# Complex numbers and physics

• physior
In summary, complex numbers were introduced because the real number system had certain limitations, they simplify certain mathematical operations, and they are used in physics for a variety of reasons.

#### physior

hello

can you tell me please why we introduced complex numbers? what was the problem that we couldn't express with rest of algebra and we introduced complex numbers?

I am basically interested in why we introduced complex number to describe and analyze AC circuits, like voltage, current and impedance.

Also, I would like to make clear why we need numbers associated with the square root of -1.

please keep it simple, no knowledge considered granted!

thanks!

Complex numbers were introduced because the set of real numbers had certain limitations. For example, there were no solutions to equations such as x^2 +1=0. The introduction of complex numbers meant that you could work with equations that had imaginary solutions .

why would you need to solve the equation such as x^2 +1=0 ?
what natural phenomenon does this question describe?

physior said:
I am basically interested in why we introduced complex number to describe and analyze AC circuits, like voltage, current and impedance.

In this case, it's purely a mathematical convenience. In principle, we can always use real numbers, e.g. ##V = V_0 \cos (\omega t + \phi_0)##. However, it simplifies certain mathematical operations to work with the complex form ##V = V_0 e^{i(\omega t + \phi_0)}## (of which the actual V is the "real part"), and then take the "real part" of the final solution. It's easier to do certain algebraic procedures with exponentials than with trigonometric functions.

physior said:
why would you need to solve the equation such as x^2 +1=0 ?
what natural phenomenon does this question describe?
It's not a matter of relating the solution of a certain equation to some natural phenomenon. That's why mathematicians were very reluctant to work with complex numbers for hundreds of years, even though it was recognized that certain equations had solutions which could only be expressed by complex numbers.

You might as well ask why we want to find the solution to equations which can only be expressed using irrational numbers. That was a big no no for the early Pythagoreans. Even though they knew that irrationals could serve as solutions to certain equations, the Pythagoreans tried to keep this knowledge secret.

Although there are mathematical reasons for using complex numbers, they find plenty of use in physics besides AC circuits. Complex numbers are used in subjects like aerodynamics and fluid mechanics, analyzing harmonic functions, etc.

Roger Penrose has a very good discussion of the matter in his book "Road to Reality".

I had the same question when I was younger. I thought of imaginary numbers as highly artificial and unnecessary, since we could always reformulate our laws to not use imaginary or complex numbers. And this bothered me, for some reason. Well, now I still think they are artificial, but it doesn't bother me anymore, since I realize that all our mathematics are just tools for understanding reality and it doesn't matter if a tool is natural as long as it works, and complex numbers work quite well for certain things.

Another reason I find: we represent the steady state solution of an AC circuit using phasors. Phasors have a y-component and an x-component. Complex algebra allows us to actually perform calculations on these phasors.

I have long had the same question.

This is the first time I've heard anyone suggest that they are a convenience - a mathematical shorthand - rather than an faithful modeling of some aspect of reality.

If true, this would go a long way toward my intuitive acceptance of them.

jtbell said:
In this case, it's purely a mathematical convenience. In principle, we can always use real numbers, e.g. ##V = V_0 \cos (\omega t + \phi_0)##. Howeve it simplifies certain mathematical operations to work with the complex form ##V = V_0 e^{i(\omega t + \phi_0)}## (of which the actual V is the "real part"), and then take the "real part" of the final solution. It's easier to do certain algebraic procedures with exponentials than with trigonometric functions.
Are you saying that, literally, phenomena that we model with complex numbers could instead use the above (much more cumbersome formula fragment)? Can you elaborate on the formula a little?

SteamKing said:
...certain equations had solutions which could only be expressed by complex numbers.
Considering the above, can this be considered an overstatement?

DaveC426913 said:
a mathematical shorthand - rather than an faithful modeling of some aspect of reality.
In physics everything is mathematical modeling. But what is "faithful modeling"?

