What is an intuitive explanation of a complex slope of a real line?

Question

On thinking about what real slopes and imaginary slopes are I have become a bit confused. We have one coordinate $x$-$y$-$z$ system that we use to specify position. I suppose the coordinates in this system will have only real values. But then again in the argand plane the $x$ axis is real and $y$ axis is imaginary. What is a complex slope of a real line? How do we find it?My book has this sentence: "If $A(z_1)$ and $B(z_2)$ are two points in the argand plane then the complex slope of line $AB$ is $\frac{z_1-z_2}{\bar z_1-\bar z_2}$." Where did they get this from? I'm not very sure whether this formula is correct either. I do not have extensive knowledge of complex numbers, I have just studied the basics.

Answer 1

Descartes has taught us how to use a real $(x,y)$-plane ${\mathbb R}^2$ to graphically display functions $x\mapsto y=f(x)$. In particular a linear function $y=ax+b$ has a graph which is a straight line, and this line has slope $a=\tan\alpha$, whereby $\alpha\in\bigl]-{\pi\over2},{\pi\over2}\bigr[\>$ is the angle between this line and the $x$-axis. The graph of a general function $f$ is a curve with varying tangents. These tangents also have slopes, that can be computed through differentiation of $f$.

Now in complex analysis the $(x,y)$-plane is used for a completely different purpose, namely as a "geometric" representation of the set ${\mathbb C}$ of complex numbers. Each point $(x,y)\in{\mathbb R}^2$ then represents the complex number $z:=x+iy$. Sometimes we do geometry with these complex numbers, and there will be figures with lines $\ell\subset{\mathbb C}$. These lines are ordinary real lines in the plane, and they have a real slope as described above.

When you start to study complex valued functions $z\mapsto w:=f(z)$ then you need a second plane (the $w$-plane) to draw the images of individual points or subsets of the $z$-plane. It is there that something as a "complex slope" arises. The official name is complex derivative. We then have $\Delta w\approx f'(z_0)\Delta z$ in the same way as in the "real world" we have $\Delta y\approx f'(x_0)\Delta x$.

Answering your comment: The quotient $q:=(z_2-z_1)/(\bar z_2-\bar z_1)$ is related to the slope of the segment connecting $A$ with $B$, but I have never seen it called "complex slope". In fact $q=e^{2i\alpha}$, where $\alpha$ is the angle described in the first paragraph. (Note that $z=|z|e^{i\alpha}$ implies ${z\over\bar z}=e^{2i\alpha}$.)

Answer 2

The Argand plane is just a way of visualising complex numbers that is often useful. It does not have to appear in real life. Simple graphs involving real numbers, e.g. $y = x^2$ are common in maths and their usefulness is rarely questioned. Nonetheless, you don't see them while out and about in the real world.

You could ask what is the use of complex numbers in the real world but I am sure that has been asked many times before. Here are two brief answers.

They don't need to be useful to be interesting. Mathematicians study many things regardless of their usefulness e.g. uncountable infinities.

Physicists find complex numbers very useful. I won't attempt to answer that as many have done so well already.

I would recommend The Road To Reality by Roger Penrose. It is a great for a fairly serious amateur. It will explain the the utility of complex numbers. It also has a chapter on infinity. Penrose seems to find it surprising that uncountable infinities are not useful as so much of maths is useful.

I think that the terms "imaginary" and "complex" are unfortunate and unnecessarily scary. When developing numbers, the step from the rationals to the reals is the most complicated yet few people worry about that. Comparatively, from the reals to the complex is simple. I blame calculators which are always happy to tell give you $\sqrt2$ and many will tell you $\pi$ but few will give you $\sqrt{-1}$.

Addition

Electrical engineering is maybe one of the simplest real world uses of complex numbers. You might know Ohm's Law: $V = IR$. This works with simple resistors. Now, consider A/C circuits with capacitors and inductors as well as resistors. It is possible to assign imaginary resistances to capacitors and inductors. Combinations of resistors, conductors, and inductors may have more general complex values. This is normally called "impedance" and represented with $Z$ rather than $R$. So, $Z$ might indicate a complex slope in a graph of $V$ against $I$. The real world interpretation of this is that not only might the current have a different amplitude (height of the A/C wave) but also a different phase (the peaks would not be in line).