real things heating up and down (i.e not an idealized particle but a thing with volume), the motion of a pendulum, electricity flowing through basically anything (same as heating, when you consider actual volume), these are a few of the things I have worked with where I literally have no option but to use imaginary numbers to model them.
Modelling here means predicting what a change would do, for example if I want to make a robot that can balance a stick upright on its hand like you might do with a broom, I use maths related to pendulums to predict the movement of the stick, which require imaginary numbers to show how moving in a certain direction will cause the pendulum to fall (in what direction & how fast). I must emphasise here that I have done these things with real robots, and they do indeed heat/electricity flows/balance as the maths predicts.
Does this imply your brain is doing the same maths, imaginary numbers and all, when you balance a broom on your hand? Is there an alternate mathematical notation that doesn't include such strange unintuitive things? probably, thats for the mathematicians to figure out. But as an engineer, they definately do work.
Modelling here means predicting what a change would do, for example if I want to make a robot that can balance a stick upright on its hand like you might do with a broom, I use maths related to pendulums to predict the movement of the stick, which require imaginary numbers to show how moving in a certain direction will cause the pendulum to fall (in what direction & how fast). I must emphasise here that I have done these things with real robots, and they do indeed heat/electricity flows/balance as the maths predicts.
Does this imply your brain is doing the same maths, imaginary numbers and all, when you balance a broom on your hand? Is there an alternate mathematical notation that doesn't include such strange unintuitive things? probably, thats for the mathematicians to figure out. But as an engineer, they definately do work.