SteamKing said:
certain equations had solutions which could only be expressed by complex numbers.
DaveC426913 said:
Considering the above, can this be considered an overstatement?
I think you are confusing physics and math here.

I think the equation x2+1 = 0 would be an example of an equation whose solutions can be expressed only with complex numbers. There are doubtless many other such similar equations.

A.T. said:
In physics everything is mathematical modeling. But what is "faithful modeling"?
This refers to the question in the opening post: is there a real word example of a complex number? I'm going to conclude that there are lots of real world examples where jtbell's formula can be seen (i.e, if you deconstruct the actual phenomenon - such as electrical signals - piece by piece, you will see the V, the cos, the w and the theta emerge.)

I guess, what I'm dancing around, is calling complex numbers a mathematical trick/convenience.

A.T. said:
I think you are confusing physics and math here.
I don't see why. Unless I misunderstand what jtbell is saying.

Could all equations not be solved (in principle) using the formula jtbell posted, instead of using complex numbers?

For equations that actually model physical processes, you could always describe the *solution* with the sin/cos etc.
However, there are many non-physical equations that really only have a complex solution.

DaveC426913 said:
is there a real word example of a complex number?
Depends entirely on the definition of "real world". Is there a "real world" example of a negative number?

DaveC426913 said:
I guess, what I'm dancing around, is calling complex numbers a mathematical trick/convenience.
Like all of math?

DaveC426913 said:
Could all equations not be solved (in principle) using the formula jtbell posted, instead of using complex numbers?
He talked about modeling (physics) nor solving all equations (math).

A.T. said:
Depends entirely on the definition of "real world". Is there a "real world" example of a negative number?
Yes. This crate's oranges, less that crate's oranges is -1.

I realize this is semantic. It comes down to what one is "used to" working with - what one accepts as a comprehensible modeling of reality. I grok how negative numbers illuminate reality for me.
A.T. said:
He talked about modeling (physics) nor solving all equations (math).
Yes. Still, my question stands.

Someone pointed out a possible analogy: irrational numbers. I believe I could do all forms of math without resorting to irrational numbers, could I not? I could use fractions, though it might make the calculations onerous.

From a maths perspective I would say complex numbers are numbers like any other. You could argue that negative numbers are in a way " imaginary", then irrational numbers are defined at the limit of sequences of rationals. Note that you cannot make a physical measurement that results in an irrational number. So you could ask why physics needs those.

Complex numbers are just the next natural extension.

As complex numbers can be represented as points in an x-y plane, what reason is there to doubt their relevance to the real world? Why shouldn't numbers be two dimensional?

No one argues against using matrices, but if we called a 3 x 3 matrix a "higher dimensional" number, then perhaps some might take issue with them.

DaveC426913 said:
Someone pointed out a possible analogy: irrational numbers. I believe I could do all forms of math without resorting to irrational numbers, could I not? I could use fractions, though it might make the calculations onerous.

Fractions can provide only an approximate representation of an irrational number, hence, that's why they are called 'irrational' (i.e., not capable of being expressed by a simple ratio). Ditto for the so-called transcendental numbers (π, e, etc.)

You could calculate approximately without resorting to transcendentals or irrationals or complex numbers, but what would be the point?

DaveC426913 said:
I grok how negative numbers illuminate reality for me.

Well, complex numbers have the geometrical explanation of the "complex plane". Is that not as intuitive as negative numbers?

I will elaborate on the issue of using complex algebra in circuit analysis. I'll focus in particular on an RLC circuit (that is, a circuit with a resistor, an inductor, and a capacitor in series).

We can represent the source voltage, the voltage across each element, and the current as sinusoids. We can write a loop law around the circuit to find the current, and it will look as follows:

$V_m \cos(\omega t + \theta) = \frac{1}{C} \int _0 ^t i dt + iR + L \frac{d i}{dt}$

This has three parts to the solution: two exponential parts and a sinusoidal part. Fortunately, as t grows, the exponential parts become negligible, and we are left with a sinusoid. Thus, we can represent the current as a sinusoid provided we are considering times a sufficient distance from t = 0. Now, sometimes we don't care about the current immediately after the switch closes. We only care about the current after this. We call this the steady state solution.

