A Visual, Intuitive Guide to Imaginary Numbers (2007)
betterexplained.com
betterexplained.com
Do you know any other examples in math where fixing terrible naming makes the concept easier to digest?
"Complex" was originally justifiable but it should be renamed to reflect changes in the English language.
Wouldn't that cause much, much greater confusion due to mathematical nomenclature being a moving target rather than remaining stable?
There may be some amount of drift over time, but you can go to the research library of any university math department and find books from pre-WWII that are still totally readable because although style has shifted somewhat, the basic nomenclature hasn't.
I agree random variable is awkward, though. I always avoided stats courses because it's full of so much jargon that collides with nomenclature used by mathematicians.
I found this doc on the origins of the name (author also agrees it's terrible): http://www.glennshafer.com/assets/downloads/talks164_The-inv...
Sounds like it got mangled as work was being translated back and forth between Russian, English, French, and German.
The actual name might be debatable, but a shortened name rather than the full definition definitely makes sense here. Random variable seems like a good choice of name to me, thinking about the intuition we're formalizing with it.
Where "complex numbers" is Chinese, "imaginary numbers" is German, and "2D numbers" is English in this analogy.
One qualifier I'd add though is that they're 'asymmetric' 2D numbers. The real vs imaginary axes have different behaviors. If I multiply by the positive unitary value on the imaginary axis (i.e. 'i'), I get CCW rotation by 90 degrees; with the negative imaginary unit I get CW rotation; positive real unit, no change occurs; negative real unit, rotation by 180 degrees.
Actually that's something I wonder about—could there be a 'symmetric' version of that? Maybe something where the real axis behaves more like the imaginary axis, but maybe some signs are flipped or something.
I guess an important aspect of how using the complex plane is beneficial is the fact that it combines these disparate elements though—we can start with something real, do certain transformations to it in our more broadly capable complex plane, then bring it back into the real line.
Really, each power corresponds to an additional 90 degree rotation. 0th = 0 degree rotation (just scaling), 1st = 90 degree rotation, 2nd = 180 degree rotation, 3rd = 270 degree rotation (equivalently -90 degree). So that seeming CW rotation is also 3 CCW rotations.
The main thing I was pointing out were the different geometric behaviors of multiplication by real vs imaginary units—not so much the difference between negative and positive units on either axis.
In the last part of my first comment I point out the benefit (that I perceive) arising from the asymmetry. I also still wonder if a symmetric version (like I describe in the first comment) would be possible.
Splitting the geometric behavior into rotation, reflection, and scaling is interesting—so thanks for the description—I just don't see how it relates to my comment.
The good thing about GA is that the same concept can be easily extended to 3D (quaternions), and in fact to 4D and nD.
In fact, geometric algebras are very general, and the one I briefly sketched is not the only possible interpretation. If you are interested, you can find many good introductions online, directed at different audiences. You can also search for the term "Clifford algebras" if you are interested in a more formal approach.
The result of the determinant might be negative, which doesn't make sense for a volume, so you need to take the absolute value to interpret it that way.
Another way to think of it is the change in volume caused by the linear transformation determined by the matrix (http://algebra.math.ust.hk/determinant/01_geometry/lecture3....)
"Signed scaling factor" is more general, and some courses (e.g. ones I teach on) use that to introduce the idea.
Anyway, 'scaling factor' still leaves the signs mysterious.
When you call them rational numbers, it sounds like you're saying they're the only numbers that are "smart" or that "make sense" -- the connection to ratios is obscured.
And all the way back in 2007, long enough ago that nickb posted it: https://news.ycombinator.com/item?id=91811.
The focus on formula memorization in schools is tragic. Once upon a time, I too have learned everything about imaginary numbers ... everything, other than why the heck they are actually useful. Can do all the calculations, don't know why I am doing them.
Are there other great math textbooks/websites (Calculus level and higher, Stats, Linear Algebra, etc.) that try to do this better? For someone older than school level who wants to learn again.
Fully understanding the math in this book requires a solid background in linear algebra, calculus, differential equations. But I find it an incredibly interesting book even without understanding all the math: the engaging writing style and numerous illustrations capture the intuitions behind differential geometry and relativity, but without sacrificing the rigorous mathematical formulations underlying it.
Most of our modern technology would not be achievable without the use of complex numbers as a tool.
He focuses on the intuitive concept of a particle having a spinny arrow attached, the arrow rotates as the particle flies through space. He only casually mentions that this is in fact a complex number, whereas the bulk of the text focuses on developing intuition around arrows.
I read that book in high school, and it certainly influenced the direction I took in university. It helped to understand that the physical universe often appears to behave in extremely non-intuitive ways, but using mathematics we can develop a model that transforms the phenomenon into something that actually does make intuitive sense.
I think some of the harder concepts in math are difficult because they act like stepping stones into aspects of our world that just don't make sense based on day-to-day experiences. But modern technology depends on this! Pedagogy is improving, but it still lags advances in technology.
