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In algebra II, USA, we teach our students complex numbers. However, after algebra II, they never use complex numbers until pretty much complex analysis. The whole point of teaching them complex numbers is to find the roots of polynomials... but, that's all we ever do with them. Sure, we do some algebraic manipulation of them, just to get a feel, but that's it. Nothing ever too deep, and often times there isn't even a geometric understanding of them.

And by the time students ever reach a point where they need to use complex numbers... it'll be very far in the future.

As far as real world applications, it'll probably be finding the real roots of something for anyone who doesn't take complex analysis or similar courses that deal with complex numbers. And at that point, you probably don't sit down and factor your polynomial. I would think most would either see it as a quadratic or go straight to a calculator (one that can solve such things with complex numbers or numerical methods)

So, my question is if its really worth it teaching students about complex numbers. Most students will forget about complex numbers even, as complex analysis is usually a far way off, if they ever get there.

Maybe to be more specific,

Why do we teach students about complex numbers if most will never reach a course that uses them? When do laymen use complex numbers in real world applications?

19Disagree with a couple of assumptions here: That "the whole point of teaching them complex numbers is to find the roots of polynomials", and that mathematics is inherently about applications. Or, if it needs be said, passing a particular high school test. – Daniel R. Collins – 2016-09-24T14:23:40.620

8How could you

nottalk about complex numbers, at least a little? In the quadratic formula, sometimes the discriminant is negative! – Adam – 2016-09-24T16:14:09.78028I would ask a different question: "why don't we teach more complex numbers?" They have vast application in trigonometry, two-dimensional vector analysis etc... much of this ought to come before calculus. – The Chef – 2016-09-24T21:29:21.687

@JamesS.Cook Good point. Maybe later as a separate question. – Simply Beautiful Art – 2016-09-24T21:38:21.437

3@JamesS.Cook: and, the formulations of physics of the last 100 years all require them in an essential way, be it electromagnetism or quantum mechanics. – Martin Argerami – 2016-09-25T01:44:17.517

1http://insti.physics.sunysb.edu/~siegel/history.html "Complex numbers don't appear till the end of quantum mechanics (itself toward the end of the book). Sin's, cos's, and complicated trigonometric identities abound, especially for electromagnetism. " – Count Iblis – 2016-09-25T06:53:32.887

2Because you can't draw Mandlebrot Sets without them. – None – 2016-09-25T13:20:57.347

That. Is. A very good point. Definitely need to draw my Mandlebrot Sets :D – Simply Beautiful Art – 2016-09-25T14:09:25.090

2They taught me complex numbers at high school, in the second year, and I've never stopped using them since then. – Massimo Ortolano – 2016-09-25T17:07:13.423

1@MartinArgerami:

the formulations of physics of the last 100 years all require them in an essential way, be it electromagnetism or quantum mechanicsHuh? I've seen more than one formulation of E&M, and none of them required complex numbers. Maxwell's equations don't, and the covariant formulation doesn't. I would actually be interested if you knew of a formulation that did require complex numbers. – Ben Crowell – 2016-09-25T18:36:26.7501@CountIblis: [quoting Warren Siegel:]

"Complex numbers don't appear till the end of quantum mechanics (itself toward the end of the book). Sin's, cos's, and complicated trigonometric identities abound, especially for electromagnetism."Complex numbers really are central to quantum mechanics. If freshman physics textbooks postpone them, it's to try to make the subject less scary for students, but it doesn't reflect the basic logic of the subject. Siegel's web page, which you linked to, is bitterly criticizing this fact about most freshman texts. – Ben Crowell – 2016-09-25T18:38:12.5472Never judge a fish by its ability to climb trees. – None – 2016-09-25T18:55:03.723

3>

2@Adam: in the Netherlands they don't mention complex numbers in high school (or they didn't use to 25 years ago, anyway), and the sign of the discriminant was used to figure out if the formula has roots. Negative discriminant -> no roots. – RemcoGerlich – 2016-09-26T08:12:07.793

1If the only things we taught were the things we could be sure pretty much everyone would need to know, school would consist of nothing but driver eduction and basic computer skills. – barbecue – 2016-09-26T16:40:00.460

Also, from my experience (admittedly many years ago) complex numbers featured prominently in my calculus, analytical geometry, and diff. eqs. courses. – barbecue – 2016-09-26T16:41:54.840

3Just a general comment that many forms of math (euclidean geometry, trig, calculus, and yes complex numbers) didn't just teach me math - they taught me how to solve problems in a different way. Geometry taught me to justify my reasons for everything; trig taught me to convert the problem from one form to another; calculus taught me to take things to the extreme and see what shakes out; and complex numbers taught me there's always another dimension to things to take into consideration. I never use complex numbers - ever - but my brain circuits are better at solving problems because I did them. – corsiKa – 2016-09-26T16:56:45.293

We used complex numbers in trig/pre-calc (Euler's formula and all that), which was the coarse right after Algebra II in my HS math curriculum. – pwcnorthrop – 2016-09-26T18:03:24.190

@corsiKa IMO, I think you summed up the reason why we teach Algebra. – Simply Beautiful Art – 2016-09-26T19:42:56.140

@SimpleArt While incredibly important in learning to think, I didn't include Algebra in the list because quite honestly, I use Algebra in every day life (my home monthly budget, for example) and at work (allocating disk space for servers based on usage trends), while I haven't exactly used "opposite angles are congruent" very much... – corsiKa – 2016-09-26T20:20:59.090

1As an engineer by degree but not by profession, I was hoping for a good old-fashioned $i$ vs. $j$ argument somewhere in the comments on this page. Then I remembered I'm on

matheducators SE. – shoover – 2016-09-26T21:10:46.947This is country-dependent, but for me the schedule of high school math topics was driven by the requirements of the other subjects; and when you teach electromagnetism and alternating current in physics, you'd want students to have covered complex numbers at that point, otherwise you'd have to teach them in physics class instead. – Peteris – 2016-09-26T21:10:58.300

Because it is a concept you come across early in algebra and might as will define it and give it a name. Sure only a fraction will use it for higher level math. – Paparazzi – 2016-09-27T02:38:21.937