Physical meaning of imaginary numbers

In summary: Or is it more fundamental, reflecting some physical reality?All numbers (except of naturals) are only abstracts used in abstract equations.
  • #1
vibhuav
43
0
Can someone give a physical meaning for imaginary numbers?

The imaginary numbers, in my opinion, are truly imaginary. What do they even represent? Irrational numbers are, well, preposterous but I can accept them. √2, π and φ have some tangible meaning, but √(-1)? What does it mean? A solution of x^2+1=0? But that equation itself is artificial, representing nothing physical, at least as of now. Complex numbers represent vectors and are useful as phasors in electrical engineering, electromagnetism and other fields. But that is all they are – a tool, a short hand notation to ease the mathematical calculations – and not really real. Phasors allay the complexity of calculations but even without them we could still do all the calculations, albeit in a convoluted way.

So do these imaginary numbers mean something in reality? Can someone give me an example? The closest real world counterparts of complex number I can think of are the probability amplitudes of quantum physics.
 
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  • #2
In antimatter based worlds, our imaginary numbers are real, and our real numbers are imaginary ;->
 
  • #3
The meaning depends on what you measure. The most commonly used measurements that I'm aware of that give complex numbers are that of electrical impedance, and various ways to measure waves.

Occasionally it's useful to measure position with complex numbers, such as in 2-D fluid flow problems.
 
  • #4
Hurkyl said:
The most commonly used measurements that I'm aware of that give complex numbers are that of electrical impedance, and various ways to measure waves.

Think of i as having the same value as "1" but in a different axis of measurement. If you find it helpful, imagine a number line; all expressible real numbers fall on this line. However, a number that departs from this line might have the same "real" component as another number but include an imaginary component. This could be the differnence between "5" and "5+2i."

Like Hurkyl mentioned, how that is interpreted in real-world applications could differ fundamentally. In electrical engineering, we use j instead of i and we use it to connote a complex effective impedance of a load.
 
  • #5
vibhuav said:
√2, π and φ have some tangible meaning, but √(-1)? What does it mean? A solution of x^2+1=0?
Well - so what is the physical meaning of π ?
I may agree that natural numbers have direct meaning (no rabbits, one rabbit, two rabbits, three rabbits, plenty of rabbits). Maybe positive rationals could also be accepted (the rabbit weights 2.5 pound).
But all the rest? What is the "physical meaning" of -3 ? Of 3/4? Or of π ?
All numbers (except of naturals) are only abstracts used in abstract equations. If your answer to the meaning of -3 is that I may walk 3 steps forward, then 3 steps back (-3 steps) - then you should accept that walking 3 teps to the left may be noted as 3i steps, and 3 steps right as -3i steps.
 
  • #6
Of course, as you suggest... mathematics [specifically, complex numbers] are tools of science.

There is a way to deal with only real numbers if you are willing to use matrices.
Let Z=aI+bJ, where a and b are real and
I is the 2x2 identity matrix and J is the matrix [itex]\left(\begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} \right) [/itex].
You can find the analogue of complex arithmetic operations in terms of matrix operations.
Do you have a problem with matrices?

In the grand scheme of things... this is probably an indication that physical quantities are not just "the counting numbers"... not just scalars... but more complicated objects reflecting symmetries or other structure. Thus, we find it convenient to use vectors, matrices, spinors, complex numbers, quaternions, ...
vibhuav said:
Irrational numbers are, well, preposterous but I can accept them.

There are actually more irrational numbers than rational numbers.
 
  • #7
xts said:
Well - so what is the physical meaning of π ?
I may agree that natural numbers have direct meaning (no rabbits, one rabbit, two rabbits, three rabbits, plenty of rabbits). Maybe positive rationals could also be accepted (the rabbit weights 2.5 pound).
But all the rest? What is the "physical meaning" of -3 ? Of 3/4? Or of π ?
All numbers (except of naturals) are only abstracts used in abstract equations.


If your answer to the meaning of -3 is that I may walk 3 steps forward, then 3 steps back (-3 steps) - then you should accept that walking 3 teps to the left may be noted as 3i steps, and 3 steps right as -3i steps.

