"Why Do I Need Complex Numbers?"

I have heard students ask this question over and over in one form or another. A recent incarnation was featured in this video by 3Blue1Brown. I think this version strikes at the heart of the vendetta that many students have against complex numbers.

"What is an application that is impossible to achieve without complex numbers? Convince me that they are needed. They are fun to work with, and maybe they make things easier, but we can do without them."

(Edited for grammar and spelling.)

At the heart of this question is a reasonable request. When introduced to a new mathematical concept, we should demand reasons, even if the answer is necessarily a bit delayed. High school students in the US are often introduced to complex numbers with little motivation. To them, this new number system can seem like a lot of complexity with little payoff. Maybe a few applications are presented, but one might argue they could be achieved without complex numbers, even if it requires sacrificing some elegance. Why put in the extra work?

 
I would like to formulate a  satisfying response to this question. While one could try responding to this challenge with the fundamental theorem of algebra (which I would challenge anyone to state with clarity while avoiding complex numbers), I think this misses the heart of the question. To the general public, this is just another abstract theorem. What people want is a reason to care.

I think a better response would be to issue a challenge of my own: convince me that I need fractions. Why can't the world get through life using only decimals? My banana bread recipe can call for 0.$\overline{3}$ cups of milk. We can call the coin a 25 cent piece instead of a quarter. Bon Jovi can sing about being 50% of the way there. Who needs the extra complexity of introducing a second system for dealing with pieces of whole numbers?

The answers to the two challenges are the same: maybe you could do without, but why would you do that to yourself? For many applications, fractions are just so much simpler. If I'm making 50% of a recipe that calls for 0.333 cups of water, the math involved is much more cumbersome than if I am halfing a recipe that calls for a third of a cup. It is much easier to remember "three sixteenths of an inch" than to retain the decimal 0.1875. And even if an exact answer is not needed, mentally dividing fractions is certainly faster and less error-prone than doing the same with decimals. There is a reason that the world uses fractions in everyday life. We use them not because of any task that is impossible without them, but because they make things easier.

If you agree that fractions are useful (and I hope you do), then you must agree that a mathematical object does not need to conquer the impossible in order to be worth learning. We should judge complex numbers by the same standard as fractions: do they make things significantly easier? This is the right question. And in the right setting, the answer is yes. They are a natural tools for dealing with rotations and oscillations, or computations in plane geometry, to name a couple of broad examples. Complex numbers can simplify computations and enhance our understanding.

To answer the original question more directly, I think the answer is no. Any direct applications of complex numbers could be achieved without them if you were willing to work hard enough to avoid them. But the same could be said of fractions, or decimals, or real numbers, or any mathematical structure. The reason we use any of these constructions is because they are the easy way out. They make things simpler to work with and understand, which is exactly what math is supposed to do.

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Footnote: A commenter on the above video with the username "Garbaz" addresses the question from an electrical engineering standpoint.

"[Once] my electrical engineering professors said that if mathematicians hadn't come up with complex numbers, electrical engineers would have. Dealing with electrical circuits that involve capacitors, inductors (and alternating currents) without complex numbers is very difficult, having to deal with differential equations and trig identities, but if you interpret inductors & capacitors like resistors, but with an imaginary resistance, you get an incredibly beautiful and simple way to work with them. In general, there is pretty much no area of electrical engineering that does not benefit greatly from using complex numbers. Especially everything involving AC."