Whether you're writing a book, code, or a school essay, I really recommend visually planning out your thoughts.

The problem with pen and paper is that correcting and re-arranging your ideas gets messy, and defeats the original idea.

There are many software tools to do mindmapping, and honestly, the few that I've tried have been more of a pain than a help.

I recentl found mindmup. I love it! And recommend it. It's free. Open source. Works with Google Docs seamlessly. Exports useful formats like PNG and PDF.

And more to the point, it is really friction free. For me it has been the best tool, just stays out of the way like a good tool should.

## Saturday, 21 June 2014

## Sunday, 8 June 2014

### Complex Numbers Are More Complete Than Reals

We've previously explained what complex numbers are, and how to work with them.

What we perhaps didn't explain so clearly is why we need complex numbers. Sure they've turned out to be very very useful for simplifying calculations about the real world, but what's a good motivation for them?

A great explanation came from this fantastic book: Elliptic Curves.

The book itself aims to do what Make Your Own Mandelbrot wants to do - share some of the most amazing and beautiful mathematics to as wide an audience as possible by taking readers through the concepts gently using clear English. I've just finished Chapter 2 and I can't wait to get through the rest of it.

Anyway, this book's explanation is super simple:

I'm really excited by this super clear explanation.

Anyway, here's a summary:

What we perhaps didn't explain so clearly is why we need complex numbers. Sure they've turned out to be very very useful for simplifying calculations about the real world, but what's a good motivation for them?

A great explanation came from this fantastic book: Elliptic Curves.

The book itself aims to do what Make Your Own Mandelbrot wants to do - share some of the most amazing and beautiful mathematics to as wide an audience as possible by taking readers through the concepts gently using clear English. I've just finished Chapter 2 and I can't wait to get through the rest of it.

Anyway, this book's explanation is super simple:

- It is nice to have a system of numbers, or things like "numbers" where a set of operations (like add, subtract, multiply, divide) on any of these numbers results in numbers that are also in the same system.
- The normal number system we all learned about at school, and use everyday, seems to be complete in this sense. That's the system of "reals" such as 1.0, 3.44, -5.6, 999.22 and so on. We can add two of these numbers and the result is also in this system.
- The problem arises when we look at polynomials whose coefficients are also taken from this same system, the real numbers. We would like the roots of these polynomials to also be found within this same system of reals. The polynomial (x-1)(x+2)(x-3) has roots that are x=1, -2 and 3. But some polynomials like x
^{2}+ 1 don't have roots from the real numbers. - So we have to extend the real number system so that these polynomials have roots that are always within the extended number system. That extended number system is the complex numbers (a+bi).
- The nice thing about this system that we appear to have invented is that polynomials with complex coefficients, always have roots also in the set of complex numbers. This now means we have a more "complete" system. The real numbers weren't complete (algebraically closed) in this sense - polynomials built from real coefficients sometimes didn't have roots in the real numbers.

I'm really excited by this super clear explanation.

**Why don't more authors do this?**Anyway, here's a summary:

## Friday, 6 June 2014

### (Fixing) Kindle Ebooks Wth Google Docs

There are many tutorials out there on making ebooks. The truth is that right now there are no good tools, the technical file formats for ebook are not great, in flux, and on top of this not well or consistently supported. Even lots of money won't buy you good tools - the Adobe Indesign plugin doesn't magically transform your works of art into perfect ebooks.

I used Google Docs (now called Drive) to type my material, and insert diagrams and images. I did use titles, headings, subtitles to give the documenty some structure. The great thing about Google Docs is that it's free, accessible from almost anywhere with no need to install software, ... very convenient and efficient. Your content in Google Docs can be exported in a range of useful formats, including Microsoft Office, ODF, PDF, and HTML.

The steps for making a Kindle ebook is simple:

You might want to use preview tools to check your epub file works and get an indication of what it might look like on real physical readers.

Just this week I fixed an annoying problem which seemed to only affect Android Kindle readers which seemed to force very wide margins, meaning the content was squished into a very thin column. The usual internet search led to messing about CSS styles to override the margin, border and padding settings. It didn't work. The answer was actually to go back to the Google Docs document and use the page setup menu to zero the page margins...voila! That worked!

Hope this helps someone else.

I used Google Docs (now called Drive) to type my material, and insert diagrams and images. I did use titles, headings, subtitles to give the documenty some structure. The great thing about Google Docs is that it's free, accessible from almost anywhere with no need to install software, ... very convenient and efficient. Your content in Google Docs can be exported in a range of useful formats, including Microsoft Office, ODF, PDF, and HTML.

The steps for making a Kindle ebook is simple:

- Export your doument as HTML. This will give you a zippled folder with the content and any images that you used.
- Import the main HTML document into Sigil. Use Sigil to add a cover, book metadata such as title and author, and contents. You might like to split up a long document into separate HTML sections. This is also your opportnity to clean up the content, remove additional spaces, blank lines, centre things that weren't.
- Export an epub from Sigil. This is an open file format for ebooks, and quite well supported by many readers but not perfectly, as I said above.
- Amazon doesn't like epubs so they convert it to their own format when you upload it to their site.

You might want to use preview tools to check your epub file works and get an indication of what it might look like on real physical readers.

Just this week I fixed an annoying problem which seemed to only affect Android Kindle readers which seemed to force very wide margins, meaning the content was squished into a very thin column. The usual internet search led to messing about CSS styles to override the margin, border and padding settings. It didn't work. The answer was actually to go back to the Google Docs document and use the page setup menu to zero the page margins...voila! That worked!

Hope this helps someone else.

## Sunday, 1 June 2014

### Feedback

Thanks for the feedback on the Make Your Own Mandelbrot ebook - keep it coming.

So far the main requests have been:

Great ideas! I think a second edition is starting to form ... And please do keep your suggestions coming in, they all contribute to an even better second edition.

So far the main requests have been:

- A super simple walkthrough of working with complex numbers with clearer examples. Perhaps as an Appendix. Someone also asked why we avoided talking about dividing complex numbers.

- A discussion of why the 3D section extends the 2D fractals into 3dimesnions but doesn't actually use 3D versions of complex numbers. It seems that several readers have naturally asked the question we discussed in a previous post.

Great ideas! I think a second edition is starting to form ... And please do keep your suggestions coming in, they all contribute to an even better second edition.

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