;; Feynman's Arrows II : OK, so what are the complex numbers? ;; requires [simple-plotter "0.1.2"] ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;; Here's some code from the previous post ;; use pomegranate to get the library, if it's not already on your classpath (require 'cemerick.pomegranate) (cemerick.pomegranate/add-dependencies :coordinates '[[simple-plotter "0.1.2"]] :repositories {"clojars" "http://clojars.org/repo"}) (use 'simple-plotter) ;; Make blackboards to draw arrows on (defn make-blackboard [title size] (create-window title 400 400 white black (- size) size (- size) size) (axes) (ink yellow)) ;; Draw an arrow shape from (a,b) to (c,d) (defn draw-offset-arrow [[a b][c d]] (let [headx (+ a c) heady (+ b d)] (line a b headx heady) (line headx heady (+ headx (* -0.086 c) (* -0.05 d)) (+ heady (* 0.05 c) (* -0.086 d))) (line headx heady (+ headx (* -0.086 c) (* 0.05 d)) (+ heady (* -0.05 c) (* -0.086 d))))) ;; Draw one of our arrows, which always have their tails at 0 (defn draw-arrow [[a b]] (draw-offset-arrow [0 0] [a b])) ;; Here's everything we know about the arrows so far: (defn add-arrows[[a b][c d]] [(+ a c) (+ b d)]) (defn multiply-arrows[[a b][c d]] [(- (* a c) (* d b)) (+ (* a d) (* c b))]) ;; That's easier to read in the standard prefix notation ;; (a,b)+(c,d) -> (a+b, c+d) ;; (a,b)*(c,d) -> (ac-db, ad+cb) ;; The addition rule says: treat arrows as if they represented vector displacements, and add them nose to tail ;; The multiplication rule says: treat arrows as if they were zooms and rotations, and use one to zoom and rotate the other. ;; We also had a few favourite arrows that we'd played with: (def arrow1 [3,4]) (def arrow2 [4,-3]) (def arrow3 [1,1/10]) ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; (do (make-blackboard "Favourite Arrows" 6) (doseq [i [arrow1 arrow2 arrow3]] (draw-arrow i)))

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;; The first thing I want you to notice is that there's a subset of ;; the arrows that works exactly like the numbers that we all got ;; comfortable with in primary school. They're the ones where the ;; arrows point due east or west. (add-arrows [10 0] [5 0]) ;-> [15 0] ;; (10,0) + (5,0) -> (15,0) is like 10 + 5 -> 15 (do (make-blackboard "Adding positive real numbers" 16) (draw-arrow [5 0]) (draw-arrow [10 0]) (ink red) (draw-arrow (add-arrows [10 0] [5 0]))) (multiply-arrows [4 0] [3 0]) ;-> [12 0] ;; (4,0) * (3,0) -> (12,0) is like 4 * 3 -> 12 (do (make-blackboard "Multiplying positive real numbers" 16) (draw-arrow [4 0]) (draw-arrow [3 0]) (ink red) (draw-arrow (multiply-arrows [4 0] [3 0]))) (multiply-arrows [-1 0] [-2 0]) ;-> [2 0] ;; (-1,0) * (-2,0) -> (2,0) is like -1 * -2 -> 2 (do (make-blackboard "Multiplying negative real numbers" 3) (draw-arrow [-1 0]) (draw-arrow [-2 0]) (ink red) (draw-arrow (multiply-arrows [-1 0] [-2 0]))) ;; You probably learned 'minus times minus is plus' as an arbitrary ;; rule, but it's obvious if you think of -1 as meaning 'leave it the ;; same size but turn it 180 degrees.' ;; We say that the 'real numbers' are 'embedded' in the arrows. ;; What we mean is that the horizontal arrows behave just like the ;; real numbers, and so in any place where we were going to use real ;; numbers we can just use horizontal arrows instead and everything ;; will work out exactly the same. ;; And so since it doesn't matter whether we think about (3,0) or 3, ;; we'll usually just forget about the difference, and sometimes write (3,0), ;; and sometimes write 3, depending on convenience. ;; So now I can say -3 * 4 -> -12, and that's a statement about arrows! ;; There's another subset of the arrows, that point due north and due south. ;; Under addition, they're just like the real numbers too. ;; (0, 7)+ (0, 3) -> (0, 10) (add-arrows [0 7] [0 3]) ;-> [0 10] (do (make-blackboard "Adding Vertical Arrows" 12) (draw-arrow [0 7]) (draw-arrow [0 3]) (ink red) (draw-arrow (add-arrows [0 7] [0 3]))) ;; But when you multiply them, they end up turning each other into horizontal arrows ;; (literally 'turning') ;; (0, 2) * (0, 3) -> (0x0-2x2, 0x3+2x0) = (-6, 0) (multiply-arrows [0 2] [0 3]) ;-> [-6 0] (do (make-blackboard "Multiplying Vertical Arrows" 12) (draw-arrow [0 2]) (draw-arrow [0 3]) (ink red) (draw-arrow (multiply-arrows [0 2] [0 3]))) ;; You can see why pretty easily. ;; (0, 10) represents 'turn 90 degrees clockwise and magnify by 10', ;; and (0, 4) means 'turn 90 degrees clockwise and magnify by 4' ;; And the product (-40, 0) means turn 180 degrees and multiply by 40. ;; In this view, (-1, 0), or just -1 means 'turn 180 degrees (no zooming!)' ;; And the (0, 1) * (0, 1) -> (-1, 0) is just the fact 'if you turn 90 ;; degrees anticlockwise, and then you turn another 90 ;; degrees anticlockwise, then that's the same as if you'd ;; turned 180 degrees'. ;; So as long as we're talking about arrows, the thing we've called ;; -1, or the pair (-1,0), or the arrow length 1 that points east, or ;; the idea of turning through 180 degrees, does have a 'square root'. ;; There is a thing, the arrow length 1 that points straight north, or ;; the pair (0,1), or the idea of turning 90 degrees, that if you ;; multiply it by itself you get -1. ;; That's important, but it's also trivial. ;; Two quarter-turns clockwise make a 180 turn. (multiply-arrows [0,1] [0,1]) ;-> [-1 0] ;; Two quarter-turns make a half-turn (do (make-blackboard "Something Whose Square is (-1,0)" 2) (draw-arrow [0,1]) (draw-arrow [0,1]) (ink red) (draw-arrow (multiply-arrows [0,1] [0,1]))))

;; We call that upwards pointing, length 1 arrow i, for historical reasons. (do (make-blackboard "The Mysterious and Magical i" 2) (draw-arrow [0,1])))

;; And those vertical numbers, again, for historical reasons, are ;; called Imaginary Numbers. ;; Why on earth would anyone call sideways-pointing arrows 'real' and ;; upwards-pointing arrows 'imaginary'? ;; Well, it's hard to say exactly. The mathematicians of the 17th ;; century weren't thinking about rotations and scalings. They were ;; thinking about roots of equations. ;; Despite what's usually said, it never bothered them in the ;; slightest that x^2+1 didn't have a solution. It was completely ;; obvious that it couldn't have, and that was fine. ;; What did worry them is that they had a formula for the cubic ;; equation, and when they used it to solve cubic equations that ;; really obviously did have three roots, like x^3=x, which is true ;; when x is -1, and when x is 0, and when x is 1, their formula kept ;; insisting that they do weird things, like taking the cube root of ;; the square root of -1. ;; And they had no idea what that meant, but some brave ones just ran ;; with the idea and found that if you followed the absurdity through ;; far enough then at the end the formula would give you the right ;; answers. ;; But they were, quite reasonably, very suspicious of the whole ;; procedure, and they called these weird things 'fictitious', or ;; 'imaginary' numbers. ;; I'm not sure why it was so important to solve cubic equations in ;; the 17th century. But they used to do it a lot, apparently. And ;; often the formula was the only way to get the answers. ;; And of course because they didn't really know what they were doing, ;; they got very confused and kept making mistakes, but nevertheless, ;; they did keep getting the right answers to their problems. ;; Eventually the breakthrough came in 1799, when a Danish ;; cartographer, Caspar Wessel, published 'On the Analytical ;; Representation of Direction', where he thought about using numbers ;; to represent directions