π is the ratio of the circumference to the diameter of a circle that I can draw on sand.
3/4 is the weight of iron (in kilogram) that is 0.75kg.
-3: can't think of physical meaning, but maybe there is one. If you interpret -3 as going backwards 3 steps, then -3 becomes only an aid to do calculations.
This is the problem I am having with i. Is i simply a tool, or is it real?
 
  • #8
vibhuav said:
This is the problem I am having with i. Is i simply a tool, or is it real?

I don't actually see the issue. Mathematics in general is simply a tool to help people understand reality, and in some sense the only mathematical reality is integers, things you can count - the rest is tools. Imaginary numbers are another tool, and can be critically important in some circumstances - roots of equations for example can be imaginary, and those roots can have real-number physical consequences even if you wish to scoff at the reality of the square root of negative one.
 
  • #9
JeffKoch said:
the only mathematical reality is integers, things you can count - the rest is tools
The things that you count are reality. The integers you use to count them are also tools.
 
  • #10
vibhuav said:
π is the ratio of the circumference to the diameter of a circle that I can draw on sand.
False. If you draw it on sand and measure it with a stick the ratio is probably 30:10. Maybe 31:10... But definitely not π. Your measurement has limited precission. So the measured ratio must be a rational number. π is an idealisation you use, as you were smart enough to understand Euclid's abstract view of geometry.


-3: can't think of physical meaning, but maybe there is one.
Try to find intuitive one - better than counting steps back as negative.

Is i simply a tool, or is it real?
It is definitely a good tool - like all other kinds of numbers and the whole mathematics.
And it is not real - it is imaginary :wink: But it may be used to describe real processes - and it is almost equally applicable for that purpose as real numbers.
 
  • #11
  • #12
vibhuav said:
Complex numbers represent vectors and are useful as phasors in electrical engineering, electromagnetism and other fields. But that is all they are – a tool, a short hand notation to ease the mathematical calculations – and not really real.
How does that differ in any substantive way from real numbers?

The real numbers are a set of mathematical abstractions that follow certain rules and transformations. These rules and transformations are useful tools for calculating the predicted results of certain physical experiments.

Similarly with complex numbers. The fact that they are called "imaginary" is merely a naming convention and not a reflection of their ontological status.
 
  • #13
xts said:
Your measurement has limited precission. So the measured ratio must be a rational number.
If you have limited precision, you don't have enough precision to measure a rational number. :tongue:

Measuring devices measure what they measure -- and interpolation is one of the ways a ruler is used, and I can certainly interpolate to [itex]\pi[/itex] if I wanted to. I could even get a ruler that has a marking of [itex]\pi[/itex] centimeters. (I have seen clocks with such markings)


It's good to remind people that their measurements have limited precision. It's bad to invoke limited precision to justify sacrificing accuracy to replace a measurement with a rational approximation, or to pedagogically cripple yourself into avoiding irrational numbers.
 
  • #14
DaleSpam said:
The fact that they are called "imaginary" is merely a naming convention and not a reflection of their ontological status.
List of bad naming choices, that will continue to stir up misguided philosophical debates for ages to come:

- real vs. imaginary numbers
- real vs. fictitious forces
- (intrinsic) curvature

Feel free to extend
 
  • #15
vibhuav said:
Can someone give a physical meaning for imaginary numbers?

The imaginary numbers, in my opinion, are truly imaginary. What do they even represent? Irrational numbers are, well, preposterous but I can accept them. √2, π and φ have some tangible meaning, but √(-1)? What does it mean? A solution of x^2+1=0? But that equation itself is artificial, representing nothing physical, at least as of now. Complex numbers represent vectors and are useful as phasors in electrical engineering, electromagnetism and other fields. But that is all they are – a tool, a short hand notation to ease the mathematical calculations – and not really real. Phasors allay the complexity of calculations but even without them we could still do all the calculations, albeit in a convoluted way.

