Complex numbers have never been so intuitive to me | Hacker News Reader

Complex numbers have never been so intuitive to me

betterexplained.com

154 pointsAndrewMoffat46 comments

Wow, 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.

So 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 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.

I don't know how Sedenions fit into the picture but "Hurwitz's theorem":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.

See https://rjlipton.wordpress.com/2011/04/13/even-great-mathema...

There 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_d... for more information.

Why 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?

> Why are there no 16, 32, 64, and so on, dimensions of numbers? Would multiplication not be invertible with them?

Yes, this is exactly why the 16 dimensional sedenions are not in the same category as the others. For it to be a division algebra, you must be able to reverse any multiplication that isn't by zero. This is just the normal concept of division in the real numbers. It also allows us to define division in the complex, quaternion and octonion numbers. However, it is possible to multiply two non-zero sedenions together and get zero. This breaks this property completely.

> 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?

Absolutely! These four numbers systems are a tiny portion of spaces studied by mathematicians. There are a lot of interesting things that can be said just by generalising a few properties of numbers (e.g. without requiring that we can divide, or without requiring that we have both addition and multiplication). There are groups, fields, rings, algebras, topological spaces, and all sorts of interesting things that aren't 'numbers' as such.

This is pretty fascinating, do you know of any good sites or books which will get me up to scratch on different number systems? I haven't studied math in a long time, so do you think it would be worth going over the basics, like calculus and geometry, again to build a foundation before venturing so far out?

(i'm replying so i get notified of an answer; good question!)

As others have noted, we don't have 3D numbers because there is a theorem saying that they don't exist (at least with norm). Interestingly, Hamilton's discovery of the quaternions was the result of a long and fruitless search for a three-dimensional analog to the complex numbers.

I wonder why mathematicians use this funny i-notation? You could write complex numbers as vectors, e.g. 2+3i <=> (2,3). Of course, it is more terse to write i instead of (0,1) and this seems to matter a lot to mathematicians.

i is commonly used as the unit vector in the x-dimension (a basically meaningless term without more context). 3-dimensional vectors are often written as:

5i + 2j + 3k.

Since you can assume that 2 is getting multiplied by the unit real vector (equal to 1), you can think of 2+3i as

2r + 3i

but why write the 'r' if we know 2*1 = 2?

I think it'd be awesome to have some sort of "decimal" notation where 2 + 3i was encapsulated as a single entity, something like 2_3. We don't write 2 + .3 when we mean 2.3, and this distinction makes it seem like a complex number is "less put together" than a real one.

in a sense it is - it involves magnitudes along two orthogonal directions. there is no qualitative difference between 2 and 0.3 in this particular context.

Thank you for posting this. I remember trying to understand quaternions a few years ago and didn't get it. Your comment combined with this article made it click.

I wonder if there's anything of numbers in infinite dimensions and, more interestingly, numbers in different degrees of infinite dimensions (yup, infinity comes in different degrees too - http://sites.google.com/site/degreesofinfinity/). It seems that math is only limited by the imagination.

You can define any dimensional number that you like, including infinite. What you can't do is get certain properties out of them. But infinite dimensional numbers are actually frequently used, under the term "n-dimensional vector".

(Infinite is often confused for "really big", but a better way of understanding it is often to use the term "unbounded". When we deal with n-dimensional vectors, we're not saying that we are always literally dealing with vectors with a millionybillionytertrillion dimensions, such that we can't even represent one in a real computer, we're saying that there isn't a firm upper limit on the number of dimensions we may encounter. It is often more the "unbounded" aspect of infinity that we are concerned with, rather than the "larger than anything else" aspect.)

From what I gather from jpallen's response to my other message the properties you're talking about are things like being able to add, multiply, subtract and divide, right? What's to stop one from coming up with novel properties which are unique to these systems? Is our commonly used system basically an arbitrary set of conditions? Do you think that some systems correlate better to how nature does things than others, and if so, which ones would they be?

Ah, so it's unbounded like how Haskell's lazy evaluation treats infinity.

> What's to stop one from coming up with novel properties which are unique to these systems?

Interestingness. Usefulness, perhaps, if you have a specific definition of 'useful' in mind.

> Is our commonly used system basically an arbitrary set of conditions?

It's an arbitrary set of conditions with interesting properties, and it's not always obvious which arbitrary conditions will have interesting properties.

> Do you think that some systems correlate better to how nature does things than others, and if so, which ones would they be?

Well, that is an interesting question, which means anyone who claims to know an absolute answer to it is a moron. We do know, for example, that vectors are a very useful tool to model a lot of what happens in physics, and that complex numbers are a compact way to talk about rotation, especially complex exponentiation (taking a real number, such as e, to complex powers).

> Ah, so it's unbounded like how Haskell's lazy evaluation treats infinity.

Yes. For example, a definition (the Peano axioms) of the set of the natural numbers (either the positive or non-negative integers; the set may or may not include zero, as conventions vary) states that the set contains 0 (or 1) and that for every number x it contains, it also contains x+1.

I wish they taught this back in school, math would have been so much more interesting. Anyway, I lack the knowledge to ask any more meaningful questions so I'm gonna go do some digging. Thanks for taking the time to respond.

Thanks for mentioning this. I gauge the value of an explanation by the follow-up questions... if you really understand imaginary numbers, then suddenly you start asking about other dimensions, etc. and get to awesome discussions like this one :). And even better, people are enthusiastic to explore these new avenues.

"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.

Yes -- 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.".

This 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.

Luckily, 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.

^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.

It'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.

Back 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.

Thanks! I took several college courses using them and didn't "get" the concept until years after I graduated.

In 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.

i 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.

> 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-...

Thanks 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-ex... * radians as the perspective of the mover: 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).

No problem, I'm happy to share something that genuinely helped me. I love what you're doing, please keep doing it :)

Wow. I feel like if I had been taught this before trigonometry, high school would have been a breeze.

It 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.

If 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

But the main website is

http://www.dimensions-math.org/Dim_download_E.htm

And I think it's chapter 5.

Given 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.

You are correct. From what I could gather from reading that write-up, "size" is used to mean the magnitude of a number.

     It 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.

This 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.

Why 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?).

Two 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.

It'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.

Why 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...

Very often, teachers don't understand the concepts very well either.

Exactly. 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?

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