Unformatted text preview: d refer the interested reader to a solid course in Calculus for a more rigorous explanation.
The number 3 = 1.73205 . . . is an irrational number3 and as√
such, its decimal representation
neither repeats nor terminates. We can, however, approximate 3 by terminating decimals, and
it stands to reason4 we can use these to approximate√ 3 . For example, if we approximate 3
by 1.73, we can approximate 2 3 ≈ 21.73 = 2 100 =
2 . It is not, by any means, a pleasant
number, but it is at least a number that we understand in terms of powers and roots. It also stands
to reason that better √
and better approximations of 3 yield better and better approximations of
2 3 , so the value of 2 3 should be the result of this sequence of approximations.5
Suppose we wish to study the family of functions f (x) = bx . Which bases b make sense to study?
We ﬁnd that we run into diﬃculty if b < 0. For example, if b = −2, then the function f (x) = (−2)x
has trouble, for instance, at x = 2 since (−2)1/2 =...
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