Hacker News Reader

Comment by tehsauce | Hacker News Reader

“Complex numbers” are rather poorly named. They are more naturally understood as simply a vector which has a magnitude, can be rotated and scaled. As geometric objects they are much more intuitive. The subject geometry algebra takes a great approach of generalizing this idea and augmenting basic linear algebra to unify complex numbers and beyond (quaternions, ect) with geometric objects and operations. This also fits in nicely with group theory, which organizes all kinds of objects which also have the same properties as numbers.

View on HN

You miss a key part of complex numbers if you think of them as just vectors: they are a field.

Because they are separate but interacting with real numbers? I don’t understand.

Multiplying vectors differs from multiplying complex numbers.

He probably means the algebraic structure of a field. "A field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do."[0]

You might be tempted to think of complex numbers as "just" being 2-dimensional real vectors (x, y). Looks pretty similar to how you can plot a complex number a + ib at point (a, b) on a 2D plane. But importantly, division is defined on a field, which is not necessarily true for vectors. For any complex number (except 0), you can find another complex number that multiplies with it to give 1, the multiplicative identity.

You _can_ think of complex numbers as being "made of" real numbers though. a and b above are just real numbers. Complex numbers are the two-dimensional normed division algebra over the reals[1].

[0] https://en.wikipedia.org/wiki/Field_(mathematics) [1] https://ncatlab.org/nlab/show/normed+division+algebra

Exactly. But I want to point out that the field character of the complex numbers is not some incidental quality which happens to distinguish them from 2-vectors. Its absolutely essential to their mathematical character and usage and it also distinguishes them from other complexes we might want to form that behave in a real number like fashion. For instance, there is no way to form a field over the three vectors. In general, one has to give up more and more structure as the dimensions go up.

I think that the obsession with quantum mechanics containing complex number is a little overblown. Quantum Mechanics is fundamentally about a defining a formalism which preserves the ability to simultaneously keep track of the physical symmetries in a system and the probabilities of particular outcomes of measurement. In many situations complex numbers provide a useful way to do this because of the symmetries involved (eg spin 1/2) but in other situations other symmetry groups are required. The appearance of complex numbers is no more (or less, I suppose) mysterious than the appearance of SU(3) in nuclear physics or SU(2)xU(1) in electroweak physics. Its just a matter of what symmetries you have and how many outcomes a measurement can have (roughly).

A vector field?

It's a different concept [0], regrettably the word "field" is vastly overworked in math (and physics)

[0] https://en.wikipedia.org/wiki/Field_(mathematics)

And not just a field, but the algebraic closure of real numbers.

IMO there is nothing “natural” in interpreting complex numbers as a vector. The fact you get a thing out of them that looks a lot like a vector is one of the stupefying ‘mysteries’ of math which make the discipline so cool.

Complex numbers afaik began as an attempt to solve polynomial equations. They begin from the agreement to invent a number i whose square is -1 so you can solve equations having sqrt(-1) in them.

The jump from sqrt(-1) to plane rotations is to my feeble mind one of the most flabbergastingly unintuitive things in ‘basic’ maths. “A rotation you say? Who ordered that!?”

Complex analysis is so much more regular than real analysis differentiability over a two dimensional quantity is so much strong than over a one dimensional quantity that you have much stronger results. Basically, if you know an analytic function in a neighborhood you know it over the entire plane. Plus you have functions like e ^ ( 1 / z ) which is pretty amazing around zero.

While what you say is true, I could never intuitively grasp that properties of analytic functions. Like I could read and understand the proofs, as in follow one step to the next, but I could never succinctly describe, intuitively, why one should expect the proofs to hold. Even the most fundamental concepts in complex analysis are more like facts rather than logical deductions (to me).

Following a proof step-by-step != understanding the proof

Reply on news.ycombinator.com