9 ms·
A Visual, Intuitive Guide to Imaginary Numbers (2007)
- dang 8y agoDiscussed in 2011: https://news.ycombinator.com/item?id=2712575 https://news.ycombinator.com/item?id=2712575 And all the way back in 2007, long enough ago that nickb posted it: https://news.ycombinator.com/item?id=91811 https://news.ycombinator.com/item?id=91811.
- deleted 8y ago[deleted]
- vinchuco 8y agoThis has also a great intuitive explanation of why sqrt(-1) is a rotation. HTTP://greatscottgadgets.com/sdr/6/
- samfisher83 8y agoI 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.
- gjm11 8y agoI 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!".
- kevin_thibedeau 8y agoThe 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.
- skybrian 8y agoI don't know how he came up with it, but I did some slides giving some geometric intuition about why it's the natural choice: http://slesinsky.org/brian/misc/eulers_identity.html http://slesinsky.org/brian/misc/eulers_identity.html
- chess19 8y agoThere 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.
- jatsign 8y agoI really enjoyed this video series on imaginary numbers. Went through the history as well as the math: https://www.youtube.com/watch?v=T647CGsuOVU https://www.youtube.com/watch?v=T647CGsuOVU
- bcaa7f3a8bbc 8y ago"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/ https://xkcd.com/2028/
- chess19 8y agoThis 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.
- mkl 8y ago> 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.
- tangentspace 8y agoI wonder, how many people downvoted this without following the link to the comic? It's relevant, intelligent, and funny.
- kowdermeister 8y agoOnce I renamed imaginary numbers to 2D numbers internally it made more sense. Do you know any other examples in math where fixing terrible naming makes the concept easier to digest?
- another-cuppa 8y agoIt's not a terrible name, it's just that we've forgotten the real meaning of "complex" and use it as a synonym for "complicated".
- mrob 8y agokowdermeister 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.
- tangentspace 8y agoWhat 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?
- mrob 8y agoMathematical 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"
- 8y ago
- forinti 8y agoI 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).
- tangentspace 8y agoMany 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.
- carapace 8y agoThe punchline for complex numbers is in Geometric Algebra...
- weiming 8y ago> 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.
- corysama 8y agoEveryone loves: 3Blue1Brown, Khan Academy, No Bullshit Guide to Linear Algebra, Linear Algebra Done Right
- chess19 8y agoYou 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.
- tangentspace 8y agoGravitation, by Misner, Thorne, Wheeler (http://a.co/dczH4eS http://a.co/dczH4eS) 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.
- kevin_thibedeau 8y agoThey 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.
- 8y ago
- disqard 8y agoI came here expecting to see Steven Wittens' excellent writeup [0] mentioned in the comments. Since it is not, I feel obligated to mention it. [0] https://acko.net/blog/how-to-fold-a-julia-fractal/ https://acko.net/blog/how-to-fold-a-julia-fractal/
- mkl 8y agoYes, I think this makes things very intuitive, and I use it (well, the first slideshow in it) to introduce the idea when I teach complex numbers.
- wruza 8y agoIf 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?
- adw 8y agoComplex numbers are not vectors. 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).
- tangentspace 8y agohttp://mathworld.wolfram.com/ComplexPlane.html http://mathworld.wolfram.com/ComplexPlane.html 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).
- lainga 8y agoIt 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?
- wruza 8y agoYeah, 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. https://m.youtube.com/watch?v=zwAD6dRSVyI https://m.youtube.com/watch?v=zwAD6dRSVyI
- ThrustVectoring 8y agoI 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.
- throwaway080383 8y agoFrom 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.
- ginnungagap 8y agoThere 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!