## 1.4. Natural Numbers, Integers, and Rational Numbers

### Example 1.4.5:

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
*x*does have a solution in^{2}- 9 = 0, but another, just as simple equation**Q***x*does^{2}- 2 = 0*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
*x*and then for each integer_{0}= 2*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 ?):*x*_{0}= 2*x*_{1}= 3/2 = 1.5*x*_{2}= 17/12 = 1.416...,*x*_{3}= 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**