I think there's definitely something to your suggestion here. For instance, there is no result to sqrt(-1) on the real line, but it becomes possible in the complex plane because you have an additional degree of freedom.
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.
You want geometric algebra. In 2D GA, you would have two unitary vectors, i and j, such that i * i = 1 and j * j = 1. The (non-commutative) product of between them (or their division) would be a bivector: i * j = ij = - j * i. You rotate 90 degrees the i and j vectors using the bivector.
The good thing about GA is that the same concept can be easily extended to 3D (quaternions), and in fact to 4D and nD.
Yes, they are. i would be the unitary vector in the horizontal direction, x, and j in the vertical one, y.
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.
Reduce these to the integral powers of i: i^0, i^1, i^2, i^3. The symmetries occur two apart (0, 2; 1, 3). Higher integral powers are equivalent to the 4 listed, i^n = i^(n mod 4).
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.
Sure, I guess that's a consequence of general rotational symmetry.
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.
Separate out the components, rotation and reflection and scaling. -1 is reflection. i is rotation by 90 degrees CCW. -i is both rotation by 90 degrees and reflection. Positive reals are scaling. So -ai is really rotation by 90 degrees, reflection, and then scaling by a.
But none of that is related to my key point: multiplication by unitary values is significantly different depending on which axis the unit comes from (real vs imaginary), hence my usage of 'asymmetric'.
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.
I don't know about math, but at least in physics, "quantum teleportation" is a terrible name; it made way more sense to me when somebody called it "quantum telegraphy" instead.
My point is that is isn't just a function, but a measurable one (with weak restriction on the measure space it has as domain) so we can't call it just "function" as you were suggesting.
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.
But you propose to simply relabel as function, which doesn't work in general because random variable corresponds to a specific type of function. You could compromise by calling it a probability function, but then you start to collide with other uses of that word.
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.
Yes, it's not just a function - there are strings attached but we deserve a better name - probability function is x100 better. "random variable" is exceptionally bad in that it leads the mind in irrelevant directions.
To follow one of the more common patterns for identifying a class of functions, it could have been named after one of the early pioneers in the field. But yeah, it would be hard to do worse than random variable, which is illogical and misleading.
Intuitively elements of R2 are not numbers, they're vectors, or points in a 2 dimensional vector space. You can also treat complex numbers as vectors in the 2 dimensional complex plane.
"Intrinsic curvature" should be called something else, because people frequently drop the "intrinsic". This implies the existence of a higher dimensional space which usually doesn't exist. "Curvinture" would be a better name.
Hmm. I only see 'intrinsic curvature' used when there is a potential confusion about whether it's intrinsic or extrinsic (otherwise it's just 'curvature'). And if that is the case, so that there is an embedding in a higher dimensional space, then intrinsic/extrinsic curvature are excellent terms (imo). They were some of the first math terms I came across where I really appreciated the selection of terminology.
But... they aren't just 2D numbers. That would be vectors. Yes, they behave like vectors under addition, but they also have a ton of other behaviour that '2D numbers' doesn't help with at all, like what it means to raise a number to a complex power.
Except they can be negative, and relate to area for 2D, not volume, and would require defining volume in dimensions higher than 3, which is not necessarily obvious or intuitive for beginners.
"Signed scaling factor" is more general, and some courses (e.g. ones I teach on) use that to introduce the idea.
I think that 'N-volume' including area, volume, and higher dimensional analogs is pretty understandable to a student. The fact that it's signed is more subtle, but still is not a big problem. The benefits to intuition of calling it 'volume' far outweigh the negatives.
Anyway, 'scaling factor' still leaves the signs mysterious.
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.
This might be one: Instead of rational numbers, call them "[integer-]ratio numbers" -- they're the numbers that can be expressed as a ratio of integers.
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.
kowdermeister wrote about imaginary numbers, not complex numbers. "Imaginary" is a terrible name, because all numbers are imaginary. Even integers are imaginary, in the sense that the percentage of integers with actual physical representations is 0%, rounded to however many decimal places you like. "Real" is a terrible name for numbers for the same reason.
"Complex" was originally justifiable but it should be renamed to reflect changes in the English language.
What would happen if we started renaming mathematical objects to reflect changes in the English language? English will continue to evolve, and the vast body of mathematical literature would have to be constantly rewritten.
Wouldn't that cause much, much greater confusion due to mathematical nomenclature being a moving target rather than remaining stable?
Mathematical nomenclature is already a moving target. And the symbols don't have to change, only the English reading of them, e.g. ∫ is from the latin "summa" (sum), but you don't need to know that to read it as "integral"
I can't think of any examples of mathematical objects being renamed, do you have any?
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.
> Focusing on relationships, not mechanical formulas.
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.
They are only really useful in the science and engineering fields that need them. That is why they remain opaque to most people. High school level pedagogy doesn't deal effectively with the useful applications of these concepts (matrices are also poorly introduced).
