Posts tagged with tristan needham

The essential prerequisite for finding the answer to a question is the desire to find it.

Tristan Needham

author of Visual Complex Analysis (the best book so far about complex numbers)




As every sci-fi geek knows, matter may travel faster than the speed of light as long as its mass is imaginary (a multiple of √−1). A so-called tachyon would not overturn special relativity—and it would provide a handy way of resolving any conflicts in a given Star Trek plot.

  • 14th Law of How to Write Star Trek: Whenever you’ve written yourself into a hole, instead of re-writing the show so that it’s better, simply make characters issue the word “tachyon” several times toward the end. Everything is magically resolved, returning all aspects of life to the way the show started with no long-term consequences for the characters—which by the way is a great lesson to teach to young adults—and then Spock or Data has an “a-ha!” moment wherein he throws around jargon to further justify the deus ex machina.

The only problem with tachyons, as any sci-fi geek can attest, is that “imaginary” mass is pure fiction! How could anything weigh an imaginary amount?

 

Well, I’m not sure that tachyons do exist—although if someone wants to post some arXiv links to relevant papers that would be awesome—but, I will say that “imaginary mass” isn’t that ridiculous of a concept.

As Tristan Needham said in the best book about complex numbers ever, the “imaginary” descriptor only reflects the historical prejudice against √−1.

Do imaginary numbers exist? No. But neither do counting numbers. Numbers are linguistic entities that humans communicate with. Sort of like how trees, flowers, bushes, shrubs, brambles, and vines all exist in nature, but those classifications, concepts, words, groupings are human-language mental constructs. “Five” doesn’t “exist” per se, but mathematical models built with the-thing-that-satisfies-the-properties making five five, do wonderfully at prediction of physics experiments.

Anyway, imaginary numbers exist just as much as other numbers. Just like rational numbers, they’re generated by an operation that comes up as a matter of course in algebra. And algebra seems to have something to do with nature. God knows why. (ohh! which way did I mean it?!)

So I’m not saying imaginary mass exists, but here are some good ways to think about imaginary numbers.

  • imaginary numbers are twisted numbers
  • imaginary numbers are phase-shifted like a sine wave versus a cosine wave
  • an imaginary current heats up a wire but does no useful work

If the mass of a particle is an imaginary number, then … that might help you make sense of tachyons.

 

Nerdy side note: E=MC² is not the real equation to describe the conversion of energy into matter or vice-versa.

  • E=MC² tells you how to convert stationary matter into energy.
  • The real equation is E² = [mc²]² + [pc]².
  • (p is momentum.)
  • (Notice that the real equation is of the form A²+B²=C². i.e., Energy is the hypotenuse (C) to the triangle sides B=mc² and A=p•c)

You can casually start/interrupt conversations with this knowledge the next time you attend a kegger / black-tie affair. Doing so will win handsome glances from potential sex partners. Also, there is a 0% chance that anyone will think you’re an insufferable know-it-all.




The essential prerequisite for finding the answer to a question is the desire to find it.

Tristan Needham

author of Visual Complex Analysis (the best book so far about complex numbers)




Here’s a mysterious fact.

e^ (i × À) = −1

What’s up with that?

The answer is forthcoming, but first I’ll remind you what all the letters mean.

e e e e e e

e = 1 + 1/1 + 1/12 + 1/123 + 1/1234 + 1/12345 + 1/123456 + 1/1234567 + 1/12345678 + …

I’m writing 1/234 to mean “One over 2×3×4.” With the (2×3×4) in parentheses. Get me?

e = £ 1/n!

Why is e useful, though?  Because the derivative of e^x is itself.  D[e^x] = e^x.  In other words, the function e^x looks like 1 to the operator D (derivative operation).  Because of that fact, answers to the differential equations that describe the interesting and complicated world we live in, are stated easiest in terms of e^x.  (More specifically, functions like e^x, e^2x, e^ix form a linear basis for the solution space of ODE’s.)

π π π π π

2π is the length around a circle.  (Assuming the circle has a radius of 1 in the chosen units.)

i i i i i i i i i

i = √-1.  What does that mean—half-negative?  Answering this question provoked a flourishing of mathematics in the 19th and 20th centuries.  √-1 is used in electrical engineering, quantum mechanics, solving equations, understanding fluid flows, non-Euclidean geometry, and according to some (Roger Penrose & friends), more and more areas of physics will ultimately be explained with complex numbers.

For now, think about this.  When the power company sends electricity to you, it is traveling in the direction “+1”.  (With AC I guess it switches between +1 and −1.)  If it traveled in the direction “i”, the wire would heat up but no useful energy would be transmitted.

the answer

In fact, e^{ i × any number of ° degrees} behaves like a unit circle — a circle with radius one — in the complex plane.

picture of the complex plane: imaginary goes north, real goes east

So e^{ i × 90° } = i, e^{ i × 270° } = −i, and e^{ i × 360° } = 1.  In terms of distance around the circle , that’s a quarter-turn ↶, a three-quarters turn ⟲, and a full rotation ⥀.  Or, in π terms:

e^{i À/2} = i   etc

Similarly, going , , 8π, or 17362 times around the circle leaves you just where you started — whether that was from a quarter-turn anticlockwise ⤿ or at the three o’clock starting point. −1 is 180 degrees ↶ or π arc-length away from the starting point of +1 so voilà,

e^ (i × À) = −1.

The Punchline

That’s all fine, for parlor tricks.  But there’s more, much more.  Seeing the gamut of numbers as concentric circles in the complex plane allows you to solve every equation, ever*.  I’ll build this up for a paragraph or two.

First, consider that there are two solutions to x²=1, which are ±1.  Meaning, x=1 satisfies the condition and x=−1 satisfies the condition, and those are the only two values of x that satisfy the condition.

Second, there are four solutions to x⁴=1:  1, −1, i, and −i.  Also known as e^{ i × 90°, 180°, 270°, and 360° ⟲}.  And what would 7, −7, 7i, and −7i solve?  x⁴ = 2401.  (Two four oh one is 7⁴.)

Following the same pattern, there are five solutions to x⁵=1 (intervals of 72°), thirteen solutions to x¹³=1 (intervals of 2π/13), and thirty-six solutions to x³⁶=1 (intervals of 10°).  And again, if you doubled the circle’s radius, the solutions would be multiplied by 2⁵, 2¹³, or 2³⁶.

ship's steering wheel

So the answer to any problem of powers is shaped like a ship’s steering wheel:  evenly spaced points on a circle, possibly a slightly wider circle if the question is x^p = 999.

The answer to polynomial problems is then built up from these sorts of objects.  (But just imagine laying two different-radius wheels on a rectangular grid, and being asked to compute the (x,y) coordinates of sums of pairs of handles.  ouch! gimme a computer)  Anyway, it’s nice to know that the answer exists, and it’s good to form your imagination around this kind of shape, in case you ever want to explore something practical that involves this lay of the land.

* That’s an exaggeration, but at least you can solve every equation you’ve ever seen.




I was delighted when I came across [Visual Complex Analysis]. As soon as I thumbed through it, I realized that this was the book I was looking for ten years ago.
Ed Catmull, Founder and Chief Technology Officer of PIXAR ANIMATION STUDIOS
I was delighted when I came across [Visual Complex Analysis]. As soon as I thumbed through it, I realized that this was the book I was looking for ten years ago.

Ed Catmull, Founder and Chief Technology Officer of PIXAR ANIMATION STUDIOS