Are complex numbers part of the real world

[Rothiemurchus]

are complex numbers part of the "real" world

The square root of -1 is used a lot in physics.
But how does it relate to what,I suspect,most people would regard
as the real world i.e real numbers (for example we speak of real
probabilities
and not imaginary probabilities - real probabilities are the "real"
world).
Complex numbers can be represented by two orthogonal axes on a sheet
of paper and so can real numbers.Since such representations are both
geometrical
entities,do complex numbers only relate to real numbers (and hence the
"real" world) in the context of geometry? And since general relativity
is a theory based on ideas of geometry, do complex numbers only relate
to the real
world in the context of general relativity i.e would an imaginary
probability seem reasonable in the theory of general relativity?

 

[matt grime]

Tell you what. Why don't you write out a definition of a real number? There is no such thing as a physical example of a real number. They're just a system that can be used to model certain things if you so choose. Same with the complex numbers, that's all. You're confusing the Real in Real number with the real in real world. They're not the same.

 

[Rothiemurchus]

Matt Grime:
They're just a system that can be used to model certain things if you so choose

This is not far from my own line of thinking that at some level complex numbers and real numbers are equivalent.This is why I metioned geometry instead of a sequence like
1,2,3 of which the square root of minus one does not seem to be an obvious constituent.

 

[NeutronStar]

Imaginary Numbers

To begin with forget about the term "real". Think instead in terms of "quantities" and ask whether complex numbers can be thought of as representing quantitative properties. I think that you'll find that complex numbers are just that,.. complex. They have what is called a "real" component and an "imaginary" component. Then what you need to ask is what do these different components represent quantitatively.

The best concrete example I can think of is an electric circuit that contains an inductor. A dynamic current flowing in such a circuit can be described by a complex number. The real part of that description refers to the electron current flow (or hole current flow if you're a semiconductor nut). The imaginary part of that description refers to the magnetic field associated with the inductor. What you lose in electron current you gain in magnetic field and vice versa.

So both the real part and the imaginary part of the complex number represent "real" quantities in the "real" world.

Think of it this way,… electron current in a circuit can only flow in two directions. They are described by the sign of the number that represents the quantity of current flow (the real part of the complex number). Current flow can never be less than zero (no current flow at all), but it can have a magnitude in either of two directions (+ or -) . So the real part of the complex number always represent a quantity of electron current greater than or equal to zero and the sign represents the direction that the current is moving.

However, when some of the electron current gets converted into magnetic field energy by the inductor we can't represent that by either positive or negative. We need a new "direction". The new direction is called i for "imaginary". This new direction can also be positive or negative. In other words, the magnetic field can either be growing (the positive imaginary direction) or collapsing (the negative imaginary direction). So the imaginary part of the complex number represents the quantity of magnetic field present as well as its dynamical state (positive or negative).

In this case the complex number isn't any more mysterious than any other number. It's really just a shorthand way of combining various coordinate systems and quantitative ideas.

I've found that no matter how abstractly we take this idea of complex numbers, if we stop to think about what we are doing, we can break down very abstract concepts into their intuitive counterparts. Assuming, of course, that we have any intuitive understanding to begin with of the concepts that we are working with.

It does appear that some mathematicians have absolutely no intuitive clue concerning the objects that they are working with. This can usually be revealed by simply asking them to offer an intuitive explanation of their project. When they start talking in axiomatic circles and can't reduce their idea to a simple intuitive explanation then you can rest assured that they, themselves, have absolutely no intuitive clue about what the heck they are doing.

Many mathematicians do indeed work entirely from this axiomatic approach. I personally like to keep the intuitive insight alive along the way. But then, I'm a scientist.

 

[Rothiemurchus]

Great explanation NeutronStar!
Also real numbers can be made from one complex number multiplied by another.
So in a sense every number is complex - it is just a question of how you want to represent that number e.g the number twenty can be 20
or (4 - 2i) (4 + 2i).
But it is easier just to write 20.

 

[selfAdjoint]

The important thing to remember about complex numbers is that they aren't just a pair of real numbers. They have additional structure. For example Euler's formula e^{i\theta} = cos\theta + isin\theta. Or complex conjugation, an involution that reduces to the identity for real numbers.

 

[Hurkyl]

I just want to emphasize what selfAdjoint said. There's already a notion of a "pair of real numbers", a thing quite literally called an ordered pair of real numbers, that we might write as (2, 3) as an example.

The complex numbers are usually used to describe things because the additional structure (usually their arithmetic or calculus) is relevant to the problem at hand.

 

[Gokul43201]

Just to play devil's advocate on this one...

Perhaps Rothiemurchus is using the fact that physically observable quantities can be written as hermitian operators ?

 

[matt grime]

Ah the intuition approach again!

Tell you what Neutron, why don't you explain why GR or SR is intuitively true? Why is it intuitively acceptable that the speed of light is constant? Many people would disagree with you there. In fact in the Theory Development they have. Your intuition is different from anyone else's, as is mine, as is everyone's.

And saying that it has been experimentally validated doesn't make it true, intuitively. Perhaps the experiments were flawed - they certainly disgree with the intuition of a great many people.

So, please stop criticizing (your interpretation of) maths for not being (your interpretation of) physics. Perhaps if you took the time to look up some of the very complicated bits of mathematics you'd see why it is *advisable* to adopt the idea encapsulated by:

Mathematical objects are their properties - their use is defined by their properties, the square root of two is a symbol that is taken to be a positive real number, but whose use is defined by the fact that when we see it squared we can replace it by 2.

If you don't believe me then by all means try and prove something in cobordisms intuitively.
Mathematicians do use intuition, though not quite in the sense you want us to, all the time: it's how one makes conjectures. Which then require proof.
Here are some interesting thoughts you may care to read:

www.dpmms.cam.ac.uk/~wtg10/equations.html