1.4. Natural Numbers, Integers, and Rational Numbers
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.