<text><span class="style10">umber Systems and Algebra (2 of 2)LOGARITHMS</span><span class="style7">Since a to the power of 3 = a x a x a, and a to the power of 2 = a x a, then a to the power of 2 x a to the power of 3 = a x a x a x a x a = a to the power of 5. This is an instance of the general rule for the multiplication of powers of the same base: ax x ay = ax + y.From this it is easy to see that a0 = 1, whence also a-x = 1/ax , and the corresponding rule for division is ax / ay = ax - y Similar considerations enable us to give a meaning to ax even where x is not an integer; for example, since x x x = x = x1, x must be x1/2.The logarithm of a number to a given base is simply the power of that base that is equal to the given number. Tables of common logarithms, which use base 10, were used in the days before pocket calculators to assist with complicated multiplications and divisions. For example, it is obviously quite difficult to multiply 135.763 by 4386.734, but it is much easier to add their logarithms, which can be found in a table. Since 135.763 is 10 to the power 2.1327 we find that the logarithm of 135.763 is 2.1327; similarly, since 4386.734 is 10 to the power 3.642l, we find that the logarithm of 4386.734 is 3.6421. We then add these logarithms to find the logarithm of the product of the given numbers. Thus the logarithm of the product is 5.7748, which we can look up in a table of antilogarithms to find the answer 595400 (since 10 to the power 5.7748 is 595400). (NB: This is only approximate because the tables are only made up to four figures; the precise answer is 595556.168042.) A slide rule is a mechanical device that applies this principle. You can add two numbers using two ordinary rulers (where the numbers are equally spaced) by placing the zero of one scale against one of the numbers and reading their sum off the other ruler opposite the second number. A slide rule has a scale that shows numbers spaced according to their logarithms, so that the same method has the effect of adding the logarithms, so that the number read off as the answer is the product of the two given numbers: Here, if we place 1 on the lower slide against 2.5 on the top slide, then the number on the top slide opposite any number on the bottom slide is the result of multiplying it by 2.5; for example, 2.5 x 3 = 7.5 as shown.</span><span class="style10">OTHER NUMBER NOTATIONS</span><span class="style7">The usual notation for numbers is a decimal place-value system. This means that there are ten distinct digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and that the position of each digit determines what it contributes to the value of the number. Each position gives a value 10 times as great as the position to its right, so, for example, 7234 can be written as 4 units (4 x 10 to the power of 0) on the right, plus 3 tens ( 3 x 10 to the power of 1) plus 2 hundreds (2 x 10 to the power of 2) plus 7 thousands (7 x 10 to the power of 3). We say that 10 is the base of the decimal place-value system.We can easily construct systems with other bases to suit particular needs. The binary system uses only the digits 0 and 1; so it has base 2. This is used in the representation of numbers within computers, since the two numerals correspond to the on and off positions of an electronic switch. In the binary system we count as follows: 1, 10 (= 2 + 0), 11 (= 2 + 1), 100 (= 4 + 0 + 0), 101 (= 4 + 0 + 1), 110 (4 + 2 + 0), 111 (4 + 2 + 1), 1000 (= 8 + 0+ 0 + 0), 1001 (= 8 + 0 + 0 + 1), etc.Sometimes, especially in computing, it is convenient to use octal arithmetic (with base 8) or hexadecimal arithmetic (base 16). In base 16, the letters A to F are used as well as the numerals 0 to 9. Obviously it is necessary to know which base is being used, so the base is indicated by a subscript, for example, 3110 = 1F16 = 378 = 111112.There are many other ways in which numbers systems can vary. Sometimes one can see vestiges of other systems in the numerical terms of a language: in French one counts up to 100 in a mixture of base 10 and base 20; for example, quatre-vingt-dix (four times twenty plus ten) equals 90. Even English retains vestiges of base 12, with the words 'eleven' and 'twelve'. The traditional Chinese abacus uses a mixture of base 5 and base 10.The system of Roman numerals is not a place-value system: the letters have fixed values and are ordered from the largest to the smallest. For example, MDCLXVI = 1000 + 500 + 100 + 50 + 10 + 5 + 1 = 1666. When, however, a letter representing a smaller value precedes a larger, it is subtracted; thus CM = 900, and IX = 9. This makes calculations very difficult, and it has been suggested that the superiority of Eastern mathematics over that of early medieval Europe was a result of the system of Roman numerals.</span><span class="style10">PRIME NUMBERS</span><span class="style7">A prime number is a natural number that has no proper factors - that is, which cannot be divided by any natural numbers other than itself and 1. We can find the primes by taking a sequence of numbers such as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 . . .and first deleting all the numbers divisible by 2 (excluding 2 itself, which is only divisible by itself and 1), then all those divisible by 3, then (since anything divisible by 4 has already been deleted) all those divisible by 5, and so on.All non-prime natural numbers must by definition be divisible by other numbers apart from themselves and 1; these other numbers can in turn be repeatedly divided until one is left with a series of prime factors. Hence, all non-prime numbers can be expressed as the product of a series of primes - in fact, for each number, the expression is unique.The prime numbers have been studied since the days of the ancient Greeks, who knew, for example, that there is no largest prime. Their proof is quite easy to understand: Suppose there is a largest prime, so that all the prime numbers can be listed in order of size. Now consider the number we obtain if we multiply all these primes together, and add 1; call this number N. Clearly N cannot be divided by any of the list of primes without leaving a remainder of 1. But since these are (we are assuming) all the primes, any other number is non-prime and so has prime factors. Therefore it cannot divide N unless its prime factors divide N - but no primes can divide N. Thus N must itself be prime. But it is a bigger prime than what we supposed was the biggest prime, so that supposition has led us to a contradiction and must be false. The largest known prime number (August 1989) is (391582 x 2 to the power of 216193) - 1, which is a number of 65087 digits.On the other hand it is not known whether or not there are infinitely many twin primes. These are pairs of successive odd numbers that are both prime, like 5 and 7, 11 and 13, or 29 and 31. Another famous conjecture about prime numbers is that of Christian Goldbach (1690-1764), who postulated that every even number is the sum of two prime numbers. It is not known whether this is true or false.Prime numbers have recently become of great interest to cyptographers: certain codes are based on the result of multiplying two very large primes together, and because even the fastest possible computer would take years to factorize this product, the resulting code is virtually unbreakable.</span><span class="style10">LAWS OF ARITHMETIC</span><span class="style7">Commutative law for addition: a + b = b + a for multiplication: a x b = b x aAssociative law for addition: (a + b) + c = a + (b + c) for multiplication: (a x b) x c = a x (b x c)Distributive law for multiplication over addition: a x (b + c) = (a x b) + (a x c)</span></text>
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<text>ΓÇó SETS AND PARADOXESΓÇó CORRESPONDENCE, COUNTING AND INFINITYΓÇó COMPUTERS</text>
