Unformatted text preview: our assumption that
S2 is rational is wrong. We say S2 is irrational. The term IRRATIONAL literally
means “ having no ratio ”. For practical mensuration the whole range of
h aving
the rationals is more than sufficient. But from the theoretical point of view this
is not enough. For example, we need to define the length of the diagonal of a
square of side of unit length.
In f act, between any two r a tional n umber s there ar e infinitely m a n y
irr
ir rational numbers. So we cannot say that the set of rationals Q is COMPLETE.
We cannot say that each and every point on the number line is some rational
number. The set of rational numbers Q i s DENSE but not COMPLETE. 10 number
Ir r a tional n umber s:
Look at the number line again and consider a portion of it, say from 0 to 1.
......
−
3 0
1
2
3
Try to visualize it as a collection of infinitely many points, so many that we cannot
even begin to ENUMERATE them. Infinitely many points represent r ational
number s and infinitely many points represent ir r a tional n umbers.
ir
1. −
2 −
1 Between any two irrational numbers there are INFINITELY many irrational
numbers. The numbers:
S2, S5, 3SS3 + S2, S3S5 + S7 and many other expressions involving rational numbers under the radical
sign S a re irrational. These irrational numbers are said to be
expressed in terms of radicals.
The decimals help us classify the rational and irrational numbers. Rational numbers
are represented by TERMINATING DECIMALS,
1
e.g. − = 0.25
4
1
or INFINITE REPEATING DECIMALS, e.g. − = 0.333 . . .
3
I r r a t i o n a l n u m b e r s a r e r epr esented by NON TERMINATING NONREPEATING DECIMALS.
Examples:
S 2 = 1 .41421356237 . . . e = 2 .718281828459045235360 . . . π = 3 .14159265358979323846264338327950 . . . 11 2. Are the irrationals COUNTABLE ? Without giving a formal proof let us try to get an intuitive feel for how many ir r ational
irr
ir
numbers there could be as campared to the COUNTABLE set of rationals Q .
How many r ational numbers can you create with TERMINATING (finitely many
digits) decimal expansion? Inf...
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 Fall '09
 TAMERDOğAN
 Limit, Δx

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