I’m writing a paper on real world applications of complex numbers and can’t understand it at all. If you have any other real world applications, that would be super helpful too. Thanks!
In: Mathematics
Complex numbers are not necessary, but they make the math of waves easier.
The main idea is that sinus and cosinus behave a lot like exponentials, but the math of exponentials is very easy, especially when solving differential equations or differentiating any function.
sin'(ax) = acos(ax) and sin”(ax)=-a²sin(ax)
exp'(ax) = aexp(ax) and exp”(ax)=a²exp(ax)
if you replace a by i*a in the second line, it almost looks like you could find a way to use the convenient properties of exponentials to solve trigonometry problems, but it makes no sense to apply exponential to an imaginary number.
exp(ix) = cos(x) +isin(x) is a new definition, that shares with exp(x) most of its convenient properties.
It’s used in any mathematical thing that is wavy, you just put your wavy function in the imaginary part of an exponential, and the math becomes super easy.
I studied physics and don’t now what you call electrical engineering exactly, we used complex numbers to study electromagnetic waves, signal processing with fourrier transforms, the responses of electrical systems to a wave function and the responses of physical systems.
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