I am going into Honors Algebra II and while I am fine at using imaginary numbers in a formulaic sense I never understood them conceptually. I can’t tell if they exist just to make math work better or because there is an actual logical way to understand them
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Imagine you had two piles of different things. You wanted to be able to keep them apart sometimes, but also combine them sometimes. Let’s call one of them imaginary for fun, and put an i in front of it. In the first pile you have x things, and in the second pile you have y things, so let’s say you have x + iy things. I might find that the correct way to add my piles is to say the square root of (x)^2 + (i*y)^2 equals the total number of things I have. This comes up a lot in power/electrical engineering.
A common use of the imaginary number that may give another helpful perspective: Euler’s formula. You know all about trigonometry like with sine and cosine, and you probably know a little about logarithms and the number e (~2.718, Euler’s number) but have you seen Euler’s formula?
e^(i*x) = cos(x) + ( i* sin(x) )
Compare this with Euler’s equation:
e^(i * pi) + 1 = 0
Euler’s formula is used all over in signals/digital processing, radio frequency, and electricity. The “Fourier transform” uses it. The signal in “signals” is information carried by time-varying voltage, current, or light: the backbone of electrical power, computers, radio, and phones.
After using the imaginary number often enough, it will become just another useful math tool you know about. It loses its mysterious/cool factor.
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