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Extensible Markup Language | 1995-08-15 | 14KB | 65 lines

<?xml version="1.0" encoding="utf-8" ?> <!DOCTYPE card PUBLIC "-//Apple, Inc.//DTD card V 2.0//EN" "" > <card> <id>42885</id> <filler1>0</filler1> <cantDelete> <false /> </cantDelete> <showPict> <true /> </showPict> <dontSearch> <false /> </dontSearch> <owner>5472</owner> <link rel="stylesheet" type="text/css" href="stylesheet_3106.css" /> <part> <id>1</id> <type>button</type> <visible> <true /> </visible> <reserved5> 0 </reserved5> <reserved4> 0 </reserved4> <reserved3> 0 </reserved3> <reserved2> 0 </reserved2> <reserved1> 0 </reserved1> <enabled> <true /> </enabled> <rect> <left>177</left> <top>19</top> <right>610</right> <bottom>339</bottom> </rect> <style>rectangle</style> <showName> <false /> </showName> <highlight> <false /> </highlight> <autoHighlight> <true /> </autoHighlight> <sharedHighlight> <true /> </sharedHighlight> <family>0</family> <titleWidth>0</titleWidth> <icon>0</icon> <textAlign>center</textAlign> Chicago <textSize>12</textSize> <textStyle>plain</textStyle> <name>HERE</name> <script>on mouseUpput the short name of this card into xTempput char 2 to 6 of xTemp & ".mov" into Movie--put Movieif Movie is not empty then-- put "Animations:" & Movie into pMovieplayMovie Movie,"Animations:"end ifend mouseUp</script> </part> <content> <layer>background</layer> <id>25</id> <text>umber Systems and Algebra (1 of 2)The natural numbers or whole numbers are those we use in counting. We learn these at an early age, perhaps pairing them with our fingers or else learning to chant their names in order: 'one, two, three, four, . . . '. These are both important features of our number system - that these numbers can be used to count sets of objects, and that they form a naturally ordered progression that has a first member, the number 1, but no last member: no matter how big a number you come up with, I can always reply with a bigger one - simply by adding 1.However, even quite simple arithmetic, as we shall see, cannot be carried out wholly within the natural numbers. Ordinarily we take the principles that govern such systems for granted, yet merely to be able to subtract and divide, for example, requires other, more complex, number systems, such as fractions and negative numbers.Natural numbers and arithmeticIf I have 3 sheep and you give me 4 more, I can count that I now have 7 sheep, or I can use the operation of addition to get the same answer: 3 + 4 = 7. If I promise to give 5 children 4 sweets each, again I can count out 20 sweets altogether, or I can use the operation of multiplication: 5 x 4 = 20. Here, we have examples of another principle of natural numbers: any addition or multiplication of natural numbers gives another natural number. Such a system is said to be closed under these operations. (A closed system is one where an operation on two of its elements produces another element of that system.) If I had 3 sheep and when you gave me your sheep I had 7, I can use the operation of subtraction to find how many sheep you gave to me: 7 - 3 = 4. If I distribute 20 sweets equally to 5 children, I can use the operation of division to find how many I gave to each: 20 ├╖ 5 = 4. Subtraction is the inverse operation of addition; division is the inverse operation of multiplication. However, the natural numbers are not closed under the operations of subtraction and division, as we shall see later.In simple algebra, we generalize arithmetic by using letters to stand for unknown numbers whose value is to be discovered, or to stand for numbers in general. Usually letters from the beginning of the alphabet are used in the latter way - for example, to express a general truth about numbers, such as a + b = b + a. The letters at the end of the alphabet are generally used to represent unknown numbers. For example, the information about the sheep can be expressed by the equation, 3 + x = 7, where x represents the unknown number of sheep you gave to me. Since the two sides of this equation are equal, they remain equal if we treat them both the same way. If we then subtract 3 from each side we get x = 7 - 3, that is x = 4. We have solved the equation.Subtraction and the integersThe set of natural numbers is not closed under the operation of subtraction; for example, 3 - 7 does not give a natural number as an answer. We need a system of numbers that is closed under subtraction. The smallest set of numbers that is closed under subtraction is the set of integers, i.e. the set ..., -3, -2, -1, 0, 1, 2, 3, ..... Here, the positive integers can be identified with the natural numbers; zero (0) is defined as the result of subtracting any integer from itself; and the negative integers are the result of subtracting the corresponding positive integers from zero (e.g. -3 = 0 - 3).Now, every subtraction has an answer within the number system of integers, that is, the integers are closed under