Unformatted text preview: initely many.
How many r ational numbers can you create with NONTERMINATING (infinitely
many digits) and REPEATING decimal expansion? Again, infinitely many.
Now, imagine how many ir r ational numbers you can create with NONTERMINATING
irr
ir
NONREPEATING decimal expansion?
1
1
Between the rationals − = 0.25 and − = 0.333 . . . we can create
rationals 4
3
innumerable ir rational numbers with NONTERMINATING NONREPEATING 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 colinear 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 (nonterminating) number of digits.
And, if it has an INFINITE number of digits, then the pattern of digits must be
REPEATING (recurring) or NONREPEATING. 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.
 Fall '09
 TAMERDOğAN

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