I’ve always thought “imaginary” vs “real” was an unfortunate naming convention.
I don’t know about other fields, but electrical engineering uses imaginary numbers with AC circuits and changing electrical fields. Since electricity moves as waves, imaginary numbers let you represent what’s coming 90 degrees later in a compact way.
Carl Friedrich Gauss would agree with you on the naming. He thought the confusion/mistery around imaginary numbers was due to naming. He said +1, -1, and root(-1) should have been called direct, inverse, and lateral units.
don’t forgot signal processing. Complex number theory is used a lot when it comes down to Fourier transforms of discrete or continuous functions.
but yea the naming convention of “imaginary numbers” is pretty bad, we did the same thing for negative numbers (we called it imaginary) when people couldn’t comprehend of a negative quantity.
negative numbers are defined from a conservative system, basically a gain/loss relationship: if I have 2 apples and I give away 1 to a friend, I have 1 apple left, that loss – that apple I gave to a friend – is the negative quantity. For imaginary numbers, we don’t really have a way to comprehend an imaginary quantity; what does it mean that I have sqrt(-1) apples.
Yeah, I think it causes unnecessary difficulties. I actually think they’re introduced at a time when you could instead teach them as two-dimensional vectors with pointwise addition and a special multiplication and division rule, and prove that (0, 1)×(0, 1) = (-1, 0) using that rule, so that sqrt(-1, 0) = (0, 1).
Then you can establish a convention that you write (a, b) as a + bi (and i = (0, 1)).
This is too abstract for younger students, but nowadays I don’t think they learn complex numbers anyway, and I think it would be less spooky for the older students.
Imaginary numbers are typically introduced in a high school “algebra 2” course in my neck of the woods, like junior year unless you are accelerated or held back. I feel like it’s really common for them to not be taught well - the naming is something that occasionally trips up the teachers.
The teaching of them is something that really interests me - they’re the kind of thing that triggers the “when am I ever going to use this?”/rants about not learning how to do taxes. You can talk about the relevance to electronics, but DC electronics is already hard enough for most to comprehend.
I like to connect it to rotation. Show them the pattern of powers of i with physical movement - quarter turns.
The big thing is that vectors seem to be the kind of “shoved into the end of the semester if there’s time after testing” from what I’ve seen. Most of the time, even when I work with calculus students they have no idea what a vector is.
A big thing to is getting them to understand what square roots even really mean. I’ve noticed a lot of students struggle with getting sqrt(x) * sqrt(x) = x, so even just the simple “hey, can you get a negative by taking a number and multiplying by itself?” is often a hurdle cognitively. (A lot of this I suspect has to do with not understanding what multiplication or division really “are” - I usually remediate with the area model)
Wow, (potentially) omitting vectors seems like a big gap. Obviously it has huge direct practical use, but it’s probably also the first introduction to how you can take a structure and augment it with operations. In that way it’s the first step on the road to abstract mathematics.
Also tbf, rational numbers are just the lattice modded out by the equivalence relations (a, b) ~ (c, d) iff ad = bc, and the equivalence classes just happen to form an ordered field. If you show an undergrad math/science student this esoteric definition of rationals, the motivation of “2D numbers” makes a lot more sense. Of course, please don’t show this to an elementary or middle schooler, a high schooler might be able to handle this if they’re passionate about math.
Of course, please don’t show this to an elementary or middle schooler, a high schooler might be able to handle this if they’re passionate about math.
Lol, I was thinking this as I was reading.
The neat thing about complex numbers defined this way is that you don’t need to understand quotient spaces because under addition ℂ is already isomorphic to ℝ²!
Quaternions are useful, but no means necessary or intrinsic to 3d graphics.
They are useful for describing rotations in 3d space, but the same can be done with a matrix, or even Euler angles. It’s just a more calculations.
That said, basically all the major game engines use quaternions, so unless you are creating your own 3d renderer, you have to learn at least the basics about them.
Quaternions really shine at spherical interpolation (Slerp - yes thats the real terminology) between two rotations.
I’d say Quaternions are necessary purely because they avoid gimbal locking common with 3 axis Euler rotations.
I know in Blender all other rotation options are really just quaterinions under the hood. But an advantage Euler rotations have over quaternions is continuous rotation. I can set 2 key frames with linear extrapolation and have them just continue along each axis forever. Everything quaternion ideally should be bounded to the 4d unit circle.
Yep, it is bad naming. I took it in highschool but it didn’t really make sense till I saw how it described electric fields with the wave 90 degrees into the flat page.
Electrical engineers also use them as a simple way of solving differential equations with laplace transforms. Much easier as long as you don’t care about what happens at infinity.
I’ve always thought “imaginary” vs “real” was an unfortunate naming convention.
I don’t know about other fields, but electrical engineering uses imaginary numbers with AC circuits and changing electrical fields. Since electricity moves as waves, imaginary numbers let you represent what’s coming 90 degrees later in a compact way.