subtraction.Division and the rational numbersThe integers, however, are still not closed under the operation of division. We can construct a system that is by defining the result of any division, a ├╖ b to be the pair of integers, a and b, written in a notation that clearly distinguishes which divides which. Thus, we write a ├╖ b as the ratio or fraction, a/b, and we have the system of rational numbers.It is important to note that rational numbers are not identical with their symbols. The same rational number may be represented by many different fractions (in fact, an infinite number of them). For example, 24/8 is the same rational number as 12/4 or 6/2. We adopt the convention of representing them, where possible, by the unique fraction in which there is no common factor that can be canceled out (thus, 14/21 becomes 2/3, where the factor, 7, has been canceled out). It should also be noted that decimals are rational numbers, since, for example, 0.5 = 5/10 = 1/2, and 1.61 = 161/100.We do have a problem, however: the rationals cannot be closed under division, because of the integer 0. We cannot give value to a/0 for any rational number a. This problem, however, cannot be avoided: we have to be content with the fact that the rationals, excluding the integer 0, are closed under division.Roots and irrational numbers6 to the power of 9, which we read as, '6 to the power 9' means 6 multiplied by itself 9 times ( 6 x 6 x 6 x 6 x 6 x 6 x 6 x 6 x 6). Generally, ab, which we read as, 'a to the power b', means a multiplied by itself b times. These are closed operations for the systems of numbers we have so far considered. However, none of these systems guarantees the possibility of the inverse operation, the extraction of roots. If b = a to the power of 2, (where n represents an integer), then a is the nth root of b, written a = nΓêÜb. For example, since 3 x 3 = 9, the second or square root of 9 (written 2ΓêÜ9 or more usually ΓêÜ9) equals 3. To give another example, since 2 x 2 x 2 = 8, the third or cube root of 8 (written 3ΓêÜ8) is 2. But none of the systems we have considered is closed under this operation. For example, ΓêÜ2, ΓêÜ3, and ΓêÜ5 cannot be expressed as fractions or as terminating decimals; they are examples of what are called irrational numbers. They have exact meaning - for example, by Pythagoras' theorem, ΓêÜ2 is the length of the hypotenuse of a right-angled triangle whose other sides are each length 1; ΓêÜ5 is the length of the hypotenuse of a right-angled triangle whose other sides have lengths 1 and 2, etc. Obviously, we need to add their rationals to our number systems to ensure closure under these calculations.All the systems we have discussed, the natural numbers, the integers, the rational numbers and the irrationals form together the system of real numbers.Imaginary and complex numbersHowever, now we have admitted the extraction of roots, we have opened up a new gap in our number system: we have not, as yet, defined the square root of a negative number. At first sight, we may wonder why this omission should be of any great importance, but without the development of a system to include such numbers, many valuable applications to engineering and physics would not be possible. Surprisingly, we need only extend the number system by one new number. Since all negative numbers are positive multiples of -1 (for example, -6 is 6 x -1, so that ΓêÜ-6 = ΓêÜ6 x ΓêÜ-1) we are concerned only with the square root of -1. The square root of -1 is denoted by the letter i, so we have i to the power of 2 = -1.Real multiples of i, such as 3i, 2.7i, 2i/3, iΓêÜ2, etc., are called imaginary numbers. The sum of a real number and an imaginary number, such as 5 + 3i, is a complex number. It can be shown that every complex number can be expressed uniquely as the sum of its real and imaginary parts.The rules for using complex numbers are the same as those for real numbers. It can be shown, for example, that (a + ib) (a - ib) = a to the power of 2 + b to the power of 2.The terms in brackets are thus the factors of a to the power of 2 + b to the power of 2. In fact it turns out that in the complex number system any algebraic expression with integer powers has exactly the same number of factors as the highest power in the expression. This result is so important that it is called the fundamental theorem of algebra.EJB</text> </content> <content> <layer>background</layer> <id>26</id> <text>he Chinese abacus usually has two beads representing 5s on each wire above the cross bar, and five beads representing 1s on each wire below the bar. The beads are moved towards the bar. Two numbers are shown here, 8654 on the left and 93 on the right.</text> </content> <content> <layer>background</layer> <id>23</id> <text>ΓÇó SETS AND PARADOXESΓÇó CORRESPONDENCE, COUNTING AND INFINITYΓÇó COMPUTERS</text> </content> <content> <layer>background</layer> <id>36</id> <text>6668334</text> </content> <name>p064-1</name> <script></script> </card>