Most of our modern technology would not be achievable without the use of complex numbers as a tool.
I always liked how Feynman dealt with complex numbers in 'QED: The Strange Theory of Light and Matter'.
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.
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.
You could look for a math history book (such as "An Imaginary Tale" by Nahin) or something like the Princeton Companion to Mathematics or frankly any other popular math book written by established mathematicians.
I always find the "algebraic closure" approach to be the best bet for explaining complex numbers. Much like how having negative numbers means that you can subtract any two numbers (closure under addition), having complex numbers means you can find the roots of any polynomial. If you don't have complex numbers, something like `x^2 + 1` has no real roots, and you have a problem.
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.
From the POV of this explanation, it's then quite fortuitous that the complexes are just two-dimensional over the reals, and can thus be easily visualized. That is, as soon as you adjoin the roots of x^2 + 1, and close under field operations, you actually get the roots of all polynomials.
There is a deep result stating that for a field F there are only three possibilities for what the dimension of its algebraic closure can be as an F-vector space: it can be 1, if F is algebraically closed, it can be 2, as happens for R and other real closed fields, or it can be infinite, as it happens for Q, but there is no other option!
I wonder if these wonderful resources we have nowadays will result in more kids getting into science and maths. It would have been fantastic to have had this when I was in school (I kept an interest in maths in spite of my teachers' efforts).
Many people have recognized the need for improvement in mathematics education, and I think it really is evolving in positive directions. I worked at DreamBox Learning for a few years, they produce an adaptive math learning program for elementary schools (and gradually reaching higher levels) which as been very popular with children.
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.
I think I first learned imaginary numbers in algebra 2 which was like 6th or 7th grade. I know they use x^2=1 as an example, but I don't know how useful that is. I think they really need to teach trig before they teach imaginary number. They also need to teach polar coordinates with imaginary numbers. Most all of its uses are for solving trig problems. You use it in EE and ME to solve sinusoidal problems, but I don't know how i=sqrt(-1) helps you understand what its used for. I think euler formula is most elegant formula in math. How did the guy even come up with it.
I don't know how Euler came up with it, but here's a fairly intuitive way: think about circular motion. If something moves around a circle at constant speed, then its velocity vector is perpendicular to the vector from 0 to the moving thing. Once you have the idea (which isn't terribly difficult) that complex numbers live on a plane and multiplication by i is rotation through a right angle, this gives you the differential equation dx/dt = ix. Solving that is easy: x = exp(it). Since it's also easy (by definition of the trig functions) to see that x = cos t + i sin t, we're done.
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 2D analogy and use of rotating vectors was alien to Euler and contemporaries. They really did think of it as an "imaginary" abstraction to make the math work.
There are two very good ways of understanding Euler's formula and one is the "circular motion" explanation given by another comment. Both are very similar.
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.
If thinking like I’m 5, it doesn’t add up. At first, “b times i” means rotation, but then ‘a+bi’ is a vector of two orthogonal components. I expected ‘bi’ to be an angle in polar coordinates and ‘a’ to be length.
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?
It still means rotation in "a + bi". If you take "a + b" you get another real number, but if you take "a + bi", the b component has been rotated by 90 degrees (i), and now it's orthogonal to a. Even if we drop complex numbers, it's not like we write out points in polar coordinates as "5 + 30 degrees" - how are you adding a length and an angle together?
Yeah, I see now. “times i” is discrete 90 degree rotation itself, not just ‘i’, nor ‘b’. Thanks everyone for making that clear.
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.
It's the multiplication by i that introduces rotation, which is the critical thing, not the b in that expression. (a+bi)i is (a+bi) rotated by 90 degrees. But note that it's a specific rotation, 90 degrees counter clockwise. Not a general "rotation by some number of degrees". To get a general rotation, you need the whole complex number a+bi. So to rotate an arbitrary complex number z by some degrees you need z(a+bi) where |a+bi| = 1 (to avoid scaling) but whose angle is the angle you want to rotate by. Simple case: 0+1i will rotate by 90 degrees CCW. More complex: 1/sqrt(2)+1/sqrt(2) i will rotate by 45 degrees CCW. Try that second one out on paper, draw out the points on paper and see where they end up.
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.
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).
I disagree. It suffices to include a rotation operator (or bivectors in general, if you want to do it in N>2). The 'full' framework of GA is, imo, foolish, because the geometric product doesn't really have a geometric interpretation.
"Does any of this really have to do with the square root of -1? Or do mathematicians just think they're too cool for regular vectors?" https://xkcd.com/2028/
This is obviously not a completely serious question but it is definitely looks like a question someone might ask when learning about complex numbers for the first time.
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.
> Mathematicians don't really use "vectors" in the traditional sense like in physics but deal with abstract vector spaces
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.
Do you know any other examples in math where fixing terrible naming makes the concept easier to digest?