Part I, Number Systems. CS131 Mathematics for Computer Scientists II Note 1 INTEGERS
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1 CS131 Part I, Number Systems CS131 Mathematics for Computer Scientists II Note 1 INTEGERS The set of all integers will be denoted by Z. So Z = {..., 2, 1, 0, 1, 2,...}. The decimal number system uses the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. A positive integer n can then be represented by a string of these digits or equivalently by a sum of multiples of powers of ten. For example the string represents the number ( ) + ( ) + ( ) + (5 10) + 8. The binary system is of great importance in computing. Here there are just two digits 0 and 1 (called bits). A string of bits then represents a sum of multiples of powers of two. For example in the binary system represents (1 2 4 ) + (0 2 3 ) + (0 2 2 ) + (1 2 1 ) + (1 2 0 ). In decimal notation this becomes = 19. Subscripts are often used to indicate the base. For example we can write two = 19 ten Binary to Decimal Conversion. The string b n b n 1... b 2 b 1 b 0 of bits represents (b n 2 n ) + (b n 1 2 n 1 ) (b ) + (b 1 2) + b 0. Decimal to Binary Conversion. Divide repeatedly by 2 to get remainders r 0, r 1, r 2,..., r n. The binary representation is then r n r n 1... r 2 r 1 r 0. Remember that the last remainder is the first digit of the representation in the new base. Problem. Convert the decimal representation 244 to binary. 1 1
2 Solution R R R R R R R 1 0 R 1 Hence 244 ten = two Problem. The hexadecimal (base 16) system uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F where A,..., F represent ten,..., fifteen. Convert the hexadecimal representation 21BAD to decimal. Solution. The decimal representations of B,A and D are 11, 10 and 13 so 21BAD represents ( ) + ( ) + ( ) + ( ) + ( ) which simplifies to Conversion algorithms. Let b > 1 be an integer. To convert a base 10 representation to base b divide repeatedly by b converting the remainders r 1, r 2,..., r n to digits in base b. The base b representation is then r n r n 1... r 2 r 1. To convert a base b representation x n x n 1... x 2 x 1 x 0 to base 10 compute x n b n + x n 1 b n x 2 b 2 + x 1 b + x 0 converting each digit to base 10 if necessary. We now look at two important algorithms concerning integers. The Division Algorithm. numbers q, r Z with If a, b Z with b 0, then there are unique a = qb + r and 0 r < b q is called the quotient and r the remainder. If a, b Z with 0 < b < a then q and r can be computed by forming the numbers a b, a 2b,..., a nb, a (n+1)b stopping when the first strictly negative value a (n + 1)b < 0 is obtained. Then q = n and r = a nb. If a and b are integers with a = qb for some integer q then we say that b divides a. The greatest common divisor of two integers is defined to be the largest integer which divides them both. The greatest common divisor of 1 2
3 m and n will be denoted by gcd(m, n). Note that gcd(0, n) = n for every n > 0. The Euclidean algorithm makes repeated use of the division algorithm to calculate the greatest common divisor of two integers. The Euclidean Algorithm. Let r 1 and r 0 be integers with r 1 < r 0. (1) For each i define r i+1 to be the remainder when r i 1 is divided by r i. The last non-zero remainder obtained is equal to gcd(r 1, r 0 ). (2) The relations r i 1 = q i r i + r i+1 (1 i N) can be used to write this last non-zero remainder in terms of r 1 and r 0 giving gcd(r 1, r 0 ) = xr 1 + yr 0 for some integers xand y. To see why the algorithm works, note that since r i 1 q i r i = r i+1 any common divisor of r i 1 and r i must divide r i+1. We may say from this argument that that all remainders r i must inherit each other s common divisors. Hence we have gcd(r i, r i 1 ) = gcd(r i+1, r i ) since each pair has the same set of common divisors. Also 0 r i+1 < r i for every i so the remainders get smaller and must eventually reach 0. Now if r N is the last non-zero remainder gcd(r 1, r 0 ) = gcd(r 2, r 1 ) =... = gcd(r N 1, r N ) = gcd(r N, 0) = r N. Problem. Find the greatest common divisor of and and determine integers x and y such that gcd(16579, 30031) = x y Solution. We have gcd(16579, 30031) = = gcd(13452, 16579) = = gcd(3127, 13452) = = gcd(944, 3127) 3127 = = gcd(295, 944) 944 = = gcd(59, 295) 295 = = gcd(0, 59) =