So do these imaginary numbers mean something in reality? Can someone give me an example? The closest real world counterparts of complex number I can think of are the probability amplitudes of quantum physics.

Multiplying by the number i rotates the plane pi/2 radians (90 degrees) counterclockwise. If you start at (1,0) in the plane and multiply by i then multiply by i again, where to you end up? At (-1, 0). So i^2 = -1. It's really that simple.
 
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  • #16
A.T. said:
List of bad naming choices, that will continue to stir up misguided philosophical debates for ages to come:

- real vs. imaginary numbers
- real vs. fictitious forces
- (intrinsic) curvature

Feel free to extend

action force - reaction force
Guess which one comes first.
 
  • #17
Ruler with pi mark? I envy you... I only have slide rule with pi, e, and roots of small integers on its logarithmic scale.

Anyway - I insist that we may only measure natural numbers. Even rational measures are secondary to them and conventional (I may say that I am 1.83m tall, but it is derived from 183 cm).
Egyptians used measuring rod and rope with equidistant knots, then they had to count how many rods (not: how much!) they had to mark along the measured distance. More precise measurement could be done with smaller (12 times shorter) rod - but then we again had integer number of 'short rods'.

Real numbers are only 18th century (pre-atomic, pre-quantum, pre-information-theory) idealisation of continuous behaviour of the Nature and are not measureable by any means.

In modern times of digital apparata such 'natural number' measurement is even more apparent.
 
  • #18
A.T. said:
List of bad naming choices, that will continue to stir up misguided philosophical debates for ages to come:

- real vs. imaginary numbers
- real vs. fictitious forces
- (intrinsic) curvature

Feel free to extend

bp_psy said:
action force - reaction force
Oh yeah, one of my favorites.

bp_psy said:
Guess which one comes first.
And which causes which.
 
  • #19
xts said:
Anyway - I insist that we may only measure natural numbers. Even rational measures are secondary to them and conventional (I may say that I am 1.83m tall, but it is derived from 183 cm).

I don't see how that's really different from using irrational numbers. To mirror your reasoning, I may say the ratio of a circle's circumference to its diameter is 3.1415..., but perhaps what I mean is, I made a measurement using a ruler with an infinite number of infinitesimal marks, and counted 3.1415...x10^inf of them.

Similarly, I might have a ruler with every ith unit marked off, and thereby measure something to be 2+3i.
 
  • #20
A.T. said:
List of bad naming choices,
My favourite: 'moment of momentum', 'moment' and 'momentum'
 
  • #21
Leveret said:
I don't see how that's really different from using irrational numbers. [...] I made a measurement using a ruler with an infinite number of infinitesimal marks
No, you can't have a ruler with infinite number of infinitesimal marks. That's what makes a difference.
You may measure the ratio as 3 (biblical), 3.14 (primary school or a tailor taking a measure) or 3.1415926 - making precise laboraty measurement. But you cannot measure it as pi.
Even if you have a ruler with mark for pi (like Hurkyl's one), you can't measure pi as something different than 3.14cm - so pi on such ruler is just a synonyme for 3.14. (rather 3.15 - 0.1mm is hard to achieve making marks on a ruler)
 
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  • #22
xts said:
Real numbers are only 18th century (pre-atomic, pre-quantum, pre-information-theory) idealisation of continuous behaviour of the Nature and are not measureable by any means.
The era you reference features the development of a rigorous foundation for real numbers, not the invention of the concept which comes straight from Euclidean geometry (although their utility has far surpassed that specific application).

In fact, from a formal point of view, elementary Euclidean geometry and elementary real number arithmetic are essentially identical theories.

And the notion of "continuum" originates with Euclidean geometry -- the only reason the term is relatively recent is because it's only recently that anyone bothered considering the alternative.
Of course, this is an aside -- the origin of the concept has no bearing on whether or not it's measurable, or that our well-tested physical theories posit that position and time are both continuums. And the question of whether a measurement results in an irrational number is mostly independent from the question of whether or not you believe you can analyze a measurement to the point of declaring it simple counting.

But I haven't found such analysis particularly convincing or useful. A length of 7 rods is not a natural number -- it is a natural number times the length of a rod.

