I Existence of experience related to i

[Cantor080]

If we have solution of an equation as x=1, it may be expressing, depending on context, 1 apple, 1 excess certain thing, etc. And, if we have solution of an equation as x=-1, it may be expressing, depending on context, 1 deficient apple, 1 deficient certain thing, etc. Is there any experience, similarly, for x=i?

 

[BWV]

One cannot have physical quantity of an irrational number either, only rational numbers can be counted

 

[Cantor080]

One cannot have physical quantity of an irrational number either, only rational numbers can be counted

Of irrational number? Though yet not clear to me, it seems that we can give physical quantity of irrational length.

 

[fresh_42]

If we have solution of an equation as x=1, it may be expressing, depending on context, 1 apple, 1 excess certain thing, etc. And, if we have solution of an equation as x=-1, it may be expressing, depending on context, 1 deficient apple, 1 deficient certain thing, etc. Is there any experience, similarly, for x=i?

This question is actually thousands of years old, only that in ancient times, people asked:
We have a square of length $1$, but is there an expression for its diagonal? We cannot tell how many parts of $1$ are needed to draw it without the square! We have a ratio between circumference $C$ and diameter $D$ of a circle, but we cannot draw a line of this length!

Kronecker once said "God made the integers, all else is the work of man."

So the question about the imaginary unit is in the same realm. It solves our equation $x^2+1=0$ which makes it useful to operate with. Same as $\sqrt{2}$ solved $1^2+ 1^2=x^2$ and $\pi$ solved $x = C/D$. People always simply bought the tools they needed to accomplish a task. $\sqrt{2}$ was as imaginary to the ancient Greeks, as $i$ was two thousand years later to our ancestors. They even called it not rational, same as we use imaginary.

 

[fresh_42]

Of irrational number? Though yet not clear to me, it seems that we can give physical quantity of irrational length.

Well, you could as well do for $i$. Have a closer look into physics: $i\cdot\hbar$ is all over the place. Only that we do not draw geometric figures with sticks in sandpits anymore. Things have evolved ever since.

 

[BWV]

Of irrational number? Though yet not clear to me, it seems that we can give physical quantity of irrational length.

You cannot measure an irrational length because ultimately you get to quantum mechanical limits on the resolution of your ruler

the geometric objects that generate irrational or transcendental numbers like circles or right triangles are idealizations that don't exist in the physical world

but if you allow these idealizations then rotations around a unit circle encompass both the imaginary unit, pi and e

 

[PeroK]

If we have solution of an equation as x=1, it may be expressing, depending on context, 1 apple, 1 excess certain thing, etc. And, if we have solution of an equation as x=-1, it may be expressing, depending on context, 1 deficient apple, 1 deficient certain thing, etc. Is there any experience, similarly, for x=i?

There's a good series of videos starting here on the history and "reality" of imaginary numbers:

 

[jbriggs444]

You cannot measure an irrational length because ultimately you get to quantum mechanical limits on the resolution of your ruler

If you measure an object and obtain a rational result, what leads you to believe that the measured attribute had a rational value in the first place. Or even a well-defined value.

 

[BWV]

Every ‘true’ length is some number of whatever the appropriate quantum unit is (Planck length?) so you can’t have an irrational length because the length is some countable number of units

 

[jbriggs444]

Every ‘true’ length is some number of whatever the appropriate quantum unit is (Planck length?) so you can’t have an irrational length because the length is some countable number of units

That is a commonly held, but incorrect belief. The Plank length is not a minimum length.

 

[PeroK]

Every ‘true’ length is some number of whatever the appropriate quantum unit is (Planck length?) so you can’t have an irrational length because the length is some countable number of units

One problem with that argument is what happens with length contraction? If an object has a length $L$ in one reference frame, and you demand that its length be a rational number, then its length in another frame is $L/\gamma$. And, it's difficult to demand that all gamma factors are rational numbers.

Lengths, therefore, in the sense of the mathematical model must be irrational. Lengths, in the sense of what you get if you measure something are limited by your experimental apparatus, which generally can only give one of a finite set of results. The raw numbers must be rational, but calcuated lengths, such as the distance if you have rational x and y coordinates can still be irrational (or even transendental).

A good example from QM is the energy levels of a finite square well, which are the solutions of a transendental equation: effectively $z = \tan z$.

 

[StoneTemplePython]

Every ‘true’ length is some number of whatever the appropriate quantum unit is (Planck length?) so you can’t have an irrational length because the length is some countable number of units

another problem: algebraic numbers are countable.

 

[pbuk]

They even called it not rational, same as we use imaginary.

I think this is confusing becuase of the multiple meanings of 'rational' and the correlation of imaginary and 'not real' in the English Language. In particular, 'irrational' and 'imaginary' have similar meanings in a sense which is entirely coincidental to their mathematical meanings and the etymology of these.

To be clear, when the Greeks called a number 'not rational' they meant that it 'cannnot be expressed by a ratio'. They did not mean that it is 'weird', or illogical. When we call a number 'imaginary' we mean that it 'cannot be expressed as a real number'. Again we don't mean that it is 'weird', or illogical - although in this case there is something weird about numbers with an imaginary part.

Or is this something that only I have been confused by?

 

[fresh_42]

I think this is confusing becuase of the multiple meanings of 'rational' and the correlation of imaginary and 'not real' in the English Language. In particular, 'irrational' and 'imaginary' have similar meanings in a sense which is entirely coincidental to their mathematical meanings and the etymology of these.

Not sure, whether this can be called a coincidence. One has to have a closer look at the original meaning, as ..

To be clear, when the Greeks called a number 'not rational' they meant that it 'cannnot be expressed by a ratio'. They did not mean that it is 'weird', or illogical.

Agreed.

When we call a number 'imaginary' we mean that it 'cannot be expressed as a real number'. Again we don't mean that it is 'weird', or illogical ...

Agreed.

... - although in this case there is something weird about numbers with an imaginary part.

Not to me, I mean, the weirdness. The numbers themselves are completely natural to me. It is a simple field extension necessary to solve an equation. Weird are theorems like Liouville's or Cauchy's, or the branching of functions.

Or is this something that only I have been confused by?

 

[pbuk]

One cannot have physical quantity of an irrational number either, only rational numbers can be counted

But √2, for instance, can easily be visualised as the length of the diagonal of the unit square. Such a visualisation for imaginary (or complex) numbers is not so easy - when you map complex numbers to the plane, you are creating an analogy, not a visualisation.

 

[BWV]

But a unit square is an idealization that does not exist in the physical world any more than the imaginary plane

Most all math concepts are abstractions to some degree

 

[RPinPA]

To answer the original question, I've spent much of my professional life working with communications signals, which are frequently described in terms of complex numbers. Relative to the carrier wave (a sine wave at the broadcast frequency), at any given time the signal has a magnitude and phase. There is usually circuitry to measure the real and imaginary components of that signal. They have real physical meaning. You can encode completely different information in the real part and imaginary part at the transmit end and recover it at the receive end.

Of course, the underlying signal is made of electric and magnetic fields which are real-valued. The "imaginary part" could also be represented as the cosine terms in a real-valued Fourier series and the "real part" as the sine terms. But it is far more common to represent the two orthogonal information streams as a complex number.

So the short answer to the original question of "does $x = i$ have a physical meaning" in the communications context is yes, it means a signal which is 90 degrees out of phase with the carrier wave.