4 and now using the identities in the second column. 59 = = ( ) = = 10( ) = = ( ) = = 53( ) = Modular Arithmetic. Let n be an integer greater than 1. We say that two integers a and b are congruent modulo n and write a b mod n or a n b if a b is an integer multiple of n, i.e. a = b + kn where k is an integer. For example: 17 1 mod mod mod mod 7 n 0 mod n 1 n 1 mod n. Two congruences with the same modulus n can be added, subtracted and multiplied just like ordinary equations. Computer Representation of Integers. In binary encoding, numbers are stored in computer locations as strings of bits. In practice these strings all have the same length so only a finite number of integers can be represented. A common method for distinguishing between positive and negative integers is known as two s complement and permits the replacement of integer subtraction by integer addition. For example, using strings of length 3 (in practice the length is likely to be 32) we can represent 2 3 integers. The 8 integers 4, 3, 2, 1, 0, 1, 2, 3 can be represented as follows: n binary string representing n
5 Here the sign of an integer can be detected from the initial bit (positive integers start with 0, negative ones with 1). If the binary string representing a positive integer m is known then, the string representing m can be found from the fact that m ( (2 N 1) m ) N where 2 N 1 is represented by a string of N 1 s. Equivalently, to find the string representing m we interchange 0 s and 1 s in the string representing m and then add 1. For example, since 3 is represented by 011, then 3 is represented by = 101 In this system arithmetic is carried out modulo 2 N for some N so the answer will be correct only if it can be represented. For N = 32 we can represent 2 32 = integers which allows us to do arithmetic with integers lying between and ABSTRACT Content Integers - Binary, conversion algorithm, division algorithm, Euclidean algorithm, modular arithmetic, 2 s complement Numbers form the basis of all measurement science. In computer science, integers have an especially important role whether as data types in their own right, with corresponding operations of addition and multiplication or, in the case of N, as a tool for analysing the notion of computation itself. Hence an insight into their nature is essential. Concepts such as negative, rational and irrational numbers (all ideas resisted strongly when first created) can be rigorously defined in terms of the natural numbers which can themselves be defined in terms of their algebraic and analytical properties. In this Note we also look at the representation of numbers in bases other than 10 and conversion from one system to another. We also introduce modular arithmetic, a system commonly used by computers to represent negative integers. Two classical but still useful algorithms concerning the properties of integers are also described. The Euclidean algorithm and modular arithmetic are at the heart of many modern data encryption algorithms and so play an important role in data security. History The ancient Egyptians, Greeks, Romans and Babylonians all evolved number systems, although none had a zero. A symbol for zero was introduced from India via Arab mathematicians in about the 8th century AD. This was a vital contribution and allowed a place-value system to be devised on which the decimal system is based. It should be remembered that a number system with base 60 was widely used and lasted from about 1700 BC until about 500 AD. At this time, the Hindus created the decimal system (but without a zero) although using different symbols from those we use today. The 60 based system still survives in our division of the units of time (hours, minutes, seconds) and angle (degrees and minutes). The binary system was first developed by the mathematician and philosopher Leibniz in 17th century. Giuseppe Peano [ ] was an Italian mathematician who was a pioneer in symbolic logic. His concise logical definitions of natural numbers were devised in order to derive a complete system of notation for logic. 1 5
6 Peano s first work in logic, published in 1888, contained his rigorously axiomatically derived postulates for natural numbers. He acknowledged his debt for some of the work to German mathematician Richard Dedekind. Peano s work was used by the English philosopher and mathematician Bertrand Russell. Peano also applied the axiomatic method to other fields, such as geometry, first in 1889 and again in A treatise on this work contained the beginnings of geometrical calculus. Peano provided new definitions of the length of an arc of a curve and of the area of a surface. Formulario mathematico , comprising his work and that of collaborators, contains 4,200 theorems. 1 6
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