I don't quite understand your question. Imaginary numbers are useful for modeling waves and particles, both foundational things in our universe.
In quantum mechanics, we use complex numbers to describe the behavior of particles. A complex number has two parts: a real part and an imaginary part. The real part represents something we can physically measure, like the position or momentum of a particle. The imaginary part represents something that's a bit harder to grasp - it's related to the probability that the particle will be in a certain state or position.
For example, let's say we're trying to describe the position of an electron in an atom. We can't know for sure where the electron is at any given moment, but we can calculate the probability of finding it in a certain region. The probability is represented by a complex number, with the real part telling us the position and the imaginary part telling us the probability.
Things like Feynman Diagrams model probability distributions of particles and all possible paths they can take from A to B. They allow us to do interesting calculus
Thanks for the explanation. In engineering circles (to use an apt word) complex numbers only assist in performing rotation, eg, calculating things such as angle changes or phase changes etc. It sounds to me that it's the same in quantum mechanics.
That is while at higher levels they allow modelling of things such as probability and state, the way they are fundamentally enabling this is also just the same thing - a shorthand trick that permits fast calculations of rotational characteristics.
I'm not sure that they're able to do anything else. A wave is a particle as I understand it, or rather, they are like views into the same thing. But waves have identical characteristics in terms of what is tracked to enable calculations with them.
Eg there's amplitude phase and frequency. Maybe they all have probabilities or are otherwise dynamic, but that's what there at the base level. The complex plane then just takes the role of enabling easier calculation and tracking of state changes.
A wave is a wave whether it is a sine wave on a scope, a wave of pressure through a solid or gas, a light wave, or the path of a particle when viewed as a wave. Or, I'm misunderstanding and there are actually two, or more types of complex numbers.
The main insight of complex numbers is the fact that it relates exponentiation (partial multiplication) to rotation. Take that away then you might as well just use matrix algebra instead.
The idea is that you can “multiply by -1” not just odd or even times, but that you can do so fractionally. You can “flip the direction” partially which corresponds to shortening the original size and adding an “impetus” attribute based on the amount you shortened it by.
The raisin d’être of complex numbers is the derivative theorem of Fourier transforms. In fact, perhaps imaginary numbers should really be renamed “impetus numbers”
In quantum mechanics, we use complex numbers to describe the behavior of particles. A complex number has two parts: a real part and an imaginary part. The real part represents something we can physically measure, like the position or momentum of a particle. The imaginary part represents something that's a bit harder to grasp - it's related to the probability that the particle will be in a certain state or position.
For example, let's say we're trying to describe the position of an electron in an atom. We can't know for sure where the electron is at any given moment, but we can calculate the probability of finding it in a certain region. The probability is represented by a complex number, with the real part telling us the position and the imaginary part telling us the probability.
Things like Feynman Diagrams model probability distributions of particles and all possible paths they can take from A to B. They allow us to do interesting calculus