A Visual, Intuitive Guide to Imaginary Numbers

[…] A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained (tags: math numbers mathematics learning visualization imaginary kids cool ** toread) […]

I prefer seeing Euler’s equation as

e^(i*pi) + 1 = 0

because then it brings together FIVE (5) really special numbers in one equation.

Good post. I learned about i as a rotational operator, and I’m surprised that it isn’t taught that way (in addition to “follow the math” ways).

Thanks Burton, glad you liked it. I like that representation of Euler’s formula also – I’m gearing up to cover it in an upcoming post (first we need a bit more on e and imaginary numbers :slight_smile: ).

I too am shocked that the “rotation” analogy wasn’t shown when I originally learned about i (in high school). For a long time I thought “i” was just an artificial abstraction used to fill in a gap in our number system (“Well, we need something to be the square root of -1, so let’s just stick i in there.”).

[…] A Thread About Whatever I asked my sister today what i was doing this time last year. She said it was ‘Terry The Wasp Christmas’ and i remembered. Im gonna click my sig to my playlist and browse the net for a bit. Seems like months since i did. Anyhow to start us off how about a link about imaginary numbers? A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained Why do i like them? As it’s all too clever for me. I think im missing out on something special i should know about. Red might mean run son but numbers dont add up to nothing. sorry that was a neil young line for those that ws wondering why i appear to be talking nonsense. I still dont know anything at all about what that last link page is talking about. I quite like the idea though of trying to think or define nothing. Quote: […]

This is a nice presentation – thanks for creating it!

I’ve successfully used this analogy to explain complex numbers to my children, and a few other elementary schoolers. I’m not sure where I first saw the seeds of it, but I recommend Hestenes’ Oersted lecture on geometric algebra for how to extend this idea and where to find lots more like this.

One thing that comes up is that a sensible answer to “What transformation x, when applied twice, turns 1 into -1?” is to subtract one. If you point out that this doesn’t work on 2, the child may reply that the answer is to subtract x, whatever it is, twice. I’ve had to clarify at this point that a transformation can only look at its input, which is a single number. The problem with “subtract again” is that it doesn’t know whether the zero that it gets after one transformation came from 1 or from 2, and it’s not allowed to remember where it started from. The problem is to find an instruction that two different people could do (in series), without sharing any information except for the intermediate number.

It’s also helpful, when explaining this in person and giving the student a chance to come up with the answer, to rotate a pencil out of (perpendicular to) the page or to rotate your arm out of the blackboard, and then back into the plane in the negative direction. This doesn’t give the answer away as much as showing a rotation within the plane, but it’s a nice intermediate clue that “primes the pump” for the explicit explanation, and also adds a somatic modality.

Mind blown here, genius way to describe imaginary numbers visually and to actually use it in real life situation without using fancy methods like sine and cosine. Thanks

@Oliver: Thanks for the insightful comment! I really like that way of looking at it: you need to do something twice, and you can’t tell different types of “zero” apart (1-1 or 2-2). Giving hints like rotating the pencil out of the paper is a nice trick as well. I think kids would be able to pick up on these ideas (better than adults even!) and it’s cool you are introducing it to your children.

@Darius: You’re welcome, I’m glad you found it useful. There are “everyday” uses of imaginary numbers, but nobody seems to talk about them!

Well, just an idea to discuss on: now we could think about a+bi+cj numbers :slight_smile: Or we could think about four-dimension numbers too: a+bi+cj+dk
And so on…

Well Done! I share share your frustration at the fact that most high school mathematics courses do not explain complex numbers adequately.

[…] Sure, some models appear to have no use: “What good are imaginary numbers?”, many students ask. It’s a valid question, with an intuitive answer. […]

@Alessio: Thanks for the suggestion. Yes, I want to learn more about quaternions, imaginary numbers extended to more dimensions :).

@Ivan: Thanks, glad you liked the article.

[…] Imaginary numbers have an intuitive explanation: they “rotate” numbers, just like negatives make a “mirror image” of a number. This insights makes arithmetic with complex numbers easier to understand, and are a great way to double-check your results. Here’s our cheatsheet: […]

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I’m glad to hear you’ll be covering Euler’s equation again in an upcoming post (I haven’t checked back until now). On the same topic, I thought you might want to check out some installation art i did on the topic a couple of years ago.
Cheers,
Burton

Hi Burton, thanks for dropping by – I like the message on that art :). Yeah, I want to cover Euler’s equation, but would like to lay a bit of groundwork (more about e & pi) to help it really sink in.

Also, I like what you said about math being a language that is self-describing to some extent; you can communicate with others and discover new ideas by using it.

[…] Sure, some models appear to have no use: “What good are imaginary numbers?”, many students ask. It’s a valid question, with an intuitive answer. […]

Hi Kalid,

Yes, I agree with the others: nice job on this page!

One comment just for fun: Did you know that engineers (at least electrical engineers) use “j” instead of “i” to denote sqrt(-1)? We need to reserve “i” for electrical current (very important!). BTW, electrical engineering makes very heavy use of complex math. So “our” version of Euler’s equation is e^(j*pi)+1=0. It’s only a difference in the use of a symbol, but I think it’s a rather interesting “cultural” difference to know about.

Peace,
Peyton

Thanks Peyton, glad you liked it! Yes, those “cultural differences” (I like that phrase) are quite interesting. Another way to set off a cultural war is to ask what base “log” refers to (e, 10, or 2).

Hello,

How can we understand e^(pi * i) = -1 ?

Nice post. As a future maths teacher I found it very interesting.

One minor point, though. When you say “complex numbers aren’t”, it’s not technically true. Complex refers to something made from more than one part (in this case the real and imaginary parts)Think of a complex of buildings.What you mean is that they aren’t complicated.

Alessio: It may interest you to know that the 3 dimensional system you suggest, a+bi+cj, has been proven not to work. The 4 dimensional system, a+bi+cj+dk, only works if you remove the insistence on associativity,ie in the quaternions ab=ba doesn’t hold in the general case.