and distances. ;; In this ground-breaking paper, Wessel both invented the idea of the ;; vector, and realized that the vectors that he was using to ;; represent directions and rotations and scalings were the same thing ;; as the 'imaginary numbers' that had been scaring and confusing ;; people for nearly two hundred years. ;; Unfortunately, "Om directionens analytiske betegning", was ;; published in Danish by the Royal Danish Academy of Sciences and ;; Letters, and so no-one could understand it, and so it vanished ;; without trace. ;; Luckily, seven years later, a bookstore manager in Paris, ;; Jean-Robert Argand, made the same discovery, and immediately ;; noticed that if you thought about arrows, and thought about ;; polynomial equations as being about arrows, then it was blindingly ;; obvious what was going on. ;; Indeed Argand realised that the question ;; 'What are the roots of x^2 + 1 =0?' ;; is the same question as 'What do you have to do twice to turn round?'. ;; And he went quickly from that to showing that any such question has ;; an answer, which is called the Fundamental Theorem of Algebra. ;; And Argand published his idea in French, which is why today the ;; pictures of arrows that people use to reason about directions and ;; rotations and scalings are called 'Argand Diagrams'. (do (make-blackboard "Wessel^h^h^h^h^h^h Argand Diagram" 10) (dotimes [i 100] (draw-arrow [(- (rand-int 20) 10) (- (rand-int 20) 10)]))) ;; Anyway, even though things that point upwards aren't really any ;; more imaginary than things that point sideways, the name Imaginary ;; Numbers has stuck, and it has the twin virtues of: ;; (a) Making the Complex Numbers look Really Cool and Mysterious, ;; adding to the Aura of Mathematics. ;; (b) Scaring and Confusing Scientists and Engineers, and indeed ;; anyone who needs to think about things that rotate and get bigger ;; or smaller. Mathematicians love doing this. ;; Engineers are practical people, and they use complex numbers all ;; the time when thinking about electricity and whether buildings will ;; fall over, and whether bridges will stay up, and eventually they ;; just accept that it's a really useful incomprehensible mystery and ;; get used to it. They say things like 'a capacitor is an imaginary ;; resistor', without even appearing to notice how strange that is. ;; But I think everyone else is puzzled and a bit frightened. I once ;; met a man with a PhD in Quantum Mechanics, who said that he ;; couldn't really believe that Quantum Mechanics was the true theory ;; of the world, because how could a thing that dealt with imaginary ;; quantities describe reality? ;; There is no need for any of this. Just Arrows. Just Rotations. Not ;; Scary. Children can do this. ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;; Anyway, if 1 is another name for (1,0), and i is another name for ;; (0,1), then notice that 5+3i is (5,0)+(3,0)*(0,1) -> (5,3) ;; So we have yet another way of describing our pairs: (defn print-arrow [[a b]] (str "the pair (" a "," b "), " "also known as the complex number " a "+"b"i, " "also known as the arrow " (cond (> a 0) (str a " north") (= a 0) "" :else (str (- a) "south")) " and " (cond (> b 0) (str b " east") (= b 0) "" :else (str (- b) " west")))) (print-arrow arrow1) ;-> "the pair (3,4), also known as the complex number 3+4i, also known as the arrow 3 north and 4 east" (print-arrow arrow2) ;-> "the pair (4,-3), also known as the complex number 4+-3i, also known as the arrow 4 north and 3 west" (print-arrow arrow3) ;-> "the pair (1,1/10), also known as the complex number 1+1/10i, also known as the arrow 1 north and 1/10 east" (print-arrow [1/11 0.25]) ;-> "the pair (1/11,0.25), also known as the complex number 1/11+0.25i, also known as the arrow 1/11 north and 0.25 east"