When I first learnt complex numbers in school, I couldn't understand their purpose or why they are called imaginary etc.
I checked where they are used and found examples of current, magnetism etc. But I couldn't understand, electricity is a measurable physical quantity, how can you use imaginary numbers there??
Then I searched internet and found a Stack Exchange answer telling the story and how, instead of, complex (sounds like difficult) or imaginary (sounds like not useful practically), if we had better terms for them, then it would have been easier for beginners to understand them more intuitively.
It's completely true in my case.
After learning usage of vectors, thanks to Godot, for rotation and movement, and then about matrix etc. I now feel like "complex" numbers etc. are the representation of data through multiple numbers where a single value (scalars) is not enough. I may not be right in strict mathematics terms, but at least that's how I was able to understand them.
I've come to understand it as something with both a magnitude and a phase. When you have a unity sine wave, it has two points per period where the magnitude is zero but those points have different phases.
I think it would have been clearer if they weren't called "numbers".
For example, look at matrices [1]: no one asks themselves whether they are "real" or not. We know that we have defined operations on arrays of numbers which are useful to some degree.
If complex numbers were just called "complexes", a lot of confusion would just wither away.
Also, interestingly, quaternions and octonions aren't called "numbers".
[1] Complex numbers may, of course, be seen as special kinds of matrices, but this is beside my point here.
Calculus is a mathematical model of the world, and it is wrong in the sense of the famous quote about "all models [being] wrong, but some are useful", so mathematicians used the imaginary unit almost like a hotfix for their model.
Whether that's an accurate/rigorous statement or not, it helped me as a learner.
You can, of course, express things like oscillation without i, using vectors and matrices. But the ability to use Euler's formula to express complicated sinusoids as compact exponentials with base e simply makes the analysis much easier. In the words of my prof, it's merely a mathematical structure to express things more easily that "regular calculus" isn't good at expressing.
To me, that made the apparent paradoxes of expressing e.g. phase angles and reactive power and stuff using i a lot easier to stomach.
I do have a soft spot for visual complex number explainers, good work. When I was 13 I credit [1] for blowing my mind about complex numbers and setting me onto my current career path
But when you represent a complex number as a+bi, what's the meaning of '+' there? In Argand diagram, the numbers are coordinates. Why do you sum up coordinates?
More fundamentally what's addition and multiplication? Is there a continuity between them, like an operator that represents both addition and multiplication meeting them half-way?
What's the significance of square and square root? Aren't they just some exponent values (2, 1/2) which have no particular significance on the real line? Why should imaginary numbers have a special place if they are tied to square roots (exponent of 0.5)?
People use sums to denote vector components, too. It's common to see ai + bj + ck (pretend there are hats on i, j, and k) to mean the vector [a b c].
So with complex numbers, you can think of it as a vector aj + bi where j and i are basis vectors, but we call j "1", because complex numbers have a notion of multiplication, and in that notion, j acts like 1, in that zj = jz = z for any complex number z.
Nothing bad happens if you write all complex numbers a + bi as vectors or tuples [a b]. It's just less convenient.
In your example 3 and 2 belong to different dimensions. you could write "3x + 2y" to express 3 steps along the x axis and 2 along the y axis. Just writing "3+2" would lead to the expectation that 5 could be calculated as reasonable result, which would not be the case.
Complex numbers would be a much easier concept to understand if they didn't call it "complex" numbers or use the term "imaginary" numbers. Both of those make it much harder to digest because now you think that it's not real. What they really are are 2 dimensional numbers, meaning they have 2 components to each number.
The same goes for the term "real" numbers. Sometimes math, and its descendants like physics and computer science, can be too cutesy with their names that make it hard for regular people to understand or accept.
I've been teaching math for 35 years, which I admit, doesn't say much, but I assure you the difficulties students have with complex numbers by and large have nothing to do with the name. It's the same with everything else: lack of preparation in arithmetic and basic algebra.