And if I had a stick as well, I could make successive measurements to see that
  • 3 sticks is between 7 and 8 rods
  • 6 sticks is between 14 and 15 rods
  • 12 sticks is between 29 and 30 rods
  • ...
If you've followed other constructions of the real numbers, it shouldn't be too hard to see how this sort of aggregation of information is yet another model of the real numbers.
 
  • #23
And the notion of "continuum" originates with Euclidean geometry -- the only reason the term is relatively recent is because it's only recently that anyone bothered considering the alternative.

I do believe that 'continuum' appears in Dante, Swift and even as far back as ancient Indian (hindu?) writings/teachings. And bear in mind that our current number system originated in ancient India.
 
  • #24
I agree that real numbers are in fact equivalent to Euclidean geometry, 18th century was just the times when they started to be widely used in physical theories. Earlier theories - up to Newton - rarely used them, preferring geometrical view.

My (and also Democritus'...) point is that mathematics is only an idealisation of real world. Especially passing from series of finite fractions (rods/sticks) to a limit of real number, and from 'small steps' to 'infinitesimally small steps' are only tools making calculations easier, but are non-physical. In order to practically apply them to reality, we always must step back to finite fractions.

It was Democritus' polemics against geometry (already developed, although it was 100 years before Euclid) - you may divide an ideal line segment by halves, then each of those halves again, and again, and such ad infinitum. You can cut a piece of wire by halves, and again, and again, but finally you must stop dividing, having in hand single atom.

Such limitations constitute difference between 'reality' and 'mathematical idealisation'.
 
  • #25
Even accepting the premise that a ruler measures a natural number of whatever unit it measures, that does not imply that the physical concept of length is better represented by the natural numbers than by the real numbers.

The question isn't one of what kind of number is measured, but what mathematical object has properties that allow you to best predict the measurement. Since you can have two rulers whose markings are an irrational multiple of each other, and since you want your theory of physics to work the same for both rulers, length is better represented as a real number than as a natural number.

Similarly, other quantities are better represented as a complex number.
 
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  • #26
There are lots of good perspectives in the above, so I'll just add one more-- a very useful aspect of complex numbers, in addition to writing them as a + bi or with the matrices we saw above, is the rule a + bi = r ei*theta, where r = (a2 + b2)1/2 is like the length of a vector and theta = tan(b/a) is like its phase relative to the real axis. The advantage of this equivalent form for reporting complex numbers is that it reveals an important message-- when you multiply complex numbers like r ei*theta times R ei*Theta, you get rR ei(theta+Theta). So multiplying complex numbers gives the behavior of a kind of multiplication and addition at the same time-- the magnitudes r multiply, and the angles theta add. When this comes in handy is whenever we are dealing with periodic phenomena, because periodic phenomena give us a concept of magnitude (how much action there is) and phase (where in the cycle is the action currently).

In many situations, physics requires a kind of combination of two periodic actions that involve a combined magnitude, and a combined phase-- the classic example of this being when the time evolution over a time t+T can be interpreted as multiplying the independent time evolution over t by the independent time evolution over T (i.e., evolution over t is always the same influence, and same for T, so evolution over t+T is first one influence and then the other). Time evolution like that is naturally interpreted in terms of multiplying complex numbers-- if we multiply by the complex number we call U(t) and interpret that as a time evolution over t, we have the following properties: U(t+T) = U(t)U(T) = U(T)U(t) and U*(t)U(t) = a real number we can get from a measurement. Here U*(t) is U(t) except reversing the sign of i, which recognizes that the sign of i is not actually determined by the solution to x2 = 1.

