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
NON-REPEATING decimal expansion?
Between the rationals − = 0.25 and − = 0.333 . . . we can create
innumerable ir rational numbers with NON-TERMINATING NON-REPEATING patterns
of decimal digits.
Again, between any two such ir r ational numbers created above we may create
innumerably more ir r ational numbers in a similar manner. There is a virtual
flood of ir r ational numbers. This should give us the feeling that the irrationals
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
ir rational number corresponds to a point on the number line. If we try to plot the
ir r ationals as co-linear points in an orderly manner we would get an “almost
continuous” line. However, the ir r ationals by themselves do not exhaust,
COMPLETELY cover, the number line.
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.
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
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.
- Fall '09