Hear, hear. "Complex" makes it sound difficult, when the original idea was that they were a complex--as in compound--of two numbers. Complex analysis turns out to be almost trivial, thus taking an orthogonal journey, compared to the gnarliness of real analysis.
For an interactive approach I built this some time ago. [1]
It allows putting complex numbers on the complex plane and doing addition, multiplication and exponentiation via drag and drop. It provides some intuition for addition==translation and multiplication==scale-rotation.
The complex exponential help to solve many common ODEs that are used to model some common problems in engineering. Usually for such problems, the real and imaginary part of exp((a+bi)t), are both solutions of the problem of type exp(at)cos(bt) and exp(at)sin(at), for this oscillatory functions the sign of a, the real part of z, determine the behavior of the amplitude and the value of the imaginary part of z, b, give us the frequency of that oscillation, this relation between complex numbers, ODE, amplitude and oscillations are at the core of many fruitful applications.
There is proof of this theorem in it that blew my mind some years ago [1]. Someday I want to finish reading the rest of the book.
> Starting with a given convex quadrilateral, construct a square, external to the quadrilateral, on each side. Van Aubel's theorem states that the two line segments between the centers of opposite squares are of equal lengths and are at right angles to one another.
The explanation is a bit confusing, since it basically reduces to "some equations don't have solutions, so we have to invent them", which, if you already accept, you don't need complicated equations, just x^2=-1.
One thing which would have worked better (which is briefly mentioned) is Cardano's casus irreducibilis, but that would have needed more work.
You could also claim that using real numbers the maximum of x*y with the constraint x+y = s is at x=y. Now the question is what happens if we try to go beyond the maximum value by introducing complex numbers? The answer is that we want two numbers whose sum is real and whose product is also real, that suggest using conjugate complex numbers and in such way you can get whatever positive product you want ( because the product of two conjugated complex numbers is just the modulus squared). So using complex numbers the old second grade equations is now able to be used to reflect solutions to problems that before did not have any solution. Now every quadratic equation in R has a pair of (conjugate) solutions, and those provide useful information no matter what the coefficients are.
Really? Of approximately one billion good, visual introductions to complex numbers, we're going to pick an article that appears to be substantially written by an LLM?
I find the historical approach is not the best way to learn about complex numbers.
Deciding square root of -1 is fine and use it in calculations as if it existed is confusing. Why does it work with sqrt(-1), and not with dividing by 0 ?
introducing the imaginary plane as: "hey someone found you could just draw them, and it works even better!" is even worse.
To teach complex numbers, you should start by defining them:
1 is here, i is here, multiplication for 1 is like in R, but multiplicating i by itself goes to -1, and then derive the calculation rules then.
And only when you went all the way to exp(i*theta) = cos(theta) + i sin(theta)
Only then you should explain what they are for. That they are a way to treat real numbers as a projection of these things that behave nicer in calculations. Deriving trig functions from that simple formula is trivial.
> Why does it work with sqrt(-1), and not with dividing by 0
This is pretty straightforward to explain.
With dividing by zero, any number you multiply by zero becomes zero. There's no other way to get zero through multiplication. So, the only sensible thing to divide by zero is zero (since "What times zero equals six?" doesn't have an answer).
But because anything multiplied by zero is zero, even if you divide zero by zero you have an undefined answer. Zero divided by zero could be anything! So that's also not meaningful.
The square root of negative one, however, only has to be defined once to get meaning out of all negative square roots.
I checked where they are used and found examples of current, magnetism etc. But I couldn't understand, electricity is a measurable physical quantity, how can you use imaginary numbers there??
Then I searched internet and found a Stack Exchange answer telling the story and how, instead of, complex (sounds like difficult) or imaginary (sounds like not useful practically), if we had better terms for them, then it would have been easier for beginners to understand them more intuitively.
It's completely true in my case.
After learning usage of vectors, thanks to Godot, for rotation and movement, and then about matrix etc. I now feel like "complex" numbers etc. are the representation of data through multiple numbers where a single value (scalars) is not enough. I may not be right in strict mathematics terms, but at least that's how I was able to understand them.