The need for this kind of multiplication comes up most often in wave mechanics of all kinds, because waves are periodic systems that obey the above algebra of complex numbers. Here multipling the amplitude A of a wave by U(t) is interpreted as "evolving the wave amplitude over time t." Note that if we have this interpretation, there is a kind of ambiguity in the sign of i, such that we should imagine that U*(t) is also an equally good way to talk about the time evolution. If we let the amplitude A be complex, then A(t) = U(t)A spawns the same time evolution as A*(t) = U*(t)A*. If A corresponds to something measurable (say a wave height), then we must take real combinations of complex A, like (A+A*)/2. This just means that if A and its time derivative, the kind of "initial conditions" we generally need to do physics, are initially real, they will remain real, even if we treat them as complex numbers, because (A+A*)/2 evolves into (U(t)A+U*(t)A*)/2 which is still real.

This tells us that the tendency for our measurements to remain real numbers is simply that they started out that way-- embedding real numbers into complex numbers does not generalize the measurable behaviors we see, because of the way physics requires initial conditions in order to make predictions. However, it sure makes the algebra of time evolution much easier. Is this just a mathematical trick, or was this convenience trying to tell us something? It seems possibly the latter, because in quantum mechanics, the concept of indeterminacy is introduced, and with it, a need for the algebra of complex numbers as something more fundamental than just a mathematical convenience.
 
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  • #27
xts said:
My favourite: 'moment of momentum', 'moment' and 'momentum'
Yeah, but that is just similarity of terms, which can cause confusion. The ones I listed are even worse, because they use common words that already have a meaning, which inspires misunderstanding and pointless philosophical interpretations of the term.
 
  • #28
xts said:
If your answer to the meaning of -3 is that I may walk 3 steps forward, then 3 steps back (-3 steps) - then you should accept that walking 3 teps to the left may be noted as 3i steps, and 3 steps right as -3i steps.

Best example!

Since you can walk 3 steps in different directions, you need to come up with some type of mathematical system that allows you to work in more than one dimension. Complex numbers may be seen as an artificial system, but it's a system that allows us to perform mathematical calculations in 2 dimensions instead of just 1 - and that's a particularly useful thing in the real world.

(And, as others have mentioned, one shouldn't put to much emphasis on the labels. In this case, 'real' is just the label for one of the axes and 'imaginary' is just the label for the other axis.)
 
  • #29
@Hurkyl

I am somewhat concerned about using the term 'measure' in respect of imaginary numbers.

Measurement implies comparison, which is one thing you can't do with imaginary numbers since they do not possesses the(well) ordering property.

You can compare by saying 3 miles is greater than 2 miles but you cannot make such a judgement about, for instance, voltages (1.5+ 3j)v and (0.3 - 0.5j)v.

You can compare the real and imaginary parts separately - but you have to resort to two sets of reals to do this - as the OP has already said.

go well
 
  • #30
Studiot -- my assertion is that a worry of your sort is entirely artificial.

Whatever your opinions on what measurement 'actually is' and whatever analysis you do to try and break down a measurement into parts, it's not really relevant to the idea that you can quantify things with complex numbers, and that the end product of your measuring process is such a complex number.
 
  • #31
So what are you 'measuring' in my voltage examples and how do you use this 'measurement' ?

I am perfectly happy using complex numbers as representations of some physical property that does not possesses the ordering structure. But surely that is another way of putting the original poster's question?
 
  • #32
Studiot said:
So what are you 'measuring' in my voltage examples and how do you use this 'measurement' ?

I am perfectly happy using complex numbers as representations of some physical property that does not possesses the ordering structure. But surely that is another way of putting the original poster's question?

You're measuring two separate things - their magnitude and their direction. I guess that does present some ordering problems. You can't say a vector with a magnitude of 5 and a direction of 0 deg is 'greater' than a vector of 5 and a direction of 90 degrees, but you can definitely say they're not equal.

Or in your example, you're measuring their magnitude and their phase, which presents the same ordering problem.
 
  • #33
You can't say a vector with a magnitude of 5 and a direction of 0 deg is 'greater' than a vector of 5 and a direction of 90 degrees, but you can definitely say they're not equal.

That's exactly it.

To say 5 miles is greater than 3 miles is meaningful.

To compare 5mph south from Stockholm with 3mph west from Rome is meaningless.

By themselves both are valid statements and can be represented (although this would be unusual) in complex format. However they cannot be combined in any meaningful manner.