The really nice part of this explanation is that it tells you why complex numbers show up everywhere. It turns out that it's rather straightforward to find physical real-world problems with input parameters that are coefficients to polynomials and behavior that depends on the roots of those polynomials. Take a slinky or another other harmonic oscillator - when you model it with a differential equation, the polynomial coefficients are how heavy the slinky is, how much speed-dependent resistance there is, and how springy it is. Factoring the polynomial gives you the behavior over time, and it pretty much always has some sort of behavior, so the roots should be some kind of number.
The kind of math that was traditionally taught in schools is still relevant and important, but I think we can leverage modern visual and interactive media to help children develop a broader class of mathematical reasoning skills, which includes much, much more than a bunch of rules, symbols, and rote procedures.
Given the sort of thing Euler was good at, though, it seems just as likely that he looked at the power series for sin, cos, and exp, and said "aha!".
The other is that "multiplication of complex numbers is rotation" (which can be demonstrated purely by algebraic manipulation) and that "exponentiation is repeated multiplication". If we know what e^(ix) is then we also know what e^(2ix) is. It is the same "vector" as e^(ix) but the length of the vector will be squared and the angle it makes with the real axis will be doubled.
It is trivial to differentiate exponents like a^(x) and we get that the derivative is simply a constant multiple of itself (depending only on "a"). We choose "e" to be the choice of real number that makes the constant 1. (We can also rigorously justify that such a choice of real number exists.)
Now, what is the value of e^(ix) for very small positive values of x? It is approximately the value of e^(ix) at zero plus x times the value of the derivative at zero. (This is just the Taylor series.) In other words, for small x, e^(ix) is essentially 1 + ix except we know our answer should have magnitude 1 so we interpret e^(ix) as having magnitude 1 and angle x for small x. The properties of exponentiation as repeated multiplication and multiplication of complex numbers being rotation justifies interpreting e^(ix) as having magnitude 1 and angle x for all x.
This is not very rigorous but it is the gist of the matter. Many tools in modern analysis were created to make arguments like this rigorous so this could definitely be considered a good way to understand complex exponentiation.
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Besides that, why did mathematicians choose this exact representation? Why not polar, spherical, hyperbolic, Hilbert-like, Minkowski-like? Did anyone explore on how that could change known problems, like e.g. Riemann-zeta?
Specifically; multiplication and division are defined on the complex numbers but not on vectors (you can multiply/divide a vector by a scalar but that's different).
The definition of a vector space says nothing about whether it does or does not have a multiplication operation defined on it. A vector space having a multiplication operator has additional structure, but is still a vector space.
Example: The space of NxN matrices. There are 2 distinct forms of multiplication on this space: between a vector and a scalar (scalar multiplication), and between vectors (matrix multiplication).
This though shows that explanations via analogies or non-strict wording may confuse one rather than enlighten. I’m not good at math, but once understood to not search analogies or geometry in things. Instead it is better to “shut up and calculate”. Not sure if imagining something is required to manage it. It’s only our brain’s faulty quirk.
Complex number notation lends itself to algebraic manipulation that isn't as easy in other representations. Additionally, it's equivalent is polar coordinates (2-d), spherical coordinates represent 3-dimensional space.
Consider the following: How would you add two polar coordinates? It turns out you need a lot of math to do this without converting, first, to cartesian and then back to polar is probably the easiest way. How do you multiply two polar coordinates? This is actually easy, multiply the magnitudes and add the angular component (division works similarly).
How would you add two cartesian coordinates? This is easy, add the corresponding components. How would you multiply two cartesian coordinates? You probably don't have a definition for this, one option is to develop a definition for it, or you can convert to polar and back.
Complex numbers, however, can be added as easily as your typical vector representation. And they have a clear definition for multiplication. i i = -1, that's the only extra fact you need as otherwise it behaves like typical real multiplication. No conversion is needed, and (a+bi)(c+di)* is easy to solve using the stated multiplication fact.
All that said, determining the complex representation for a rotation is not as easy as doing so with polar coordinates. In fact, I basically did the first example by starting with a concept of (1,45 degrees) (using (r,theta) notation) and computed the complex equivalent. However, once you've got your initial system in place, it's very easy to continue doing the rest of your math using the complex notation rather than switching back and forth between other representations.
The answer is completely historical in nature. Imaginary numbers began as being interpreted as the square root of -1 for the purposes of solving polynomial equations (hence the name.) Later, their field structure and their interpretation as vectors-with-multiplication became their primary use but the name remained.
Mathematicians don't really use "vectors" in the traditional sense like in physics but deal with abstract vector spaces where a "vector" is simply a member of a "vector space" which is "a set of things with addition and scalar multiplication and a few other nice properties".
However, if something needs to be done with vectors in a plane, complex numbers are extremely useful because scaling and rotation can be represented as multiplication. Therefore natural operations in the complex numbers often correspond to natural operations in whatever you are trying to study with complex numbers.
This is not at all true in general: many mathematicians use non-abstract vectors too. My (maths) PhD, for example, uses vectors throughout but doesn't mention vector spaces once.