Pretend you only know what \(\N\) and \(\Z\) are, together with their operations (\(\pm, \cdot\)).

1. Why are we not happy with \(\N\) and \(\Z\)? Why develop further types of numbers like \(\Q\)?
2. hard/phylosophical Can you give a non-algebraic answer to the above question?
3. important What is \(\Q\)? You have set operations, functions, and all you expect from basic math, and you know what \(\N\) and \(\Z\) are. You need to describe \(\Q\) (e.g. as a set). How do you go about it?
4. What are the operations on \(\Q\)?
5. optional/not analysis If our goal is to make \(\Z\) into a field, is \(\Q\) the best way possible? Why?
6. good review How do you construct \(\Z\) basing yourself only on \(\N\) and addition on \(\N\)?
   • How do you define \(\Z\) as a set?
   • How do you define addition?
   • What should you check about the definition of addition?
   • What are the properties that you should check to make sure that \((\Z,+)\) is a group?

Now let us suppose we know what \(\Q\) is. We want to talk about limits

1. Can numbers in \(\Q\) "go" anywhere?
2. Can a number in \(\Q\) be "close" to another number?
3. Define a sequence (valued in \(\Q\)).
4. Define a limit of a sequence valued in \(\Q\) using quantifiers.
5. Define a limit of a sequence valued in \(\Q\) using english (but in a way that can be translated to quantifiers easily). You could use the words neighborhood, eventually, for any, there exists, etc.

1. Why is \(\Q\) not good enough for us to do analysis?
2. important What is \(\R\)?
3. What is a Cauchy sequence?