I think the OP was seeking better physical examples than mine, rather than platitudes.
 
  • #34
vibhuav said:
-3: can't think of physical meaning, but maybe there is one. If you interpret -3 as going backwards 3 steps, then -3 becomes only an aid to do calculations.
This is the problem I am having with i. Is i simply a tool, or is it real?

If you can accept that "-3" is "only an aid to do calculations", then you can just think of "i" the same way if you like.

As another post said, in many "practical" applications of complex numbers in science and engneering, you don't really NEED to use n complex numbers. You could replace them by 2n real numbers, but the complex numbers are a neat way to automatically keeping track of the "n" similar relationships between each pair of real numbers. In those situations, often there is no phyiscal difference between what the "real" and "imaginary" parts of the numbers mean, and it would make no difference if you swapped over the the real and imaginary parts of all the complex numbers, except that you would alos have to flip the sign of one of the real numbers in each pair. But that goes back to the reason for using complex numbers in the first place - they keep track of details like that without you having to think about them all the time.

For some purposes, numbers that are "even more imaginary" than complex number are useful. Google quaternions and octonions, for example.

The reason that the imaginary parts of complex numbers are call "imaginary" is just a historical accident. The different meanings of "real" and "imaginary" in ordinary English don't have any mathematical significance, and neither do the different meanings of "rational" and "irrational" in ordinary English.
 
  • #35
AlephZero said:
The different meanings of "real" and "imaginary" in ordinary English don't have any mathematical significance, and neither do the different meanings of "rational" and "irrational" in ordinary English.

"Rational" means that the number can be formed by the ratio of integers. "Irrational" means that it cannot be formed by the ratio of integers.

"Imaginary", though, does not mean what it sounds like. ;-)

BBB
 
<h2>1. What are imaginary numbers? </h2><p>Imaginary numbers are numbers that can be written in the form a+bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1.</p><h2>2. What is the physical meaning of imaginary numbers?</h2><p>Imaginary numbers do not have a direct physical interpretation, but they are used in various fields of science and mathematics to represent quantities that cannot be expressed using only real numbers. They are particularly useful in describing oscillations, such as in electrical circuits and waves.</p><h2>3. How do imaginary numbers relate to the real world?</h2><p>Imaginary numbers are often used in combination with real numbers to describe complex phenomena in the real world. For example, in physics, they are used to describe the behavior of quantum particles and in engineering, they are used to analyze electrical circuits and signals.</p><h2>4. Can imaginary numbers be visualized?</h2><p>Unlike real numbers, imaginary numbers cannot be represented on a traditional number line. However, they can be visualized on a complex plane, where the real numbers are represented on the horizontal axis and the imaginary numbers on the vertical axis.</p><h2>5. What is the significance of the imaginary unit, i?</h2><p>The imaginary unit, i, is a fundamental concept in mathematics and is essential for solving complex equations. It has numerous applications in science and engineering, and its use has led to groundbreaking discoveries and advancements in various fields.</p>

1. What are imaginary numbers?

Imaginary numbers are numbers that can be written in the form a+bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1.

2. What is the physical meaning of imaginary numbers?

Imaginary numbers do not have a direct physical interpretation, but they are used in various fields of science and mathematics to represent quantities that cannot be expressed using only real numbers. They are particularly useful in describing oscillations, such as in electrical circuits and waves.

3. How do imaginary numbers relate to the real world?

Imaginary numbers are often used in combination with real numbers to describe complex phenomena in the real world. For example, in physics, they are used to describe the behavior of quantum particles and in engineering, they are used to analyze electrical circuits and signals.

4. Can imaginary numbers be visualized?

Unlike real numbers, imaginary numbers cannot be represented on a traditional number line. However, they can be visualized on a complex plane, where the real numbers are represented on the horizontal axis and the imaginary numbers on the vertical axis.

5. What is the significance of the imaginary unit, i?

The imaginary unit, i, is a fundamental concept in mathematics and is essential for solving complex equations. It has numerous applications in science and engineering, and its use has led to groundbreaking discoveries and advancements in various fields.

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