how to use barcode in c#.net Copyright 2008 by The McGraw-Hill Companies, Inc Click here for terms of use in Software

Printer QR Code in Software Copyright 2008 by The McGraw-Hill Companies, Inc Click here for terms of use

Copyright 2008 by The McGraw-Hill Companies, Inc Click here for terms of use
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The Number Hierarchy
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Figure 9-1 An interval on the rational-number line can be cut in
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half over and over, and you can always find infinitely many numbers in it
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The length of the diagonal of a square that measures 1 unit on each edge The length of the diagonal of a cube that measures 1 unit on each edge The ratio of a circle s circumference to its diameter The decimal number 001001000100001000001
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Whenever we try to express an irrational a number in decimal form, the result is an endless nonrepeating decimal (The last item in the above list has a pattern of sorts, but it is not a repeating pattern like the decimal expansion of a rational) No matter how many digits we write down to the right of the decimal point, the expression is an approximation of the actual value A pattern can never be found that allows us to convert the expression to a ratio of integers The set of irrationals can be denoted S This set is disjoint from the set Q of rationals No irrational number is rational, and no rational number is irrational In set notation, S Q=
Real numbers The set of real numbers, denoted by R, is the union of the set Q of all rationals and the set S of all irrationals:
R=Q S
126 Irrational and Real Numbers
We can envision the reals as corresponding to points on a continuous, straight, infinitely long line, in the same way as we can imagine the rationals But there are more points on a real-number line than there are on a rational-number line (Whether or not the real numbers can be paired off one-to-one with the points on a true geometric line is a question that goes far beyond the scope of this book!) The set of real numbers is related to the sets of rational numbers Q, integers Z, and natural numbers N like this: N Z Q R The operations of addition, subtraction, and multiplication can be defined over R If # represents any of these operations and x and y are elements of R, then: x # y R This is a fancy way of saying that whenever you add, subtract, or multiply a real number by another real number, you always get a real number This is not generally true of division, exponentiation (raising to a power), or taking a root You can t divide by 0, take 0 to the 0th power, or take the 0th root of anything and get a real number Also, you can t take an even-integer root of a negative number and get a real number
Russian dolls Now we can see the full hierarchy of number types We started with the set of naturals, N Then we built the set of integers, Z, by introducing the notion of negative values From there, we generated the set of rationals, Q, by dividing integers by each other Now, we have found out about the set of irrationals, S, and the set of reals, R The sets N, Z, Q, and R fit inside each other like Russian dolls:
N Z Q R The set S is a proper subset of R, but it s standoffish in the sense that it does not allow any of the rationals, integers, or naturals into its realm The Venn diagram of Fig 9-2 shows how all these sets are related Later on, we ll learn about a set of numbers that s even larger than the reals Those are the imaginary numbers and complex numbers They result from taking the square roots of negative reals and adding those quantities to other real numbers
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