For example, look at matrices [1]: no one asks themselves whether they are "real" or not. We know that we have defined operations on arrays of numbers which are useful to some degree.
If complex numbers were just called "complexes", a lot of confusion would just wither away.
Also, interestingly, quaternions and octonions aren't called "numbers".
[1] Complex numbers may, of course, be seen as special kinds of matrices, but this is beside my point here.
[1] https://www.youtube.com/watch?v=T647CGsuOVU&list=PLiaHhY2iBX...
Calculus is a mathematical model of the world, and it is wrong in the sense of the famous quote about "all models [being] wrong, but some are useful", so mathematicians used the imaginary unit almost like a hotfix for their model.
Whether that's an accurate/rigorous statement or not, it helped me as a learner.
You can, of course, express things like oscillation without i, using vectors and matrices. But the ability to use Euler's formula to express complicated sinusoids as compact exponentials with base e simply makes the analysis much easier. In the words of my prof, it's merely a mathematical structure to express things more easily that "regular calculus" isn't good at expressing.
To me, that made the apparent paradoxes of expressing e.g. phase angles and reactive power and stuff using i a lot easier to stomach.
[1]: https://acko.net/blog/how-to-fold-a-julia-fractal/
More fundamentally what's addition and multiplication? Is there a continuity between them, like an operator that represents both addition and multiplication meeting them half-way?
What's the significance of square and square root? Aren't they just some exponent values (2, 1/2) which have no particular significance on the real line? Why should imaginary numbers have a special place if they are tied to square roots (exponent of 0.5)?
People use sums to denote vector components, too. It's common to see ai + bj + ck (pretend there are hats on i, j, and k) to mean the vector [a b c].
So with complex numbers, you can think of it as a vector aj + bi where j and i are basis vectors, but we call j "1", because complex numbers have a notion of multiplication, and in that notion, j acts like 1, in that zj = jz = z for any complex number z.
Nothing bad happens if you write all complex numbers a + bi as vectors or tuples [a b]. It's just less convenient.
The same goes for the term "real" numbers. Sometimes math, and its descendants like physics and computer science, can be too cutesy with their names that make it hard for regular people to understand or accept.
It allows putting complex numbers on the complex plane and doing addition, multiplication and exponentiation via drag and drop. It provides some intuition for addition==translation and multiplication==scale-rotation.
[1]: https://static.laszlokorte.de/complex-plane/
> Starting with a given convex quadrilateral, construct a square, external to the quadrilateral, on each side. Van Aubel's theorem states that the two line segments between the centers of opposite squares are of equal lengths and are at right angles to one another.
https://en.wikipedia.org/wiki/Van_Aubel%27s_theorem
One thing which would have worked better (which is briefly mentioned) is Cardano's casus irreducibilis, but that would have needed more work.
It can be used to convince students that they are useful.
https://pratik.deoghare.com/complex-mars-rover.html
Deciding square root of -1 is fine and use it in calculations as if it existed is confusing. Why does it work with sqrt(-1), and not with dividing by 0 ?
introducing the imaginary plane as: "hey someone found you could just draw them, and it works even better!" is even worse.
To teach complex numbers, you should start by defining them:
1 is here, i is here, multiplication for 1 is like in R, but multiplicating i by itself goes to -1, and then derive the calculation rules then.
And only when you went all the way to exp(i*theta) = cos(theta) + i sin(theta)
Only then you should explain what they are for. That they are a way to treat real numbers as a projection of these things that behave nicer in calculations. Deriving trig functions from that simple formula is trivial.
This is pretty straightforward to explain.
With dividing by zero, any number you multiply by zero becomes zero. There's no other way to get zero through multiplication. So, the only sensible thing to divide by zero is zero (since "What times zero equals six?" doesn't have an answer).
But because anything multiplied by zero is zero, even if you divide zero by zero you have an undefined answer. Zero divided by zero could be anything! So that's also not meaningful.
The square root of negative one, however, only has to be defined once to get meaning out of all negative square roots.
Explain X to me prompt now gives me pictures, quizes . so good.