This is a soft question but I’m willing to ask.
There are few ways to introduce the field of complex numbers, but if You had the opportunity to write an elementary textbook, what would be the most intuitive introduction for a very sceptical audience?
Example of an “bad” answer: write down all the axioms and then quote Hilbert “existence in mathematics means freedom from contradiction“.
$\mathbb{C}$ has very little to do with axioms. Just let $\mathbb{C} = \mathbb{R} \times \mathbb{R}$ and then define the appropriate operations on it. In particular:
$$(x,y)(x’,y’) = (xx’-yy’,xy’+yx’)$$
Then prove that arithmetic inside $\mathbb{C}$ behaves as expected.
Other possible approaches:
Okay, but what if the audience is uber-skeptical?
If they don’t accept that $\mathbb{R}$ exists, you can use an axiomatic set-theory (say, ZFC) to prove its existence. Similarly, if they don’t accept that from $\mathbb{R}$ we can construct $\mathbb{R} \times \mathbb{R}$ with the aforementioned operations, then you may need to appeal to an axiomatic set-theory (or other foundation).
If they don’t accept the usual axioms of set theory, applaud them for being very skeptical. Then challenge them to come up with their own ideas and their own formal system on which to secure mathematics. This is a surprisingly hard challenge, and is likely to increase their respect for existing foundations! Hopefully not so much that they stop pondering their own foundations, though. The rules of the game are:
I don’t like the introduction via $x^2+1=0$, because it’s too easy for a skeptical person to say, why should that equation have a solution, anyway? I prefer to look at something like $x^3-6x-4=0$. Now, that has a real solution; the left side is negative for $x=2$ and positive for $x=3$, so there’s a solution somewhere between 2 and 3. And you can express that solution as $$x=\root3\of{2+2\sqrt{-1}}+\root3\of{2-2\sqrt{-1}}$$ as you can check by cubing it and doing some algebra.
Historically, complex numbers came in as solutions of cubics, not quadratics.
If the audience is very skeptical, there will be no short way to introduce complex numbers. I think they would be suspicious of any “axiomatic” approach.
For such an audience, I would take a longer way, which happens to be historically relevant. I would present (and prove) the Cardano’s algebraic formulas to solve third and fourth degree equations.
Then, I would show that those formulas produce correct REAL roots even when there are square roots of negative numbers involved in the calculation that cancel out. So this fact suggests that the square roots of negative numbers can be treated as numbers which really exist.
Then, I would remark that square roots of negative numbers, IF THEY EXIST, can be reduced to $a\sqrt{-1}$ where $a$ is a real number.
Then, I would algebraically deduce what would be the properties of such “imaginary” numbers when combined with real numbers.
Finally, I would come back and show that those “strange” numbers (or, we may call them “complex” numbers), resulting from the combination of real and “imaginary” numbers, work perfectly with Cardano’s formulas and also with second degree equations.
It is a looong way to present complex numbers, but it corresponds (at least partially) to how historically the concept of complex numbers appeared.
I’ve had some fun and some success with middle school kids using this approach.
They know about the number line. They can understand that addition of a fixed quantity can be thought of as translation, right for positives and left for negatives. Note that translation by $a$ takes the number $0$ to $a$. They can understand a fraction like $a/2$ as what you need to translate by twice to translate by $a$.
Then we move on to multiplication by positive numbers. As a transformation, multiplying by a fixed $a$ is a change of scale. Note that scaling by $a$ takes the number $1$ to $a$. That leads to a definition of $\sqrt{a}$ as the number you have to scale by twice to scale by $a$.
Next we note that multiplication by $-1$ is a reflection about $0$ on the line. (Incidentally, that helps with “why is a negative times a negative a positive?”) But what is half that reflection? What can you do twice to end up moving $x$ to $-x$? Just rotate the number line half a turn (counterclockwise is the right convention). Kids really like that idea. By analogy, that’s a rescaling by the number that $1$ moves to – so we have $i$ in the right place in the newly introduced number plane. The additive structure of the rest of the number plane follows easily when interpreted as translation.
The full multiplicative structure is much harder – both historically and for now tired middle schoolers. You can do it with the distributive law, but the geometry is harder. You can’t quite get to polar coordinates. But they will be delighted to find $\sqrt{i}$ as a rotation and then its real and imaginary parts since they know the length of the hypotenuse of an isosceles right triangle with side $1$ is $\sqrt{2}$.
Just a soft answer – perhaps wrong.
means that $x = \pm \sqrt{2}, \mathbf{1} = 1$, but we can also write
$$
x =
\pm \left( \begin{array}{cc} \sqrt{2} & 0 \\ 0 & \sqrt{2}\end{array} \right),
\mathbf{1} =
\left( \begin{array}{cc} 1 & 0 \\ 0 & 1\end{array} \right).
$$
We can now consider objects written as
$$
\mathbf{c}(x,y) = x + \mathbf{i} y,
$$
keeping the rotation over 90 degrees in our mind.
While for real numbers, we cannot have $x^2 < 0$,
for rotations $\mathbf{R}$, we CAN have $\mathbf{R}^2 = -1$.
Note that
$$
\mathbf{c}(x,y) =
\left( \begin{array}{cc} x & -y \\ y & x\end{array} \right).
$$
And note that
$$
x^2 + 2 \cdot \mathbf{1} = 0,
$$
means that
$$
\mathbf{x} =
\pm \left( \begin{array}{cc} 0 & -\sqrt{2} \\ \sqrt{2} & 0\end{array} \right) = \pm \mathbf{i} \sqrt{2}.
$$
Here are the main points I would consider:
I. Motivation: Define a complex number, then state some reasons for studying complex numbers. It is assumed that the students know about real numbers. So tell them that equations like $x^2+4=0$ has no solution in the real numbers; as a consequence introduce the concept of $i=\sqrt{-1}$, and then give them the solution to the equation above.
Algebra of Complex numbers: Addition and Subtraction of C.N, equations in C.N, Multiplication of complex numbers, Conjugate of a complex number, then division in C.N.
Representation, Forms and Applications: Talk about Visualisation (Argand’s diagram), then talk about the three forms of a complex number (Standard: z=a+ib, Polar: $z=r(Cos\theta+iSin\theta)$ and Euler: $z=re^{i\theta}$) and how to go from one to another. Finally, talk about DeMoivre’s theorem et-cetera.
C.N means Complex Numbers.
For me one of the best examples comes from the Feynman Lectures. First he starts with positive integers and the operations you can use on those numbers. However gradually you run into problems that make it necessary to introduce other numbers to find solutions. Such as 2-5=x has no solution unless you introduce negative integers. Or 2/5=x doesn’t work unless you introduce rational numbers. Eventually this results in x^2+1=0 to introduce the existence of imaginary numbers. Caltech has made these lectures available to read online for free and if you would like to see how Feynman explains it.
suppose you have an equation like this:
x² = -1
now you want to solve it. how?
x = sqrt(-1)
which can NOT be answered for $x \in \mathbb{R}$
thus to obtain the solution, we introduce the Complex numbers, with their main axiom being: i = sqrt(-1)
…whenever there is something that can’t be solved with the mathematics of the time, it grows into mathematics.
e.g. Real numbers couldn’t solve equations like this: x = 1/3
so in order to solve them, Rational numbers have been introduced. The same logic applies to Complex numbers, quod erat demonstrandum.