Example:
Why does one need another number system more complicated than the rational
numbers Q?
There are several reasons, many of which are explored in detail in the
next chapters. Here are a few of them (which may use terms we are not yet
familiar with).
- a simple equation like x2 - 9 = 0 does have a solution
in Q, but another, just as simple equation x2 - 2 = 0
does not have a solution in Q.
- If we construct a right triangle for which two sides have length 1, then
we could not measure the length of the remaining side if all we knew were
rational numbers.
- We could not measure the circumference of any circle if all we knew
were rational numbers.
- if we set x0 = 2 and then for each integer n > 0
compute the number
successively, then each resulting number is a rational number, the sequence of
numbers is getting smaller and smaller, but they seem to get closer and closer
to some limit. However, this sequence of numbers does not converge to a
rational number. The sequence looks like this (do you know its limit ?):
- x0 = 2
- x1 = 3/2 = 1.5
- x2 = 17/12 = 1.416...,
- x3 = 577/408 = 1.414215686 , ...
- and so on ...
- There are sets consisting of rational numbers that are bounded, but do
not have a least upper bound in Q.
- Equations such as sin(x) = 1/2 or cos(x) = 0 do not have
solutions in Q.
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