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Physical meaning of imaginary numbers 
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#1
Sep2311, 01:02 PM

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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. 


#2
Sep2311, 01:26 PM

P: 163

In antimatter based worlds, our imaginary numbers are real, and our real numbers are imaginary ;>



#3
Sep2311, 01:28 PM

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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 2D fluid flow problems. 


#4
Sep2311, 01:39 PM

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Physical meaning of imaginary numbers
Like Hurkyl mentioned, how that is interpreted in realworld 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
Sep2311, 01:45 PM

P: 882

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
Sep2311, 01:54 PM

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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, .... 


#7
Sep2311, 01:55 PM

P: 18

3/4 is the weight of iron (in kilogram) that is 0.75kg. 3: cant 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
Sep2311, 02:08 PM

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#9
Sep2311, 02:12 PM

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#10
Sep2311, 02:19 PM

P: 882

And it is not real  it is imaginary But it may be used to describe real processes  and it is almost equally applicable for that purpose as real numbers. 


#11
Sep2311, 02:41 PM

P: 5,632

See here:
http://en.wikipedia.org/wiki/Imaginary_numbers under "geometric interpretations " and "applications". imaginary numbers are widely used in electrical phase measurements, feedback and signal processing analysis. 


#12
Sep2311, 03:14 PM

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P: 17,214

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
Sep2311, 03:22 PM

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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
Sep2311, 03:25 PM

P: 4,018

 real vs. imaginary numbers  real vs. fictitious forces  (intrinsic) curvature Feel free to extend 


#15
Sep2311, 03:28 PM

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#16
Sep2311, 03:42 PM

P: 452

Guess which one comes first. 


#17
Sep2311, 03:51 PM

P: 882

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 (preatomic, prequantum, preinformationtheory) 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
Sep2311, 04:15 PM

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