6 ms·
Complex numbers have never been so intuitive to me
- baddox 15y agoGiven a complex number a + bi, the square root of a^2 + b^2 is called the norm. The norm shouldn't be thought of as a measure of the "size" of a complex number, since complex numbers are not well-ordered. It makes little sense to say that 2 + 3i is equal to 3 + 2i.
- hrasm 15y agoYou are correct. From what I could gather from reading that write-up, "size" is used to mean the magnitude of a number.
- bad_user 15y agoIt makes little sense to say that 2 + 3i is equal to 3 + 2i As the article correctly points out the norm is "the distance from 0". 2 + 3i is not equal to 3 + 2i, but both numbers are on the same circle. It is generally considered in mathematics that a norm is used to describe length, size, or extent.
- baddox 15y agoThis may just be a difference in terminology, but I'm not familiar with mathematicians calling the norm of a complex number its "size." To me, if you call the norm the size, then you could say that one complex number is larger than another, which I believe is nonsensical. Of course, you could just define a relation < on the set of complex numbers by "x < y if and only if norm(x) < norm(y)". This relation is obviously trichotomous, and it could be considered an ordering of the complex numbers. In my experience, however, this assumption is generally not held or used by mathematicians.
- jules 15y agoWhy wouldn't you be able to call it size? Equal size doesn't imply equality elsewhere in mathematics or elsewhere in the world (Jimmy and Paul are the same height, so they must be the same person?).
- baddox 15y agoTwo people of equal height aren't the same person, so it would be odd for an article to claim that the way to convert a person to a real number would be to simply measure their height, and even odder to call that the "size" of a person.
- natemartin 15y agoThis is a fantastic explanation. I've always had trouble "getting" imaginary numbers.... even though I've had to use the fairly often as an Electrical Engineer. This is the first time they've made intuitive sense to me.
- Daniel_Newby 15y agoLuckily, my AC circuits class was basically the applied version of this article, with lots of conversions between the complex and phase-amplitude descriptions. What blew my mind at the time was the exponential notation for the unit phasor: e^(i * x). It turns out that e^(i * x) = cos(x) + i * sin(x) because that's just the way the math works out, and it's trivial to work it out yourself by looking at the Taylor series expansions of the three terms.
- fastfinner 15y ago^Same here, I have taken countless tests using imaginary numbers, but it never went beyond a really pointless exercise in my mind. Now I get it.
- checker 15y agoIt's a shame I paid for all that college and knowing this would have greatly expanded my understanding of a lot of what I learned in my math classes. EDIT: I guess I learned it in high school though, and the math teacher probably didn't know this either. Point being, everyone should come across this at some point. It would be beneficial to many math students.
- kruhft 15y agoBack in school, about halfway through my course in DSP, my teacher realized that none of us had any idea what he was talking about...so he went back to basics and explained Imaginary numbers much like this article for a whole day. Things started to make a lot more sense after that, but he was shocked that nobody had taught us that in any of our classes before.
- kalid 15y agoThanks! I took several college courses using them and didn't "get" the concept until years after I graduated.
- mvzink 15y agoWow. I feel like if I had been taught this before trigonometry, high school would have been a breeze.
- CallMeV 15y agoIt was for me. But then, I had a good maths tutor. Ironically, I cannot recall his name. But the lessons he taught stayed with me, so at least he's remembered for something.
- vain 15y agoi wish someone had explained this to me about 15 years ago. i had sought, and settled on the unsatisfying explanation that it is a conventional notation with useful ways to do coordinate geometry. non essential, but a way of doing it. though i have looked at euler's identity with awe, it's mostly been a mystical sort of awe. I hate to admit it but i had started using a line of argument to certain theist friends, that if god helps you, as a concept, no need to be bothered, think no more of it than a concept such as the the mathematical concept of i, its a number that does not exist but has real consequences. now I feel stupid for doubting the existence of i. I keep trying to relearn my fundamentals, and this article does that beautifully. i would otherwise have died a disbeliever.
- AndrewMoffat 15y ago> though i have looked at euler's identity with awe, it's mostly been a mystical sort of awe. betterexplained also has a fantastic visual explanation of euler's identity: http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ http://betterexplained.com/articles/intuitive-understanding-...
- kalid 15y agoThanks so much for helping share this! :) My goal is to get all these concepts out of awe (which I had too) into a real sense of "Ah, I get it!". It doesn't help anyone to memorize incantations. Two other insights I love: * e as continuous growth: http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/ http://betterexplained.com/articles/an-intuitive-guide-to-ex... * radians as the perspective of the mover: http://betterexplained.com/articles/intuitive-guide-to-angles-degrees-and-radians/ http://betterexplained.com/articles/intuitive-guide-to-angle... I realized that I didn't really get what e and radians were about -- I memorized them, but I wasn't comfortable. Once you have the right analogies, Euler's formula starts making sense (without resorting to "Oh, take the Taylor series expansion of each and see how they match up", which is essentially an incantation).
- AndrewMoffat 15y ago
- cormullion 15y agoIf you'd like some sexy French graphics to accompany that, there's a good Dimensions episode devoted to imaginary numbers. I found it on YouTube at http://www.youtube.com/watch?v=egIPnwcJuZ8 http://www.youtube.com/watch?v=egIPnwcJuZ8 But the main website is http://www.dimensions-math.org/Dim_download_E.htm http://www.dimensions-math.org/Dim_download_E.htm And I think it's chapter 5.
