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> Also wtf are complex numbers really about? Complex numbers have an intuitive way of thinking about them, they’re just not frequently given sufficient exposit

by dsacco 9y ago

> Also wtf are complex numbers really about? Complex numbers have an intuitive way of thinking about them, they’re just not frequently given sufficient exposition for clarity. The fundamental concept to internalize for complex numbers is the square root of a negative number. In the real world, calculations involving square roots of negative numbers are practical. If all you have are the real numbers, you cannot resolve equations involving the square root of a negative number, like x^2 = -5. This isn't arbitrary - it becomes very handy when you're dealing with problems that can run into complex numbers. For example, we can use complex numbers to model phase and magnitude in physics and electrical engineering. Complex numbers resolve this problem by segregating the discrepancy and reducing it to an imaginary unit i, where i^2 = -1. If you can agree to the definition of i, you can usefully model any square root of a negative number; if you recall that 1 is the multiplicative identity (1x = x), -1 can be considered to be something like the complex multiplicative identity. In other words, definining it axiomatically allows you to extrapolate the square root of arbitrary negative numbers using real numbers and i. In this way, the complex numbers extend the real numbers and encompass them. If you’re with me so far, this next part might improve your intuition. A complex number is typically written z = a + bi, where a and b are real numbers and i is the imaginary square root of -1. This is presumably what you’re familiar with. For the complex number z, we have a real part, a, and the imaginary part, b. More formally, Re(z) = a and Im(z) = b. Instead of reasoning about complex numbers using this form, you can instead represent them as ordered pairs of the form (a,b), where a and b are real numbers and the same real and imaginary parts of z, respectively. This form doesn’t represent i explicitly, which might be more straightforward to understand. Moreover, complex numbers of the form (a,b) can be modeled as points in a plane, because they are coordinates. Recall that a plane is a two dimensional coordinate space; a line is a one-dimensional space, and you can generalize this conceptually to k dimensions, represented by R^k. An ordered pair is a 2-tuple, which generalizes to a k-tuple, such that the points in a k-dimensional space R^k are k-tuples, (x1, ..., xk). Therefore, when we’re talking about ordered pairs, we’re talking about planes, and each element in an ordered pair is a dimension of the plane. For any given real number a, the real number’s ordered pair is (a,0). i = (0,1); for any x = (a,b) and y = (c,d), xy = (ac - bd, ad + bc). This defines multiplication on the set of all complex numbers (recall, then, that any set defined with operations for addition and multiplication becomes a field, so this along with addition establishes the complex field, extending the real field). Because any complex or real number can represented as an ordered pair of real numbers (a,b), we use R^2 to represent the complex plane, where R traditionally denotes the set of all real numbers. Finally we get to our result: by modeling the already intuitive real numbers and the complex numbers as points in a plane, we can practically model and reason about real world 2-dimensional problems using geometric and topological methods that would be unresolvable without a square root of negative numbers. Naturally, this features heavily in mathematical analysis :) Now that we've gotten this far, we can look at a basic, practical example. Let's say we want to model a voltage or current on a two dimensional plane. Let z be a complex number representing the voltage or current on the complex plane R^2. Further let the vertical axis be Im(z) and the horizontal axis Re(z). Then we have line segments x and y representing the real and imaginary parts of z respectively, and drawn from z to the vertical Im(z) and horizontal Re(z) axes, respectively. The phase is initially represented by 0 (the origin), and the magnitude is a line segment drawn from the origin to the point of z, where the real and imaginary parts of z intersect on the plane. You could use trigonometry to reason about phase differences in a current, but it is much simpler to instead represent them as linear equations involving complex numbers. I hope that helps a bit to build your intuition. If you can sit through it and work through the proofs, I have found that Rudin’s Principles of Mathematical Analysis is really excellent for building up the number systems from first principles in the first chapter. It’s dry and might take you 10 minutes per page, but it’s self-contained and will really enhance your understanding of why we have systems of numbers from a unifying theoretical perspective. For example, you can (hopefully) see how complex numbers are really just a resolution to the problem of square roots of negative numbers, and that they have a practical application for modeling problems in physical space. You can think of irrational numbers in a very similar way: just as complex numbers extend real numbers with this resolution, irrational numbers extend the rational numbers with a resolution to the problem of square roots that have an infinite decimal expansion. We can look at the square root of 2: there is no rational p to satisfy p^2 = 2. The intuitive theory of irrational numbers is not that they're "infinite decimals" (which leads to the question, "What are infinite decimals and why do we care about them?"). The intuitive theory of irrational numbers is that they are a set of numbers which resolve equations that we cannot resolve using only rational numbers. Similarly, the intuitive theory of complex number is not that they're numbers with some arbitrary "imaginary" part embedded in them, they're a system of numbers that express ideas we cannot reason about using only the reals. Putting all of this together, we have: 1. The set of all natural numbers N: {0, 1, 2, 3, ...}. These are useful for counting. 2. The set of all integers, Z: {..., -2, -1, 0, 1, 2, ...} ("Z" stands for "Zahlen"). 3. The set of all rational numbers, Q, where Q stands for quotient. 4. The set of all irrational numbers, sometimes denoted by I. 5. The set of all real numbers, R, which contains both the rational and irrational numbers. 6. The set of all complex numbers, sometimes denoted by C, which contains both the real and complex numbers. Each number system is an extension of the number system preceding it, and expands the system in a way that allows new axiomatic operations to be defined such that we can resolve problems observed (and unsolvable) in the previous number systems.

