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2 improved word choice and sentence structure

This answer doesn't show how the complex numbers are useful, but I think it might demystify them for students. This is an answer that most Most are probably already familiar with its content, but it might be useful to point state it out again. Since the question was asked two months ago and Professor Zudilin started teaching a month ago, it's likely this answer is also too late.

If they have already taken a class in abstract algebra, one can remind them of the basic theory of field extensions with emphasis on the example of $\mathbb C \cong \mathbb R[x]/(x^2+1).$

It seems that most introductions give complex numbers as a way of writing non-real roots of polynomials and go on to show that if multiplication and addition are defined a certain way, then we can work with them, that this is consistent with handling them like vectors in the plane, and that they are extremely useful in solving problems in various settings. This certainly clarifies how to use them and demonstrates how useful they are, but it still doesn't demystify them. A complex number still seems like a magical, ad hoc construction that we accept because it works. If I remember correctly, and has probably already been discussed, this is why they were called imaginary numbers.

If introduced after one has some experience with abstract algebra as a field extension, one can see clearly that the complex numbers are not a contrivance that might eventually lead to trouble. Beginning students might be thinking this and consequently, resist them, or require them to have faith in them or their teachers, which might already be the case. Rather, one can see that they are the result of a natural operation. That is, taking the quotient of a polynomial ring over a field and an ideal generated by an irreducible polynomial, whose roots we are searching for.

Multiplication, addition, and its 2-dimensional vector space structure over the reals are then consequences of the quotient construction $\mathbb R[x]/(x^2+1).$ The root $\theta,$ which we can then relabel to $i,$ is also automatically consistent with familiar operations with polynomials, which are not ad hoc or magical. The students should also be able to see that the field extension $\mathbb C = \mathbb R(i)$ is only one example, although a special and important one, of many possible quotients of polynomial rings and maximal ideals, which should dispel ideas of absolute uniqueness and put it in an accessible context. Finally, if they think that complex numbers are imaginary, that should be corrected when they understand that they are one example of things naturally constructed from other things they are already familiar with and accept.

Reference: Dummit & Foote: Abstract Algebra, 13.1

If they have already taken a class in abstract algebra, one can remind them of the basic theory of field extensions with emphasis on the example of $\mathbb C \cong \mathbb R[x]/(x^2+1).$
Multiplication, addition, and its 2-dimensional vector space structure over the reals are then consequences of the quotient construction $\mathbb R[x]/(x^2+1).$ The root $\theta,$ which we can then relabel to $i,$ is also automatically consistent with familiar operations with polynomials, which are not ad hoc or magical. The students should also be able to see that the field extension $\mathbb C = \mathbb R(i)$ is only one example, although a special and important one, of many possible quotients of polynomial rings and maximal ideals, which should dispel ideas of absolute uniqueness and put it in an accessible context. Finally, if they think that complex numbers are imaginary, that should be corrected when they understand that they are one example of things naturally constructed from other things they are already familiar with and accept.