A lot of people have asked the perfectly reasonable question "how long should an essay be?" I'm sorry, I don't have a good answer. "Length" of an essay in maths is very difficult to measure, and not all ways of measuring it are at all sensible.
For example, projects and theses in this university usually have a word limit. How many words in a formula? Do some formulas contain more words than others? Is a word in maths? What about ? One word or two? Same problem for diagrams. I know of no good way to convert a maths essay into "number of words", though there are literally hundreds of computer programs that try to do this, mainly for the satisfaction of the bureaucrats that think each essay must be constructed out of words and should have a word limit. Every student writing an essay in maths has the headache of trying to work out how many "words" it contains, and this exercise is only for the benefit of the the bureaucrats - it doesn't help the marker or the student at all.
In contrast, mathematics publishers rarely use word count as a measure of the "size" of a piece of work. Instead, if you have a contract to write a mathematics book, your contract will specify the number of pages. But page count is not a particularly good measure either: this depends on the font size, margins, and so on.
In this module, I don't want to prescribe the form of your "essay", i.e. I don't want to tell you or imply that you must write it mostly in words, though that is what I generally expect. Therefore neither a word count nor a page count seems appropriate.
I strongly suspect that when you start writing you will "know" what is a reasonable length, based on a reasonable length of time to do this work. By the end of week 6 you are officially supposed to have done about 50 hours work on this module. If you are a long way off that figure you probably haven't done enough. (You should probably think of 50 hours as an upper limit, and it includes absolutely everything: lectures, reading books and web pages, writing, and submitting your work. 30-40 hours may well be enough.) Something similar applies to the week 11 task.
That said, the best essays are condensed and short, with a lot of good points in a short amount of space, and shortening an essay to improve it takes time. I would always recommend that essays are written in "long form" first and then condensed to something shorter towards the end. (I.e. avoid "waffle" and irrelevant sections that you might be tempted to put in to bring the word count up. But do make sure all relevant explanations are present!) I know it takes time to assess an essay you are writing and make it short and succinct. That time is well worth taking.
The idea in this module is that everyone should be able to see other students' drafts, give feedback to others, and also get feedback on your draft, so you will get a good idea of whether you are in the right ballpark by comparing with others after the first draft is put online at the end of week 4. (Please bear in mind that there are 2 weeks between the draft and the essay, so there is plenty of time for further work.)
I hope these guidelines will help all students get an idea of what is required for good marks, especially in cases where you are attempting something unusual (e.g. something that has a lot of formulas or diagrams or both). (Don't forget to check out the other web page on essay writing that focuses on content. In fact you should remember that your essay's content is always much more important than its presentation, despite what bureaucrats might tell you.)
If you are still not sure after reading all this, I would imagine that (depending on font size, density of your text, and not counting diagrams etc.) four or five good A4 pages of mathematical text should be OK. You will need more if your font is large or if there is a lot of spacing or there are a lot of diagrams. For an essay that is all in words 4-5 pages is about 2000 words. But these are not meant to be strict rules. The more general rule, "Do enough work to show you have explored and can explain something relevant to Gowers' book, the module and your essay title" should be a sufficient guide in most cases.