In mathematics involving waves, it is common to express cosine (or sine) with a different representation using the inverse Euler formula:
$$\cos(\theta)=\frac{e^{i\theta}+e^{-i\theta}}{2} \tag{1}$$
For example, the following expression can be rewritten using the inverse Euler formula:
$$\cos(5\pi t)\cos(6\pi t) = \left(\frac{e^{i5\pi t} + e^{-i5\pi t}}{2}\right)\left(\frac{e^{i6\pi t} + e^{-i6\pi t}}{2}\right)$$
And this makes the simplification/decomposition of this expression much simpler.
I was just wondering why Euler's formula can be used to seemingly convert a real number to a complex number. When I think of real numbers, I think of just a single number/scalar. I think of imaginary numbers, however, as a tuple of sorts, $(a,b),$ where $a$ is the real part and $b$ is imaginary part (but both $a$ and $b$ are real numbers of course).
In other words, I see the left side of $(1)$ as just a number and the right side as an ordered pair. How can a single number in any expression be replaced by this ordered pair and not "change the result"?
Of course, the imaginary part of the ordered pair in $(1)$ is $0,$ and I have heard people say that real numbers can be treated as complex numbers with an imaginary part equal to $0.$ However, how does one go about proving this? How does one know that any operation performed on the real and complex number will have the same result? I feel comfortable proving this on a case-by-case basis, but is there a general way to show that this can always be done?
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