Fundamental complex number formula kept secret?

Question

I recently started self-studying of complex analysis.
I checked several popular online courses, and all of them, of course, start with the basic definitions:

• real and imaginary parts of a complex number,
• complex conjugation and the module of a complex number,
• addition and multiplication of complex numbers.

This set of definitions is usually followed by a set of properties that includes the connection with vectors.

The connection with vectors is trivial for all the terms, except for the multiplication.
I was surprised I was not able to find it in those courses.
Furthermore, for some time I could not find it anywhere, including available AIs.

But the formula is known (e.g. Tristan Needham "Visual Complex Analysis", 2023):

$$\tag{1}zw = (\bar{z}w) + i[\bar{z}w]$$

where $\bar{z}$ is the complex conjugate of $z$,
$(xy) = Re(x)Re(y) + Im(x)Im(y)$ is the vector dot product,
$[xy] = \begin{vmatrix} Re(x) & Im(x) \\\ Re(y)&Im(y) \end{vmatrix}$ is the vector cross product.

We can also use the fact that $[xy] = (xy^{\perp})$,
where $\begin{pmatrix} a \\\ b \end{pmatrix}^{\perp} = \begin{pmatrix} b \\\ -a \end{pmatrix}$ is a vector orthogonal to $\begin{pmatrix} a \\\ b \end{pmatrix}$:

$$\tag{2}zw = (\bar{z}w) + i(\bar{z}w^{\perp})$$

In this form we can find it in the proof of the Cauchy's integral theorem:
https://en.wikipedia.org/wiki/Line_integral#Relation_of_complex_line_integral_and_line_integral_of_vector_field

The formula looks beautiful and deep to me, so I cannot understand why it is not mentioned in the first chapter of every course on complex numbers.

What is the reason of not including the formula into the basis of the complex numbers theory?
Is it considered not fundamental enough?
Or, maybe, I am taking wrong courses?
Are there other interesting applications of the formula?

Answer 1

Probably one of the reasons the formula is not considered in many courses is because it belongs to a different "world": to the Geometric Algebra.

Let us consider a 4-dimensional space $\{1, e_1, e_2, e_1e_2\}$ with the following multiplication rules:

• $e_1{\cdot}e_1 = e_2{\cdot}e_2 = 1$
• $e_1{\cdot}e_2 = e_1e_2$
• $e_2{\cdot}e_1 = -e_1e_2$

From the rules above:

• $e_1{\cdot}e_1e_2 = e_1{\cdot}e_1{\cdot}e_2= 1{\cdot}e_2 = e_2$
• $e_2{\cdot}e_1e_2 = -(e_1{\cdot}e_2){\cdot}e_2 = -e_1(e_2{\cdot}e_2) = -e_1$
• $e_1e_2{\cdot}e_1e_2 = e_1(e_2{\cdot}e_1)e_2 = -e_1(e_1{\cdot}e_2)e_2 = -(e_1{\cdot}e_1)(e_2{\cdot}e_2) = -1$

Thus, the imaginary unit $i$ of complex numbers corresponds to the element $e_1e_2$ of the 4-space.

The product of 4-vectors in this space is defined as the multiplication of polynomials:

$(a_1,b_1,c_1,d_1)(a_2,b_2,c_2,d_2) = (a_1{\cdot}1{+}b_1{\cdot}e_1{+}c_1{\cdot}e_2{+}d_1{\cdot}e_1e_2)(a_2{\cdot}1{+}b_2{\cdot}e_1{+}c_2{\cdot}e_2{+}d_2{\cdot}e_1e_2)$

Applying the rules above (in column vector form for convenience):

$\begin{pmatrix}a_1 \\\ b_1 \\\ c_1 \\\ d_1\end{pmatrix}\begin{pmatrix}a_2 \\\ b_2 \\\ c_2 \\\ d_2 \end{pmatrix} = \begin{pmatrix} a_1a_2 + b_1b_2 + c_1c_2 - d_1d_2 \\\ a_1b_2 + a_2b_1 + d_1c_2 - c_1d_2 \\\ a_1c_2 + a_2c_1 + b_1d_2 - b_2d_1\\\ a_1d_2 + a_2d_1 + b_1c_2 - b_2c_1 \end{pmatrix}$

Using the matrix formula it is easy to check that the subset of elements of the form $z = (x, 0, 0, y)$
is isomorphic to complex numbers:

$zw = (x_1, 0, 0, y_1)(x_2, 0, 0, y_2) = (x_1x_2 - y_1y_2, 0, 0, x_1y_2 + x_2y_1)$,

whereas the product of $2D$ vectors $\overrightarrow{z} = (0, x, y, 0)$
is a map onto the complex numbers:

$\overrightarrow{z}\overrightarrow{w} = (0, x_1, y_1, 0)(0, x_2, y_2, 0) = (x_1x_2 + y_1y_2, 0, 0, x_1y_2 - x_2y_1)$

Comparing the last two expressions one can find the "true" connection between multiplications of vectors and complex numbers:

• $\overrightarrow{z}\overrightarrow{w} = z^*w$
• $zw = \overrightarrow{z^*}\overrightarrow{w}$

where $z,w$ are the "complex", and $\overrightarrow{z}, \overrightarrow{w}$ the corresponding $2D$-vector elements of the 4-space,
and $(a,b,c,d)^* = (a,b,-c,-d)$