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Real numbers are for straight number lines, and imaginary numbers are for circular number lines. That is about all there is to it. It's really a shame we intro
by dexwiz 5y ago
Real numbers are for straight number lines, and imaginary numbers are for circular number lines. That is about all there is to it.
It's really a shame we introduce imaginary numbers as quirky work around for negative square roots. I would say that is more of a side effect. Multiplying positive numbers always results in positive numbers, but due to the circular nature of imaginary numbers you can get positive or negative results.
Complex roots are like saying this equation has no solution in a traditional Cartesian/Euclidean world, but does have solutions in a rotational one.
I think we if introduced imaginary numbers as rotations initially, the seemingly magical things like Euler's Identify and Formula would look much more trivial.
EDIT: I would also say the lie that we live in a 3+1D world (X, Y, Z + Time), helps confuse the mater. We probably live in a 6D + 1 world: X, Y, Z, Pitch, Yaw, Roll, and Time. Just as you can conceive of a flatland 2D universe without a Z axis, you could conceive of a 3D world without rotational orientation.
- mmcdermott 5y agoI really like this explanation. The negative square roots explanation never sat right with me in high school and I think this touches on why.
- CamperBob2 5y agoThe way I think about it: - To express direction in Cartesian space, you need signed numbers - To express 2D rotation, you need imaginary numbers - To express 3D rotation, you "need" quaternions (in the sense that they're the smallest way to represent an arbitrary rotation)
- dls2016 5y ago> introduced imaginary numbers as rotations initially How would you do this without developing the exponential first?
- zwkrt 5y agoWell any complex number a + bi can be described as a transformation matrix |a -b| |b a| So you could start by just asking what it means to transform the plane
- deleted 5y ago[deleted]
- dls2016 5y agoOk but in a college algebra class it’s very easy to ask “what is the inverse of the squaring operation?” Not so clear a path to the action of a matrix on the plane. I have successfully discussed this in an advanced calculus class.
- orra 5y ago> quirky work around for negative square roots. I would say that is more of a side effect. I agree it's not a perfect motivation, for teaching purposes. When you introduce an answer to the square root of a negative number, questions of soundness might arise. Obviously it's sound, but is that a distraction at the level where complex numbers are introduced? Besides, many people like geometric interpretations: they might feel more 'real' or relevant than an algebraic motivation. /Nonetheless/, I can't agree well defined square roots is just a side effect. The complex numbers are (up to isomorphism) the one and only algebraic closure of the reals. That's a very natural and and useful way look at them for, say, university level maths.
- wyager 5y agoOversimplifications like this are not helpful. There are a ton of ways that complex numbers are interesting, and they have important generalizations as e.g. an even subalgebra of G^2 (leading to quaternions as the even subalgebra of G^3, and so on). They’re not just “a circle”. > EDIT: I would also say the lie that we live in a 3+1D world (X, Y, Z + Time), helps confuse the mater. We probably live in a 6D + 1 world: X, Y, Z, Pitch, Yaw, Roll, and Time. If your model doesn’t treat time the same way as space up to the metric tensor, it’s probably not very well generalized. There should be an even number of parameters. Rotation including the time dimension is just acceleration. The math is simpler if you’re in a ++++ metric - rotations are a bit weird in +++- because rotations into the time dimension are hyperbolic. I suggest reading Dichronauts, a book covering a ++-- tensor.
- dexwiz 5y agoWhat does the ++++ and +++- notation mean? Looking at info about tensors I don't see much. I did order the book out of interest.