Real World Applications of Imaginary Numbers

[AzonicZeniths]

Something has been puzzling me...what is an imaginary number in real life? I know that engineers sometimes use it but how do they apply it to real world situations? How is it anything but a mathematical constant?

 

[Hurkyl]

Something has been puzzling me...what is an imaginary number in real life? I know that engineers sometimes use it but how do they apply it to real world situations? How is it anything but a mathematical constant?

The situation is no different than for any other number.

 

[AzonicZeniths]

Could you give me an example for how it is used for a real life problem (not mathematically)?

 

[Hurkyl]

Could you give me an example for how it is used for a real life problem (not mathematically)?

It's most commonly used in situations that can be described by an 'amplitude and phase'; for example, in signal processing.

The analytic properties of the complex numbers also make them very useful for describing two-dimensional fluid flow.

 

[AzonicZeniths]

Ok, that's where it is applied, but not how, I really could use how it is applied. I'm still not grasping how an imaginary number can be used in real life, not just in mathematics.

 

[FrogPad]

Lets say you have an audio signal from a microphone and you want to get rid of some really low frequency. When you perform a Fourier transform, you get numbers that represent the signal in a different way. These numbers happen to be complex. You can then filter some of these numbers away and then perform an inverse Fourier transform that goes from complex-numbers to real-numbers and you get a new signal.

edit:
I'm thinking more about what you said, and I believe you will interpret the process I described above as being some type of mathematical machinery using complex numbers to perform something useful. However, (in your mind) there is no physical connection of a complex number to something real.

Well I will ask you, what physical connection does any number hold?

edit 2:
I believe the Schrödinger equation MUST be formed with the complex numbers.

 

[CRGreathouse]

I imagine the question is motivated by the fact that you can imagine measuring with real numbers, but not with complex numbers. But there's no good reason to think that real-life distances properly represent real numbers: quantum mechanics suggests a minimum distance that makes physical distances more like the integers than the real or even rational numbers. But the real numbers are attractive in that they provide a good model of distances, glossing over the quantum aspects that we recognize but need not interact with directly.

The use of complex numbers for electric amplitudes is likewise for convenience: electricity 'just happens' to work very much like a complex number in several respects. In fact, that representation is probably 'closer' to reality than measuring physical distances with real numbers.

In short, you underrate the complexity of the ordinary things: real numbers aren't any less weird than complex numbers!

 

[AzonicZeniths]

Lets say you have an audio signal from a microphone and you want to get rid of some really low frequency. When you perform a Fourier transform, you get numbers that represent the signal in a different way. These numbers happen to be complex. You can then filter some of these numbers away and then perform an inverse Fourier transform that goes from complex-numbers to real-numbers and you get a new signal.

edit:
I'm thinking more about what you said, and I believe you will interpret the process I described above as being some type of mathematical machinery using complex numbers to perform something useful. However, (in your mind) there is no physical connection of a complex number to something real.

Well I will ask you, what physical connection does any number hold?

edit 2:
I believe the Schrödinger equation MUST be formed with the complex numbers.

Well the example you gave me was great, thank you. And yes, you got what i was thinking perfectly, I was trying to make a physical connection of a complex number with something real, when there is nothing. And, yes, now that I think of it, no numbers hold any physical connections. *mind blows* The main reason I was thinking about this, is that I am more of a physics guy, physics to me is just common sense, and I heard that had physical applications.

 

[AzonicZeniths]

I imagine the question is motivated by the fact that you can imagine measuring with real numbers, but not with complex numbers.

Not really, I was just perplexed by how a imaginary number can be applied to practical situations. Basically a number that does not exist being applied to real situations. I'm still struggling to grasp the concept.

 

[Hurkyl]

Not really, I was just perplexed by how a imaginary number can be applied to practical situations. Basically a number that does not exist being applied to real situations. I'm still struggling to grasp the concept.

The technical modifiers "real" and "imaginary" have absolutely nothing to do with their usual layusage -- they are merely a historical artifact of the attitudes of a darker time.

 

[mgb_phys]

Another use is that trig functions can be written in terms of complex numbers, so any application that uses trig is often easier to handle with complex numbers.
Trig functions are no more 'real' but probably easier to picture as a real application

 

[LukeD]

It's not that an imaginary number does not exist. No numbers "exist". There are just applications of them in the real world: situations where a certain number system describes certain properties of objects.

Let's say that you have a robot that can walk in any direction on your floor but only turn in 90 degree increments (I.e., it can only go forward, backwards, left, and right, but this can be oriented in any way so that forward could mean at an angle of 32 degrees). Then instead of lying down an x-y plane, we can have a Complex plane (where the real axis takes the place of your x-axis and the imaginary axis takes the place of your y axis). We can let V be the velocity that your robot (but as a single complex number instead of as 2 real numbers) is moving currently. Then note that the only velocities in which he can move are
V, -V, iV, and -iV

In other words, multiplying by i rotates your robot by exactly 90 degrees. So it turns out that complex numbers actually do describe the situation. In fact, if you want to to rotate your robot by angle $$\theta$$, then you'd just have to multiply V by $$\cos(\theta) + i \sin(\theta)$$ (note that this always has absolute value 1). And to change the speed, multiply by a real number.

---

Note: This also gives a physical significance of the square root of a number: If x^2 = y, then rotating by x twice is the same as rotating by x. So if we want to know what we can rotate by twice to get your robot to turn around, this a solution to the equation x^2 = -1. Your solutions are i and -i. So if you rotate by one of those twice, it's the same as turning around 180 degrees.

 

[AzonicZeniths]

Ok, thank you everyone for helping me understand this. I now have it. :)

 

[shamrock5585]

Could you give me an example for how it is used for a real life problem (not mathematically)?

i find this to be a funny thing to say... almost anything you do with numbers is mathematical, and then to talk about the number "i" which is not only a negative 1 but it has a square root symbol around it (which is a mathematical operation) how can you talk about that in a non-math scenario

 

[FrogPad]

i find this to be a funny thing to say... almost anything you do with numbers is mathematical, and then to talk about the number "i" which is not only a negative 1 but it has a square root symbol around it (which is a mathematical operation) how can you talk about that in a non-math scenario

I thnk the OP meant mathematical in the sense of pure math. As an example (there are many) the Cantor set may have come on the seen as something that is interesting from a pure math standpoint. However, is one going to build a bridge with that concept?
It doent have the direct utility as say geometry or vectors, where n can visualize tension as a vector that has a length and direction.

It's interesting to note that the Cantor set becomes useful when one desribes Chaotic systems.