In my math class, we understand how to work problems with imaginary numbers in them, but do not understand the purpose or reasoning behind them.
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see also:https://math.stackexchange.com/q/170334/347317 – Logan Luther Nov 14 '17 at 16:30
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Not an answer to the question but a recourse for answering it: I highly recommend the YouTube channel Welch Labs. They have a ten-part series all about complex numbers and they explain the uses of them quite well. I also would like to recommend 3blue1brown, who makes use of complex numbers in a number of other videos, though I don't know if they have a dedicated series all about complex numbers. (I thought that they did, but it turns out I was thinking of the Welch Labs video). – Tom Dacre Nov 14 '17 at 16:36
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Maybe it helps knowing why they were first introduced: Cardan's formula for third degree equations involves square roots of negative numbers when the equation has three real roots. – egreg Nov 14 '17 at 16:41
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I like to think of them in terms of scaling and rotations on the plane. On the real line we have scaling, but only two 'rotations' (positive and negative). In the plane we have a richer collection of rotations. Multiplying by $i$ corresponds to rotating a point by $90^\circ$ in an anti clockwise direction. Do it twice and you have reversed direction (essentially multiplied by -1). – copper.hat Nov 14 '17 at 17:39