Again infinitely many now imagine how many ir r

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Unformatted text preview: initely many. How many r ational numbers can you create with NON-TERMINATING (infinitely many digits) and REPEATING decimal expansion? Again, infinitely many. Now, imagine how many ir r ational numbers you can create with NON-TERMINATING irr ir NON-REPEATING decimal expansion? 1 1 Between the rationals − = 0.25 and − = 0.333 . . . we can create rationals 4 3 innumerable ir rational numbers with NON-TERMINATING NON-REPEATING patterns irr ir of decimal digits. Again, between any two such ir r ational numbers created above we may create irr ir innumerably more ir r ational numbers in a similar manner. There is a virtual irr ir irr flood of ir r ational numbers. This should give us the feeling that the irrationals ir are NOT COUNTABLE. Since we cannot even begin to COUNT them, there is no sense in talking about a subscript to enumerate them. If the r ationals are DENSE then the ir r ationals are more than DENSE. Each irr ir irr ir rational number corresponds to a point on the number line. If we try to plot the irr ir r ationals as co-linear points in an orderly manner we would get an “almost “almost continuous” irr continuous” line. However, the ir r ationals by themselves do not exhaust, ir COMPLETELY cover, the number line. −2 S −3 − 2 − 1 0 12 1 π e S2 2 3 3. REALS , COMPLETE , C OMPLETE CONTINUOUS C ONTINUOUS From the decimal expansion point of view, any number must have either a FINITE (terminating) number of digits or an INFINITE (non-terminating) number of digits. And, if it has an INFINITE number of digits, then the pattern of digits must be REPEATING (recurring) or NON-REPEATING. Can you think of any other possibility? Also, we can compare any two numbers (rational or irrational) and plot them on the number line in an orderly manner. Numbers: R eal Number s: It should now be at least intuitively clear that any point on the number line is either a r a tional o r an ir r a tional n umber. And together irr ir these two sets of numbers COMPLETELY cover or exhaust the whole number line extending in both direc...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.

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