Physics and the 'i' (mathematical term)

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In summary, imaginary numbers are used in various fields of physics, such as quantum mechanics and electronics, to represent orthogonal quantities and to simplify calculations. They are not the only way to represent physical systems, but they are often a useful mathematical shortcut. In certain areas of physics, such as quantum field theory, they are necessary for obtaining realistic results. While some may have a prejudice against complex numbers, they are simply a more fundamental mathematical construct and are used to make calculations easier. Ultimately, all numbers are mathematical shortcuts and physical measurements will always be real.
  • #1
Pengwuino
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Do imaginary numbers ever manifest themselves in the physical world as we know it? And when I say that, i mean do they ever appear in solutions to problems that we know have physical meaning and aren't in the solutions or derivations purely as mathematical shortcuts?

Hopefully this question makes sense as I am not well versed on that advanced of mathematics.
 
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  • #2
Pengwuino said:
Do imaginary numbers ever manifest themselves in the physical world as we know it? And when I say that, i mean do they ever appear in solutions to problems that we know have physical meaning and aren't in the solutions or derivations purely as mathematical shortcuts?

Hopefully this question makes sense as I am not well versed on that advanced of mathematics.


In quantum mechanics and in Electronics. They are used to separate orthogonal quantities.
 
  • #3
Pengwuino said:
Do imaginary numbers ever manifest themselves in the physical world as we know it? And when I say that, i mean do they ever appear in solutions to problems that we know have physical meaning and aren't in the solutions or derivations purely as mathematical shortcuts?

Hopefully this question makes sense as I am not well versed on that advanced of mathematics.

As far as the appearance of anything physical, usually an "i" will show up in something, such as an exponential term, that represents a quantity leaving a system. In particular it often shows up in a damping term in oscillatory motion or as the source term of an energy "bleed" where energy is leaving a system.

-Dan
 
  • #4
topsquark,

Are imaginary numbers the only way to correctly show the system or is it just a mathematical shortcut?

I got to stop asking questions where i don't expect to understand half of the responses...
 
  • #5
Pengwuino said:
topsquark,

Are imaginary numbers the only way to correctly show the system or is it just a mathematical shortcut?

I got to stop asking questions where i don't expect to understand half of the responses...

I suppose if we knew everything we would be able to have a theory to track, say, what happens to the energy of an electric field as it penetrates a nonconductor and attribute the energy loss to the molecules in the non-conductor. That's a perfect world and as there are rather many molecules in even a small sample of matter, I doubt anyone is going to work on the problem in the near future! :tongue:

Still and all, there are many places in which "i" is useful...so many that I don't know where Physics would be without it. One of my favorite spots in Physics is when we try to describe a Quantum Field Theory for spin 1/2 particles. There is something called a "Grassman Algebra." A Grassman algebra consists of a set of "anti-commuting complex numbers." That is to say all of the elements of the set may be represented by a+ib. But in this algebra (a+ib)(c+id) is not the same as (c+id)(a+ib), in fact one is the negative of the other! The use of this algebra is not a mere mathematical trick: in order for the field theory of spin 1/2 particles to give any sort of realistic results we MUST have such a set of numbers. In this part of Physics, at least, "i" is a very real number. (Sorry, I couldn't help the pun! :biggrin: )

-Dan
 
  • #6
So the answers yes and I've learned to stop asking questions where the answers will leave me speechless adn confused haha
 
  • #7
Complex functions are sometimes useful, but physics quantities must be real. In QM, observables are hermitian operators, so its eigenvalues are real. We use, for example, imaginary exponentials to make it easier cause it is hard to work with trigonometric functions.
 
  • #8
Pengwuino said:
topsquark,

Are imaginary numbers the only way to correctly show the system or is it just a mathematical shortcut?
They are just a math shortcut. It could all be done with trigonometry, but physicists hate trig for good reason.
 
  • #9
Of course, real numbers aren't the only way to represent things either. You can express everything in terms of the number "0" and the "increment" operation... and arranging them in clever ways.

Of course, it would be very cumbersome to do that -- so we use real numbers for things that are well described by real numbers, and complex numbers for things that are well described by complex numbers.

The predjudice against complex numbers is just a historical thing from back in the day when numbers really meant lengths of curves, areas of shapes, and other similar things.
 
  • #10
All numbers are "mathematical shortcuts"!
 
  • #11
One could say that any oscillation is a manifestation of imaginary numbers. We use Trig functions to hide the fact that imaginary exponentials are the more fundamental mathematical construct.
 
  • #12
Actual physical measurements are always real, as has been said.

Complex numbers make life easier for many fields, like quantum mechanics, but one is not required to use them. You can always just cast your system in terms of a two-dimensional variable (that's all a complex number really is -- a two-dimensional vector) and proceed as usual.

- Warren
 
  • #13
Of course, a real number is nothing more than a special kind of complex number, so you're working in the complexes anyways. :tongue:
 

1. What is 'i' in physics and mathematics?

'i' is a mathematical term representing the imaginary unit, which is defined as the square root of -1. In physics, 'i' is used to represent the imaginary part of complex numbers, which are often used to describe physical quantities such as electric and magnetic fields.

2. How is 'i' used in physics and mathematics?

In physics, 'i' is used in conjunction with real numbers to represent complex numbers, which are useful in analyzing physical phenomena such as oscillations and waves. In mathematics, 'i' is used in various fields such as complex analysis, number theory, and quantum mechanics.

3. What is the significance of 'i' in physics and mathematics?

The use of 'i' in physics and mathematics allows for the representation and manipulation of complex numbers, which are essential in solving many problems in these fields. It also plays a crucial role in understanding quantum mechanics and the behavior of waves and oscillations.

4. Why is 'i' sometimes referred to as an imaginary number?

'i' is referred to as an imaginary number because it cannot be expressed as a real number and is often used to solve problems that involve the square root of negative numbers. However, despite its name, 'i' is a fundamental and essential part of mathematics and physics.

5. Can 'i' have a physical meaning or interpretation?

While 'i' itself does not have a physical meaning, it is often used in physical equations to represent physical quantities such as electric and magnetic fields. Additionally, in quantum mechanics, 'i' is used to represent the imaginary part of wavefunctions, which have physical significance in describing the behavior of particles.

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