And in optics, the “real” portion of a material’s refractive index represents scattered light and the “imaginary” part represents absorbed light.
Carl Friedrich Gauss would agree with you on the naming. He thought the confusion/mistery around imaginary numbers was due to naming. He said +1, -1, and root(-1) should have been called direct, inverse, and lateral units.
Lateral numbers is my preferred term
don’t forgot signal processing. Complex number theory is used a lot when it comes down to Fourier transforms of discrete or continuous functions.
but yea the naming convention of “imaginary numbers” is pretty bad, we did the same thing for negative numbers (we called it imaginary) when people couldn’t comprehend of a negative quantity.
negative numbers are defined from a conservative system, basically a gain/loss relationship: if I have 2 apples and I give away 1 to a friend, I have 1 apple left, that loss – that apple I gave to a friend – is the negative quantity. For imaginary numbers, we don’t really have a way to comprehend an imaginary quantity; what does it mean that I have sqrt(-1) apples.
It means you have an orange.
Yeah, I think it causes unnecessary difficulties. I actually think they’re introduced at a time when you could instead teach them as two-dimensional vectors with pointwise addition and a special multiplication and division rule, and prove that (0, 1)×(0, 1) = (-1, 0) using that rule, so that sqrt(-1, 0) = (0, 1).
Then you can establish a convention that you write (a, b) as a + bi (and i = (0, 1)).
This is too abstract for younger students, but nowadays I don’t think they learn complex numbers anyway, and I think it would be less spooky for the older students.
Imaginary numbers are typically introduced in a high school “algebra 2” course in my neck of the woods, like junior year unless you are accelerated or held back. I feel like it’s really common for them to not be taught well - the naming is something that occasionally trips up the teachers.
The teaching of them is something that really interests me - they’re the kind of thing that triggers the “when am I ever going to use this?”/rants about not learning how to do taxes. You can talk about the relevance to electronics, but DC electronics is already hard enough for most to comprehend.
I like to connect it to rotation. Show them the pattern of powers of i with physical movement - quarter turns.
Yeah. I think the vectors-first approach allows you to get straight to rotations, too.
The big thing is that vectors seem to be the kind of “shoved into the end of the semester if there’s time after testing” from what I’ve seen. Most of the time, even when I work with calculus students they have no idea what a vector is.
A big thing to is getting them to understand what square roots even really mean. I’ve noticed a lot of students struggle with getting sqrt(x) * sqrt(x) = x, so even just the simple “hey, can you get a negative by taking a number and multiplying by itself?” is often a hurdle cognitively. (A lot of this I suspect has to do with not understanding what multiplication or division really “are” - I usually remediate with the area model)
Wow, (potentially) omitting vectors seems like a big gap. Obviously it has huge direct practical use, but it’s probably also the first introduction to how you can take a structure and augment it with operations. In that way it’s the first step on the road to abstract mathematics.
Also tbf, rational numbers are just the lattice modded out by the equivalence relations (a, b) ~ (c, d) iff ad = bc, and the equivalence classes just happen to form an ordered field. If you show an undergrad math/science student this esoteric definition of rationals, the motivation of “2D numbers” makes a lot more sense. Of course, please don’t show this to an elementary or middle schooler, a high schooler might be able to handle this if they’re passionate about math.
Lol, I was thinking this as I was reading.
The neat thing about complex numbers defined this way is that you don’t need to understand quotient spaces because under addition ℂ is already isomorphic to ℝ²!
Right, it’s just an orthogonal basis. You can extend this to many dimensions, fields and geometries.
Not just electricians.
3d graphics are all about about quaternions (4d version of imaginary numbers).
Quaternions are useful, but no means necessary or intrinsic to 3d graphics.
They are useful for describing rotations in 3d space, but the same can be done with a matrix, or even Euler angles. It’s just a more calculations.
That said, basically all the major game engines use quaternions, so unless you are creating your own 3d renderer, you have to learn at least the basics about them.
Quaternions really shine at spherical interpolation (Slerp - yes thats the real terminology) between two rotations.
I’d say Quaternions are necessary purely because they avoid gimbal locking common with 3 axis Euler rotations.
I know in Blender all other rotation options are really just quaterinions under the hood. But an advantage Euler rotations have over quaternions is continuous rotation. I can set 2 key frames with linear extrapolation and have them just continue along each axis forever. Everything quaternion ideally should be bounded to the 4d unit circle.
Yep, it is bad naming. I took it in highschool but it didn’t really make sense till I saw how it described electric fields with the wave 90 degrees into the flat page.
Electrical engineers also use them as a simple way of solving differential equations with laplace transforms. Much easier as long as you don’t care about what happens at infinity.
Infinity is too far away. We’ll deal with it when it gets closer.
Is that just an EE thing?