- Deestan 15y agoSo real numbers are 1-dimensional, and complex numbers are 2-dimensional. Going along the same lines, we also have Quaternions, 4-dimensional numbers: http://en.wikipedia.org/wiki/Quaternion http://en.wikipedia.org/wiki/Quaternion Further again we have 8-dimensional Octonions, and 16-dimensional Sedenions. I'm curious as to why we don't have any useful numbers for the non-power-of-2 dimensions. E.g. 3-dimensional numbers.
- mreid 15y agoI don't know how Sedenions fit into the picture but "Hurwitz's theorem":http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(normed_division_algebras) http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(normed_div... may be one explanation as to why there are essentially only 1,2,4, and 8 dimensional numbers.
- gmartres 15y agoSee https://rjlipton.wordpress.com/2011/04/13/even-great-mathematicians-guess-wrong/ https://rjlipton.wordpress.com/2011/04/13/even-great-mathema...
- jpallen 15y agoThere are no three dimensional numbers because it's impossible to construct such a system that behaves like 'numbers'. The real, complex, quaternion and octonion numbers are the only 'normed division algebras'. This basically means they are the only spaces with a notion of length in which mutliplication is invertible (division). These properties are needed for anything that we want to call 'numbers'. The fact that there are only 4 such spaces is quite a profound result and leads to a lot of other important results. See http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_%28normed_division_algebras%29 http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_%28normed_d... for more information.
- Produce 15y agoWhy are there no 16, 32, 64, and so on, dimensions of numbers? Would multiplication not be invertible with them? Is there a possibility of things that are distinct from our concept of numbers but are none-the-less useful being discovered in the future?
- iwwr 15y agoIn the same vein, you can have split-complex numbers which represent the 2d hyperbolic plane. Also, there are the 4-dimensional generalizations like quaternions, split-quaternions or other related algebra systems. What's amazing about these systems is that there is usually an Euler relation that holds. Example: e^(i*t) = cosh(t) + sinh(t) for the split-complex.
- PetrolMan 15y agoWhy do I feel like my teachers intentionally made math harder than it needed to be? Some concepts just lend themselves to simple visual representations and just never seem to be taught that way...
- lotharbot 15y agoVery often, teachers don't understand the concepts very well either.
- kalid 15y agoExactly. And ironically, we expect that to be the norm. "Oh, nobody gets imaginary numbers, let's just memorize it and move on." To me, that's a huge canary in the coal mine! Why aren't we stopping here and making sure we get it? It's like reading a sentence, not understanding the key vocabulary word, and moving on. Yes, you "read" it but did you get anything from it?
- aristus 15y ago"It’s a testament to our mental potential that today’s children are expected to understand ideas that once confounded ancient mathematicians." Ex-fucking-actly. Math is hard. Compsci is hard. But if it remains hard, then we adults have failed to do our job.
- kalid 15y agoYes -- things are hard, now, because we don't have the right models. Multiplication is hard when you're working with Roman numerals. Is it a problem with multiplication, or our thinking? What's funny is that we think it stops there. "Oh, we made made multiplication easy, and negatives, and decimals, but Calculus... well that needs to remain difficult forever and ever.".
- dkokelley 15y agoIt's unfortunate that our schools in general haven't been able to convey with as much clarity and passion this concept. I suspect that any passion or enthusiasm for a subject quickly gets destroyed when it's turned into a job (particularly a job in a system run by bureaucrats). This is why I am excited about programs like Khan Academy. One of the things he's been able to do that has eluded most public schooling is explain a concept simply, and with enthusiasm.
- johnconroy 15y agoreally, really nicely done
- johnconroy 15y agoreally, really nicely done
- T-hawk 15y agoWow, I got this too. Here's my followup question and answer. We know i is the square root of -1, but what is the square root of -i ? I've always thought that you'd need another dimension to describe that, and another dimension for the square root of that unit, and so on. But no. (Let's tackle sqrt(i) as a simpler case first.) We can answer sqrt(i) in terms of rotation as described in the article. i is a rotation from the unit vector by 90°, so applying that twice turns 1 into -1. What operation applied twice would result in i? This just clicked: a 45° rotation. Thus the unit vector at a 45° angle is the square root of i: 0.5√2 + 0.5√2 * i. The mathematical approach bears that out. Follow the rules of complex number arithmetic to square 0.5√2 + 0.5√2 * i (multiply it by itself) and you do indeed get i. And we can solve sqrt(-i) the same way. -i is a 270° rotation from the unit vector 1. So the square root of -i is a 135° rotation, or -0.5√2 + 0.5√2 * i. Finally, a 270° rotation is equivalent to a -90° rotation. So -45° should also be a square root of -i, and indeed it is. Multiplying 0.5√2 + -0.5√2 * i by itself also gives you -i. We've arrived back at the axiom that all numbers have two square roots of opposite signs. 135° and -45° are the same vector pointing in exactly opposite directions. Last question: What's the cube root of i? Easy: a 30° rotation. The 30° unit vector is 0.5√3 + 0.5i, and cubing that does indeed get you i.