Yeah, honestly I just didn’t pay enough attention in Complex Analysis to get a real sense of why any of it worked.
Looking at Wikipedia, mathematicians were disgusted at both negative numbers and imaginary numbers. And wrote treaties on how stupid it was to use them.
The use of negative numbers was like you suggested-- motivated in part by debts and credits. And in part by wanting to easily manipulate equations. Imaginary numbers didn’t show up until we started solving cubic equations, and it became clear that you could get a real number answer if you just didn’t mind some nonsensical bullshit on the way.
But again, a convenient way to solve an equation is not a “real thing” to me. But then there are “real things” that follow. Maybe I need to think more on Euler’s equation or something.
I guess the issue comes down to the same question twig poses-- whats a real life example of multiplying complex numbers…
This is more application - but I seem to recall in a 3Blue1Brown video he talked about how complex numbers are very useful in rotations (specifically he was talking about animation). Matrices can also work, but they are more clunky.
Actually - come to think of it - I think it was this video about Quaternions - which are useful in 3D rotations. (I’m not going to rewatch it to make sure though…)
Perhaps another approach to use with “negative” and “complex” numbers is to think in terms of vectors . . . including their offshoots of speed/velocity and acceleration. And if you’re really good, the jerk.
I suppose one example of a negative * a negative is flipping a thermometer upside down.and looking at how much it’s dropped to know the temperature increase.
By that approach, multiplying by i or -i would mean turning you thermometer clockwise or counterclockwise, and then looking at how much changed sideways.
Guess I should look at those links before I come up with even stupider ideas…
It had to do with the PE candidates. I got three emails today encouraging me to “Meet PE candidate [Bob Ama].”
Unless I’m bad at reading, they are all men. That led me to think about addressing them here as “Messrs. Dewey, Cheatem, & Howe” and how I like to use the abbreviation/word “Messrs.”
