Is the complex plane meaningful?

[1MileCrash]

I'm not sure I understand the complex plane very well.

For the cartesian plane, or other planes such as polar, points are plotted by a function. One value of x coresponds to a value of y. (or r to theta, or whatever.)

The complex plane isn't a plot of functions, just of a single number, it's imaginary part and it's real part.

Is the complex plane just a choice of how we want to graphically represent a complex number? Or does it have more innate meaning?

Do identities such as euler's completely rely on this graphical representation of a complex number to make sense?

 

[homeomorphic]

The complex plane is a way to picture complex numbers. Alternative, you could take the viewpoint that complex numbers are useful for studying plane geometry. It turns out any similarity can be achieve by a rotation, a dilation, and a translation. So, complex numbers give you all the similarities of the plane, and similarity is sort of what plane geometry is concerned with (or maybe congruence, depending on what you want). The fact that the complex numbers are a plane, rather than just numbers with two parts is useful in other situations, at least conceptually. But formally, there's really no difference between numbers with two parts and a plane.

If you want to "plot" a complex function, you need a domain and a range, so that means you have to have two complex planes. You can't really draw a graph of it, but you can think of it as a mapping from one plane to another.

Euler's equation makes sense whether or not you draw pictures of the complex plane or not, but the pictures can shed some light on it. It's basically a conceptual thing. It doesn't really affect the math, formally, whether you want to think of it as numbers or a plane. The math doesn't care. But people care because it helps to be able to picture things.

 

[AlephZero]

Is the complex plane just a choice of how we want to graphically represent a complex number? Or does it have more innate meaning?

It's a useful choice of how to graphically represent complex numbers, because for example addition and multiplication of complex numbers are equivalent to simple geometrical operations.

It has a many uses in advanced math, for example calculus involving functions of complex variables, and in engineering topics like signal processing and vibration analysis.

 

[Lukas_23]

There is no other way to visualy represent a complex number. A line has ordering while complex numbers have not, so a plane is the only choice.

The complex plane provides a challenge when trying to visualize geometrically complex functions because they occupy a 4 dimensional space.

It is also the battlefield of one of the main endeavours in math, conquering mount Riemann.

 

[Deveno]

I'm not sure I understand the complex plane very well.

For the cartesian plane, or other planes such as polar, points are plotted by a function. One value of x coresponds to a value of y. (or r to theta, or whatever.)

the plane is not the plot. the plot (or graph of a function) is a subset of the plane, not all of it.

The complex plane isn't a plot of functions, just of a single number, it's imaginary part and it's real part.

one can specify regions of the complex plane as functions of the real part of a complex number. but the fact that complex numbers have two pieces of information attached to them (rather than just one, as with a real number) gives us more flexibility in defining regions of the plane. rather than defining a unit circle as {(x,y) : x²+y² = 1}, we can say {z : |z| = 1}, which is obviously a more compact definition, and more intuitively expresses a unit circle as "all points of distance 1 from the origin".

Is the complex plane just a choice of how we want to graphically represent a complex number? Or does it have more innate meaning?

i'm not sure what you're asking here. as a (real) vector space, the complex plane is identical to R². but we have a way to "multiply" vectors in C, that we don't have for other kinds of vectors, in general. it turns out that multiplying by a point (complex number) on the unit circle, corresponds to rotating by the angle that point (vector) makes with the x-axis. this explains why we have the (non-intuitive) fact that a negative real times a negative real is a positive real, multiplying by -1 is the same as rotating 180 degrees (an about-face). it is perhaps a happy accident that complex numbers have such rich geometrical interpretations, but we would be fools not to exploit that.

Do identities such as euler's completely rely on this graphical representation of a complex number to make sense?

no. but they do give us an idea as "how to guess the behavior" of complex numbers...we can draw pictures! all of what we know about complex numbers, could just be formulated algebraically, without ever referring to a plane at all. but pictures help, we can talk about complex numbers using "spatial" concepts: closeness, "in-line" (collinear), angle, direction, distance. we can imagine boundaries of sets of complex numbers as acting like fences in a field, or boundaries on a political map, and use our every-day intuitions of these things, to help see the internal structure of complex numbers. don't you think that might be useful?