jacobolus 9y ago
IMO the “proper” way to learn about / think about complex numbers is as quotients of Euclidean planar vectors. http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf Your version as an alternative gives much less insight about their structure. Rudin’s book is full of good problems for analysis students who want to be inducted into the exclusive club of mathematicians, but is not really the best way to learn about the meaningful mathematical modeling potential of number systems, IMO.
- dsacco 9y ago
  Actually I agree with you. That’s a good point. To clarify, the reason I like my approach is not because it’s comprehensive, but because it’s better than the traditional exposition for letting students build initial intuition. In my experience people dislike learning about complex numbers because the notation of z = a + bi feels arbitrary to them, and there’s not enough context in which to motivate them. To your point, representing them as quotients of vectors is definitely better for overall understanding. But for a quick and dirty approach that still involves some theory for building intuition of why complex numbers exist and are useful, representing them as ordered pairs is better than what’s usually done, which is the introduction of complex numbers in computational courses instead of theoretical ones. Ordered pairs are extremely intuitive because they can be plotted on two axes, and topological reasoning in general is fairly intuitive in lower dimensions. Were I not confined to a small comment I’d definitely go with your approach, though. I’m just optimizing for solving a student’s “what even is?” initial intimidation, not for really working with them comprehensively.
  - jacobolus 9y ago
    The problem I have with just an “ordered pair” pedagogy is that it confuses (in students’ minds) the concepts of Euclidean point, displacement vector between points, and “complex number” (which I think of in most contexts as a similarity transformation of Euclidean vectors). In an arbitrary Euclidean vector space, the coordinates chosen are somewhat arbitrary, except for being typically orthogonal and of the same length (a sometimes arbitrary chosen or sometimes meaningful “unit”). But similarity transformations are in some sense “unitless” (insofar as they don’t involve meters or inches or whatever, because they are quotients of vectors and the units cancel): the value 1 is very meaningful and not arbitrary at all (it represents the identity transformation), the value –1 represents a reflection of vectors through the origin, the value i is a unit bivector (in this case representing a specific quarter-turn rotation), and all values of unit magnitude (e.g. √3/2 + i/2) represent pure rotations. To teach someone who has done some analytic geometry in Cartesian style (which is itself sort of jumbled conceptually, because students are taught that the grid is the plane instead of being an arbitrary coordinate system laid down over the plane), explaining this takes quite a bit of nuance and clarity on the part of the instructor. And even many mathematicians and technical professionals elide these distinct concepts all the time. For example, they represent points (e.g. points on plane curves), vectors, and transformations all as complex numbers. Or they represent points as “position vectors” without always making it obvious that the point per se also involves adding the displacement to an arbitrarily chosen origin.

by dsacco 9y ago

jacobolus 9y ago

IMO the “proper” way to learn about / think about complex numbers is as quotients of Euclidean planar vectors. http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf Your version as an alternative gives much less insight about their structure. Rudin’s book is full of good problems for analysis students who want to be inducted into the exclusive club of mathematicians, but is not really the best way to learn about the meaningful mathematical modeling potential of number systems, IMO.

dsacco 9y ago
Actually I agree with you. That’s a good point. To clarify, the reason I like my approach is not because it’s comprehensive, but because it’s better than the traditional exposition for letting students build initial intuition. In my experience people dislike learning about complex numbers because the notation of z = a + bi feels arbitrary to them, and there’s not enough context in which to motivate them. To your point, representing them as quotients of vectors is definitely better for overall understanding. But for a quick and dirty approach that still involves some theory for building intuition of why complex numbers exist and are useful, representing them as ordered pairs is better than what’s usually done, which is the introduction of complex numbers in computational courses instead of theoretical ones. Ordered pairs are extremely intuitive because they can be plotted on two axes, and topological reasoning in general is fairly intuitive in lower dimensions. Were I not confined to a small comment I’d definitely go with your approach, though. I’m just optimizing for solving a student’s “what even is?” initial intimidation, not for really working with them comprehensively.
- jacobolus 9y ago
  The problem I have with just an “ordered pair” pedagogy is that it confuses (in students’ minds) the concepts of Euclidean point, displacement vector between points, and “complex number” (which I think of in most contexts as a similarity transformation of Euclidean vectors). In an arbitrary Euclidean vector space, the coordinates chosen are somewhat arbitrary, except for being typically orthogonal and of the same length (a sometimes arbitrary chosen or sometimes meaningful “unit”). But similarity transformations are in some sense “unitless” (insofar as they don’t involve meters or inches or whatever, because they are quotients of vectors and the units cancel): the value 1 is very meaningful and not arbitrary at all (it represents the identity transformation), the value –1 represents a reflection of vectors through the origin, the value i is a unit bivector (in this case representing a specific quarter-turn rotation), and all values of unit magnitude (e.g. √3/2 + i/2) represent pure rotations. To teach someone who has done some analytic geometry in Cartesian style (which is itself sort of jumbled conceptually, because students are taught that the grid is the plane instead of being an arbitrary coordinate system laid down over the plane), explaining this takes quite a bit of nuance and clarity on the part of the instructor. And even many mathematicians and technical professionals elide these distinct concepts all the time. For example, they represent points (e.g. points on plane curves), vectors, and transformations all as complex numbers. Or they represent points as “position vectors” without always making it obvious that the point per se also involves adding the displacement to an arbitrarily chosen origin.