Are complex numbers just vectors defined by cosine (x axis) and sine (y-axis)?

Question

There is no clear and standard definition of a complex number, I recall a prof being at a loss when a freshman aske him what is a complex number.

I read somewhere that a great mathematicians was not ashamed to say, re the formula $e^xpi = -1$ I think, that nobody knows what it means , but they are proud of it, is that true?*

I suppose nobody has a clear answer, I am just asking to specify what is more or different from the regular operations on a vector field taken as a 360° circle, where points are identified by cosine of the relative angle (real part) and sine (imaginary part)

Edit

The linked question/answer resembles only superficially my question. I stressed the crucial feature that the vector is defined by sin and cos and consequently obeys the rules of duplication etc of angles. I am not familiar with all details of that theory and I am asking , if any, about the differences. If there are none or minor ones, then complex numbers ar not numbers and not complex at all.

*I found the quote:

Benjamin Peirce, a noted American 19th-century philosopher, mathematician, and professor at Harvard University, after proving Euler's identity during a lecture, stated that the identity "is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth

*

Answer 1

First of all, there IS a clear and standard definition of complex numbers. It is just that you have not seen one, and those people you asked happens to not know one/forgot. It could be that your curriculum rushes to teach many contents without ensuring that you understand concepts well.

Actually, there are multiple isomorphic ways to define complex numbers, but I suppose you are nowhere near understanding what "isomorphic" is.

Here is a list of definitions I know of:

1. Write down the Cartesian product $\Bbb R\times\Bbb R$, which can be visualised as the plane. A complex number is an element of $\Bbb R\times\Bbb R$. Addition of complex numbers $+'$ is defined by $(a,b)+'(c,d):=(a+c,b+d)$, while multiplication of complex numbers $\cdot'$ is defined by $(a,b)\cdot'(c,d):=(ac-bd,ad+bc)$.

2. Write down the set of all two by two matrices with real entries of the form $$\begin{bmatrix}a&b\\-b&a\end{bmatrix}.$$ A complex number is such a matrix. Addition of complex numbers is defined as usual addition of matrices. Multiplication of complex numbers is defined as usual multiplication of matrices.

3. The set of complex numbers is defined as the splitting field of the real polynomial $x^2+1$. Explicitly, let $\Bbb R[x]$ by the ring of all real polynomials. Then the set of complex numbers is defined as the quotient ring $\Bbb R[x]\big/\langle x^2+1\rangle$ by the ideal generated by $x^2+1$.

The third one is not elementary. You will need to learn abstract algebra to understand it. But the first and second one should be elementary enough for you to understand, unless you have not yet learned elementary set theory or linear algebra.

Using definition 1, $\sqrt{-1}$ would be defined as (0,1), while using definition 2, $\sqrt{-1}$ is defined as $$\begin{bmatrix}0&1\\-1&0\end{bmatrix}.$$

But to understand the formula $e^{\pi i}=-1$, the above definition is not enough. You will need some calculus/analysis. You need to give the set of complex numbers a norm, or metric, etc., so that you can talk about convergence of an infinite sum. For a complex number $z$, $e^z$ is defined as $$e^z=\sum_{n=0}^\infty \frac{z^n}{n!}.$$ You can also define $$\sin z=\sum_{n=0}^\infty (-1)^n\frac{z^{2n+1}}{(2n+1)!}$$ and $$\cos z=\sum_{n=0}^\infty (-1)^n\frac{z^{2n}}{(2n)!}.$$ Then the identity $$e^{zi}=\cos z+ i \sin z$$ is a theorem which has a proof, not just stated.

Edit: As you may know already, every complex number can be written in the form $r(\cos\theta+i\sin\theta)$, or $re^{\theta i}$ due to the above identity, for some nonnegative real number $r$ and a real number $\theta$. This is not just for fun. This makes multiplication of complex numbers easy and geometric.

The product of $r_1e^{\theta_1 i}$ and $r_2e^{\theta_2 i}$ is given by $r_1r_2e^{\theta_1 i}e^{\theta_2 i}=r_1r_2e^{(\theta_1+\theta_2)i}$. You can see a picture illustrating this on Wikipedia: https://en.wikipedia.org/wiki/Complex_number#Multiplication_and_division_in_polar_form.

Answer 2

The standard definition of complex number is: a complex number is an ordered pair of elements of $\mathbb{R}$. It is due to Hamilton.

And of course that $e^i\neq \pi i$. What you meant was $i^i=-\frac\pi2$.