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
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
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.
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.
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,
e.g. − = 0.25
or INFINITE REPEATING DECIMALS, e.g. − = 0.333 . . .
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.
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
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
- Limit, Δx