40 differ by 18, which is a multiple of 6. It is important to realize that the congruence x ≡ a(mod m) defines an infinite arithmetic sequence of integers. For example, the congruence x≡1(mod 2) defines the sequence of odd numbers: ... − 7,−5,−3,−1,1,3,5,7, ... This sequence may be written as xn = 1+2n; n∈Z. Notice that we transform the congruence x≡1(mod 2) into an equation by introducing a parameter n. 4.2.3 Arithmetic Operations We have seen that a modulus is simply a positive integer to be thought of as a divisor. If we think of it this way, then we can imagine that it generates a quotient and a remainder. For modular arithmetic, we are interested only in the remainder (residue). Without loss of generality, we deal with positive divisors in this chapter. Clearly, zero cannot be a divisor, and we are not interested in m=1 either, since remainders are always zero when we divide by 1; thus, m≥2. Now suppose a,b,c,d are any integers. We wish to add, subtract, multiply and divide these numbers with respect to a particular modulus m≥2. What are the rules? It turns out that, for the most part, they are about as simple as we could hope for. For example, suppose a ≡ 2(mod 7) and b≡6(mod 7). Then a+b≡(2+6)(mod 7), a−b≡(2−6)(mod 7), a×b≡(2×6)(mod 7). More generally, If a ≡ c (mod m) and b≡d (mod m) then, for n≥0,we have a+b ≡ (c+d)(mod m) a−b ≡ (c−d)(mod m) ab ≡ cd(mod m) a n ≡ c n (mod m)
Page 2 and 3: Applications of Spreadsheets in Edu
Page 4 and 5: To my wife Keddy and daughter Helga
Page 6 and 7: Part III - Management Sciences 7. P
Page 8 and 9: ii PREFACE This e-book is devoted t
Page 10 and 11: iv The chapters contained herein we
Page 12 and 13: vi Wee Leong Lee School of Informat
Page 16 and 17: Fault Analysis in Power Systems App
Page 18 and 19: Use of Spreadsheets for Analyses Ap
Page 22 and 23: Optimal Control of Dynamical System
Page 26 and 27: 84 Applications of Spreadsheets in
Page 36 and 37: Teaching Portfolio Theory Applicati
Page 40 and 41: Forecasting with Innovation Diffusi
Page 46 and 47: Graduates’ Use of Technical Softw
Page 52: 276 Applications of Spreadsheets in

chapter 3 - Bentham Science

Create successful ePaper yourself

Delete template?

Save as template?