# SP20:Lecture 14 prep

We will show that and are functions. We will also use these to define the Base b representation (which is an important bijection between the natural numbers and the set of strings of digits).

I will use the following notation:

We often want to prove that there exists a sequence of values , all in , satisfying some property. Formally, we would say "there exists and values such that ".

This takes a lot of writing, and also requires us to introduce the variable (which often just adds complexity). So, we will abbreviate this to "there exists such that ".

We will also abbreviate sums and products of all the values: denotes the sum of the and denotes their product. We won't worry too much about the indices; unless otherwise specified, we just mean add (or multiply) all of them.

Note: there is ambiguity in this notation about whether we allow finite or infinite sequences. I will only use this notation for finite sequences.

Come to lecture knowing the following definitions and how they relate to your existing notion of quotient and remainder:

Definition: Quotient
We say is a quotient of over if for some with . We write (note that quot is a well defined function).
Definition: Remainder
We say is a remainder of over if for some and . We write (note that rem is a well defined function).
Definition: Digit
numbers satisfying are called base b digits.
If are all natural numbers satisfying for all , then the base b interpretation of , written is given by