Now, calculating this sinusoid can be difficult and time consuming, so is there an easier way?

That's where complex numbers come in. Instead of imagining a sinusoid so wave as the voltages and currents, let's just imagine a vector. This will rely on two things: 1. We can consider this to be the same as a sinusoid in that it has a magnitude and a phase (angle). It will rotate around, forming a circle. 2. The frequencies of everything will be the same.

If we have these two conditions, then there's no problem with treating this all as a vector that rotates (called a phasor). That is, when all we care about is the steady state solution, it's much easier to represent everything as mo in vectors. It's not even necessary to take time into account! Since we assume everything has the same frequency, all we need to worry about is the magnitude and the phase. So do we NEED to use complex numbers in circuits? Not in this case. We could simply add the phasors (using a set of rules for how to determine each phasor) just like we add vectors. But complex algebra allows us to do algebra on the x and y axes. It's the same thing as breaking a regular vector up into its x and y components. We use complex numbers because it makes it easier to do algebra on them. It's much easier to add two complex numbers together and find their magnitude and angle than it is to combine two sinusoids into one.

This is particularly useful because capacitors and inductors like to shift the phase angle of the current.

In other words, we don't have to use them. But it makes it easier to work in two dimensions. Given their relationship to sinusoidal functions, they're just another mathematical tool to be able to represent phenomena more easily.

It's basically a matter of opinion if complex numbers are "real" or not since nobody knows what "real" means outside of the mathematical context. I suppose there's three camps of thought: the first considers mathematics to be just a tool that we use to describe reality; the second sees all mathematics as existing in some abstract reality; the third sees some portions of math as real (like the whole numbers) and others as artificial. It's a philosophical question with no bearing on physics.

Then we have to distinguish between our conventional language used for complex numbers (a+ib (or a+jb if you are an engineer)) and the abstract concept. The latter says if it acts like a duck and quacks like a duck, it's a duck. Check out the matrix representation of complex numbers here: http://www.sosmath.com/matrix/complex/complex.html
These matrices are complex numbers in the sense that if you wrap each of these matrices into a box and call it a number, the algebra matches that of complex numbers. It is definitely possible to write any laws of physics without using the conventional language of complex numbers, but it might not be possible to write our laws of physics without accidentally incorporating the abstract concept of the complex number into the equations somehow.

PeroK
DaveC426913 said:
Yes. This crate's oranges, less that crate's oranges is -1.
So, how do those negative oranges taste?

DaveC426913 said:
I grok how negative numbers illuminate reality for me.
Your orange example introduces an operation (subtraction) to yield a result outside of positive numbers. So you introduce negative numbers. The operation sqaure root yields results outside of real numbers. So you introduce imaginary numbers.

DaveC426913 said:
I believe I could do all forms of math without resorting to irrational numbers, could I not? I could use fractions, though it might make the calculations onerous.
How would you express the ratio of circumference to diameter as a rational number?

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DaveC426913 said:
Someone pointed out a possible analogy: irrational numbers. I believe I could do all forms of math without resorting to irrational numbers, could I not? I could use fractions, though it might make the calculations onerous.

One problem is that you lose the square root function. Also sines and cosines. These functions cannot be defined on Q.

And e wouldn't exist, which is probably more of an issue than pi.

You can construct the real numbers from the rational numbers and the rational numbers from the whole numbers and the whole numbers from sets of sets, so in principle, you can do all mathematics with just sets.

It's easier to construct the real numbers and think of them as limits of a sequence than it is to try to avoid them when they show up.

But this ultimately comes down to how "real" math is, which does not have a satisfactory answer to everyone.

