Formal Notation and Language in Mathematical Proof

In a previous article, we showed how the phrase

For all $n \in \N$ , $9^n - 1$ is divisible by $8$ .

Could be written formally as

$\forall n \in \N \quad (\exists N \in \Z: \quad 9^n - 1 = 8N)$

It might feel as if we've introduced a lot of new and complicated notation, just to say something we were already able to communicate quite easily. These formal statements may appear rather cryptic at first, and in this particular case, it's much shorter just to write the statement in prose instead. However, there are a few reasons we use this notation:

As statements become more complicated, writing these in prose can become unweildy
In some cases, writing the statement in English can be ambiguous
When communicating with people who speak other languages, writing in prose can result in miscommunications, while the use of mathematical notation should be much more stable and consistent.

In this section we will cover a few examples which highlight the value of writing logical statements in formal notation, which we will need when writing mathematical proofs.

...