Thread: complex numbers, and complex planes

currently im studying complex numbers in complex planes, with polar forms and argument of trigonometric functions, y question here is what is the use of complex planes and complex numbers in everyday applications if applicable?

Electrical engineers use this concept to describe the relationship between current, voltage and power in alternating current circuits.

If you multiply the voltage and current in a circuit, you get a value with the units of volt-amps. These are the same units as power (watts), but power is not the same as volt-amps, because there is an imaginary component, as a result of the current and voltage being out of phase.

Learn more about it here:

http://en.wikipedia.org/wiki/Reactive_power

Originally Posted by Heinsbergrelatz

currently im studying complex numbers in complex planes, with polar forms and argument of trigonometric functions, y question here is what is the use of complex planes and complex numbers in everyday applications if applicable?

Complex numbers are integral to quantum mechanics.

Complex numbers are necessary for Fourier analysis which itself is central in electrical engineering and the theory of mechanical vibrations. As Harold14370 noted, electrical engineers work daily with complex numbers -- as "phasors" in elementar circuit theory, in Fourier transforms in communication theory, in Laplace transforms in control theory, again as Fouriere transforms in the theory of electromagnetic radiation and antennas, .......

Complex numbers are necessary for the solution of many algebraic equations.

Complex analysis is used for the theory of airfoils.

If you have any desire to understand physics or almost any branch of engineering you will need to be very familiar with complex numbers.

oo i see,
and one more question to add, when we learn complex planes (well at least for beginners like me), there is the imaginary axis, what does this "imaginary" indicate?

Simply that it isn't "real".

If you have a polynomial equation with complex solutions, it doesn't "really" have solutions in the conventional sense. e.g. if you throw up a ball and want to calculate when the ball crosses a certain height, you don't have any 'real' solutions if you don't throw hard enough, but you can solve the equation to get imaginary solutions. In this example the imaginary solution is useless, but in a lot of other domains, it is useful, and sometimes necessary.

e.g. the roots of the denominator of a transfer function of a system are called the poles. They are plotted in the complex plain to determine whether the system is stable or not. You can also plot the movement of these poles through this plain when you change parameters to stabilise the system.

wow... it seems like i still have a long way to go,
but thank you for all the help

And let's not forget fractals.


can give wonderful results if the divergence speed is plotted in the complex plane.

yes Mandelbrot set is indeed astounding

Originally Posted by Heinsbergrelatz

oo i see,
and one more question to add, when we learn complex planes (well at least for beginners like me), there is the imaginary axis, what does this "imaginary" indicate?

The complex numbers are a real vector space, which is the content of the representation of an arbitrary complex number as a +bi where a and b are real numbers. b is called the imaginary part of the number.

The axis in the complex plane that consists of all multiples of "i" is called the imagiinary axis, by convention. The terminology doesn't mean anything more than that.

The origin of the terminology "imaginary" is simply that when the complex numbers were developed the notion of a number whose square was negative was a bit unfamiliar and unsettling to some people. The name "imaginary" was coined to describe these numbers, a bit whimsically.

Originally Posted by Bender

Simply that it isn't "real".

If you have a polynomial equation with complex solutions, it doesn't "really" have solutions in the conventional sense. e.g..

It most certainly does have solutions in the conventional sense. It simply has no solutions from the set of real numbers.

There is nothing more "real" about real numbers than complex numbers. That is a misconception. Real numbers are applicable to models of the geometry of the physical world, but complex numbers are better suited to modeling other aspects of physics.

People were as much taken aback by the realization of the existence of irrational real numbers as they were later by imaginary numbers. The real numbers are not more "real" than other numbers, and in the minds of some less "real" than the rational numbers. The terminology is unfortunate and one ought not read so much into it.

Originally Posted by DrRocket

It most certainly does have solutions in the conventional sense. It simply has no solutions from the set of real numbers.

There is nothing more "real" about real numbers than complex numbers. That is a misconception. Real numbers are applicable to models of the geometry of the physical world, but complex numbers are better suited to modeling other aspects of physics.

People were as much taken aback by the realization of the existence of irrational real numbers as they were later by imaginary numbers. The real numbers are not more "real" than other numbers, and in the minds of some less "real" than the rational numbers. The terminology is unfortunate and one ought not read so much into it.

That's purely a matter of opinion. In a lot of applications I agree that the terminology makes no sense, but is as good as any other. In some applications, however, it makes perfect sense.

It might be easier to understand for a starter if it had a different name. Or more difficult. It never bothered me.

It might be easier to understand for a starter if it had a different name. Or more difficult. It never bothered me.

sorry but i dont understand this part of your statement.

Originally Posted by Heinsbergrelatz

It might be easier to understand for a starter if it had a different name. Or more difficult. It never bothered me.

sorry but i dont understand this part of your statement.

For someone who starts to learn a new concept, an unfortunate name might slow down the process of understanding. I have no idea whether this is the case for "imaginary" in the context of complex numbers

Originally Posted by Bender

Originally Posted by Heinsbergrelatz

It might be easier to understand for a starter if it had a different name. Or more difficult. It never bothered me.

sorry but i dont understand this part of your statement.

For someone who starts to learn a new concept, an unfortunate name might slow down the process of understanding. I have no idea whether this is the case for "imaginary" in the context of complex numbers

The name "imaginary" when taken as the usual everyday meaning of the term is confusing to a lot of people when they first encounter complex numbers.

If you read definitions as mathematicians read definitions there is no problem. But it you think the definitions have anything to do with the ordinary English definitions then problems can arise.

The real numbers are no more and no less real than other numbers

Irrational numbers have no link to insanity, and rational numbers have no link to sanity.

Complex numbers are not more complicated than other numbers.

Imaginary numbers have no relation to fantasies.

Hilbert spaces were invented by John von Neumann (he named them for David Hilbert).

In the a-c power example I referred to earlier, real power is the power that registers on your watt-hour meter. Reactive power (the component on the imaginary axis) does not. So, in that application, the name "imaginary" seems to make some sense.