The whole discussion of "what is real" is unlikely to bear any more fruit than the last hundred times this has been discussed. If you want to ponder that question, I recommend "The Velveteen Rabbit" and maybe taking a philosophy class or three.

Complex numbers provide a way to replace (in certain, interesting, cases) a set of coupled differential equations in two variables with a single differential equation on one complex variable. This, in turn, allows for some very elegant solutions to certain problems - problems involving entirely real variables. I think that is a good rationale for why we use them.

M Quack
Could it be (loosely) analogous to the way we conventionally represent the geometry of virtual images when working with reflections?

I mean, when we represent an object as seen in a plane (or curved) mirror, we use geometry describing things that don't really exist (because there are no light rays behind the mirror), yet it is an excellent way of getting correct results, as long as we learn and obey the rules of virtual reflective geometry. (It would be possible to draw the entire diagram using angle-of-incidence/angle-of-reflection constructs, but why bother?)

I don't know. Is the image really real? Or merely real?

Arguments about whether this is real or not are not just pointless, they get in the way of understanding.

That looks like a good analogy. I will note that in my experience with circuits, complex numbers are just as "real" as real numbers. It helps to think of them as a number that shifts the phase of a vector (or phasor) by 90 degrees. If we can represent the current of a circuit as a complex number, assuming that it is really a sinusoid with the same frequency as the source, then it makes it much easier to calculate it. It's simple to find the magnitude and angle of a complex number. If I have a voltage $$v(t) = 5 \cos( \omega t)$$ then I can easily represent it as a vector that rotates with a certain frequency. We know that the other elements' voltage will rotate with that frequency as well, so all we care about is their relationship to each other. Thus, we can just write $v = 5$ at an angle of 0 degrees. Or as a complex number, V = 5. If we were to find that, as a complex number, i = 1 + j, then we can easily see that $i = \sqrt{2}$ at an angle of 45 degrees. Since we technically know the frequency, that means we know $$i(t) = \sqrt{2} \cos(\omega t + 45)$$ where 45 is in degrees. So we can easily transform everything to the complex plane, do some easy algebra, and convert everything back to sinusoids instead of trying to combine sinusoids from the beginning. Why complex numbers? In this case, it's just as possible to do this without complex numbers by just using vectors. Complex numbers just make it simpler. This is especially due to the fact that the impedance of an inductor is shifted up 90 degrees ($+j$) and for a capacitor, it's back 90 degrees ($-j$).

Quite a useful tool for simplifying calculations. It's as real as any mathematical tool is. Certainly we can never "carry a charge off to infinity", but that's just another example of a mathematical method that helps us model the world.

Ah. I am beginning to see that, once one is familiar with any new type of math tool, it becomes second-nature, just like happens with logs or trig or bases (such as binary).

But I do get asked the OP's question all the time. I think I will use my 'virtual image' analogy next time I describe it.

Thanks.

To me, it helps that one of the first things one learns in complex analysis is that a complex number is nothing more than an ordered pair (which can also be represented by a + bi). They're just ordered pairs that we know how to do operations with. Or from a physics viewpoint, it's almost like a unit vector that points upwards (with the exception of that little $i^2 = -1$ thing) if we assume that a number not multiplied by i points horizontally.

Complex numbers weren't invented with any specific application in mind. They were introduced to solve certain equations which had no real solutions. The fact that they are useful in physics was , I reckon, a coincidence. They weren't designed to be that way.

I recommend a very entertaining book An Imaginary Tale: The Story of i [the square root of minus one] By Paul J. Nahin

The science and math communities resisted the need for complex numbers for centuries. They had the same question as the OP. The book tells the story of how they were eventually persuaded.

Another book, QED: The Strange Theory of Light and Matter By Richard Feynman explains quantum mechanics using very clever graphics instead of complex numbers. Indeed instead of any number or equations. Studying Feynman's graphics makes it obvious that quantum mechanics can never be correctly described by just real arithmetic. Both the magnitude and phase are mandatory to get the correct answers.