# Learning Clojure

## Thursday, October 24, 2013

### Feynman's Arrows : What are the Complex Numbers?

;; Feynman's Arrows : What are the Complex Numbers?

;; I've lost count of the number of times I've met physicists and even
;; professional mathematicians who think there's something a bit
;; spooky and unreal about the complex numbers.

;; There isn't.

;; Imagine that you're sitting in front of a photo-manipulation
;; program, and you're able to enlarge a photograph by a factor of
;; 2, and rotate it 30 degrees clockwise.

;; I don't think anyone is going to find that spooky. If you do, go
;; find a photo manipulation program and do it a couple of times.

;; If you do it twice, then you'll find that that gives you the same
;; effect as enlarging the photo by a factor of four and rotating it
;; 60 degrees clockwise.

;; In fact, how would we get our original photo back? My first guess
;; would be that I should shrink it by a factor of four and rotate it
;; 60 degrees anticlockwise. And that's the right answer. Go and try
;; it if it's not obvious.

;; And if you have your head round that, then you understand the
;; complex numbers.

;; If the mathematicians of the 16th century had been thinking about
;; how to use photoshop, instead of worrying about how to solve cubic
;; equations, then they'd have come up with the complex numbers in
;; about fifteen minutes flat, and they'd have thought they were the
;; most obvious thing in the world, and it would never have occurred
;; to them to talk about 'imaginary numbers', and an awful lot of
;; terror and confusion would have been avoided over the years.

;; And if the complex numbers had been found that way, then I think
;; we'd teach them to little children at about the same time we teach
;; them about fractions, and well before we teach them about the
;; really weird stuff like the square roots of two.

;; And the little children would have no problem at all with them,
;; and they would think that they were fun, and easy.

;; Look at what happens if we keep multiplying arrow1 by arrow3 (ink red) (draw-arrow (multiply-arrows arrow3 arrow1)) (draw-arrow (multiply-arrows arrow3 (multiply-arrows arrow3 arrow1))) (draw-arrow (multiply-arrows arrow3 (multiply-arrows arrow3 (multiply-arrows arrow3 arrow1)))) ;; and so on (def arrows (iterate (fn[x] (multiply-arrows arrow3 x)) arrow1)) (draw-arrow (nth arrows 3)) (draw-arrow (nth arrows 4)) (draw-arrow (nth arrows 5)) (doseq [ i (take 100 arrows)] (draw-arrow i)) ;; By repeatedly applying very small turns and very small zooms, we've ;; made a beautiful expanding spiral. ;; And I think, I genuinely think, that you might be able to get small ;; children to play with these arrows and to make this pretty spiral, just ;; after you've taught them about fractions.

#### 1 comment:

1. This series is a great contribution to mankind, thanks for making it.

I found that I had to use:
(use 'simple-plotter.core)
When I just had "(use 'simple-plotter)", I got a FileNotFound.

I also found that, like most other sites with Blogger.com stuff, I couldn't post a comment at all with FireFox