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> On the other hand, there is no straightforward intuitive mapping for complex numbers. Au contraire! Complex numbers are just rotations, translations, and sca

> On the other hand, there is no straightforward intuitive mapping for complex numbers. Au contraire! Complex numbers are just rotations, translations, and scalings in the plane. In the same sense that real numbers capture 1D rigid transformations and scaling, complex numbers capture 2D rigid transformations and scaling. When you treat negative numbers as "distance in an opposite direction", you're more generally treating real numbers as a transformation of the 1D line, i.e. translation (addition) or scaling (multiplication) by some amount. Try thinking about the same but for the 2D plane and complex numbers. Addition is 2D translation and you'll see that multiplication corresponds to a rotation and a scaling. The first section of Needham's "Visual Complex Analysis" walks you through this perspective in detail, if you're interested and also want lots of good exercises.

crazygringo 3y ago
I'm very familiar with Needham, thank you. But perhaps I wasn't clear enough. When engineers need to handle rotations, translations, and scaling, they don't use complex numbers. They use vectors, usually. There's no intuitive mapping for the concept of the square root of negative one in real life. Not in the way there is for negative numbers. And complex numbers are not about 2D geometric representation in general, the way vectors are. They are much more specifically about rotation or spiral motion.
- xelxebar 3y ago
  > There's no intuitive mapping for the concept of the square root of negative one in real life. Not in the way there is for negative numbers. I really think you might be a victim of bad pedagogy. Admittedly, the notation confuses complex numbers as points vs. complex numbers as operations. Let me drive home the point. You can represent complex numbers as matrices: e := [[1 0] our identity element, i.e. what you typically write as 1. [0 1]] i := [[0 -1] our imaginary unit [1 0]] These 2x2 matrices operate over 2D vectors, obviously. Furthermore, notice that <e,i> = [[0 0] the inner product of e and i [0 0]] in other words e and i are orthogonal, meaning that their span is a 2D subspace of the underlying matrix space. 2D... Meaning every matrix like this can be written as v = a×e + b×i. Also, um, notice that i^2 = -1×e, or more on the nose, i = sqrt(-1×e). We've just rediscovered complex numbers! Now, using the above matrix representation uses 4 parameters, but you really only need 2. Indeed, any general matrix in our 2D algebra looks like this a×e + b×i = [[a -b] [b a]] where the repetition is obvious. Why not throw away the slop and just directly use e and i? Also, notice that e acts just like the identity 1, meaning that a+bi as a notation makes sense. Then we end up writing that i^2 = -1 and unfortunately invite all sorts of confusion about the meaning of "imaginary" numbers. However! Even though these are 2D things, it's very important to not confuse them with the 2D vectors they manipulate. The matrix representation above makes that more manifest. Said another way, it doesn't make sense to compose points in space, but it does make sense to compose operations which operate on that space. It's an unfortunate and confusing quirk that we often write a+bi to mean a 2D point in space, a 2D translation operation, or a 2D rotation and scaling, despite all these really being completely different things.

crazygringo 3y ago

I'm very familiar with Needham, thank you. But perhaps I wasn't clear enough. When engineers need to handle rotations, translations, and scaling, they don't use complex numbers. They use vectors, usually. There's no intuitive mapping for the concept of the square root of negative one in real life. Not in the way there is for negative numbers. And complex numbers are not about 2D geometric representation in general, the way vectors are. They are much more specifically about rotation or spiral motion.

xelxebar 3y ago
> There's no intuitive mapping for the concept of the square root of negative one in real life. Not in the way there is for negative numbers. I really think you might be a victim of bad pedagogy. Admittedly, the notation confuses complex numbers as points vs. complex numbers as operations. Let me drive home the point. You can represent complex numbers as matrices: e := [[1 0] our identity element, i.e. what you typically write as 1. [0 1]] i := [[0 -1] our imaginary unit [1 0]] These 2x2 matrices operate over 2D vectors, obviously. Furthermore, notice that <e,i> = [[0 0] the inner product of e and i [0 0]] in other words e and i are orthogonal, meaning that their span is a 2D subspace of the underlying matrix space. 2D... Meaning every matrix like this can be written as v = a×e + b×i. Also, um, notice that i^2 = -1×e, or more on the nose, i = sqrt(-1×e). We've just rediscovered complex numbers! Now, using the above matrix representation uses 4 parameters, but you really only need 2. Indeed, any general matrix in our 2D algebra looks like this a×e + b×i = [[a -b] [b a]] where the repetition is obvious. Why not throw away the slop and just directly use e and i? Also, notice that e acts just like the identity 1, meaning that a+bi as a notation makes sense. Then we end up writing that i^2 = -1 and unfortunately invite all sorts of confusion about the meaning of "imaginary" numbers. However! Even though these are 2D things, it's very important to not confuse them with the 2D vectors they manipulate. The matrix representation above makes that more manifest. Said another way, it doesn't make sense to compose points in space, but it does make sense to compose operations which operate on that space. It's an unfortunate and confusing quirk that we often write a+bi to mean a 2D point in space, a 2D translation operation, or a 2D rotation and scaling, despite all these really being completely different things.