Imaginary numbers

An Fear Aniar

They can't be much use if they're only imaginary.

What practical use have they ever been?



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Thomas_S_Hunterson

Here's some info:
http://en.wikipedia.org/wiki/Imaginary_numbers#Applications_of_imaginary_numbers

The subject of mathematics as a whole is an abstract one. One could argue that the whole field is imaginary, bounded by arbitrary rules.

Richard W

You could also argue that negative numbers don't exist, and therefore have no practical use. As you could hardly have -3 apples or something.

Fremen

In a sense, all numbers are imaginary...

Complex numbers have some very useful properties when applied to the analysis of functions.
One of the first applications you'd come across in college maths is the Laplace transform. The idea here is that if you have a calculation which is a bit awkward, sometimes you can transform it in some way and do the calculation, then transform the result back, which gives your answer. The transform which often works involves complex numbers.

Another useful application is what's known as the residue theorem, which allows you to express the sum of certain series as an integral over a function from the complex plane to the real numbers.

the GALL

yaaawn.... the who and the what now?:)

Fremen

yaaawn.... the who and the what now?

Bah. Close your brain, go back to watching neighbours.

An Fear Aniar

This is well explained in the Wiki:

For most human tasks, real numbers (or even rational numbers) offer an adequate description of data. Fractions such as ⅔ and ⅛ are meaningless to a person counting stones, but essential to a person comparing the sizes of different collections of stones.

Negative numbers such as − 3 and − 5 are meaningless when weighing the mass of an object, but essential when keeping track of monetary debits and credits[1].

Similarly, imaginary numbers have essential concrete applications in a variety of sciences and related areas such as signal processing, control theory, electromagnetism, quantum mechanics, cartography, and many others.

I get it now (kind of).

Thanks.

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chris85

Also used a lot in electrical terms. And stability of control systems

Michael Collins

I know people who go on about how the square root of -1 just doesn't exist so why do they invent an imaginary number for it?

The answer is
1) a whole branch of maths is based on it - complex analysis.
2) it simplifies calculations in loads of areas, especially in oscillatory type situations e.g. linear system analysis, electronic circuits etc. It's also fundamental in quantum mechanics, allows a simple time-harmonic description of Maxwell's equations, and helps in loads of other areas.

Plus there's the beautiful equation(!):

e^(i.pi) + 1 = 0

Fallen Seraph

I feel the most appropriate answer is: we use imaginary numbers because they have physical meaning. I'm always most impressed when they turn up alongside oscillating.

Son Goku

The first thing about the complex numbers is that all equations have roots in them. This is the main reason they became important. Secondly they have important links to 2D geometry.

As somebody already mentioned, their main "advanced" application is in complex analysis. A branch of mathematics with a tremendous amount of applications. Complex Analysis is kind of the integration and differentiation of complex functions.

Finally, they have a direct physical meaning in Quantum Mechanics where they are responsible for destructive and constructive intereference that you wouldn't get with real numbers.

The higher order number systems, such as the quaternions, octonions and sedenions, have far less applications.

mawk

im also led to believe that complex turns count in electrical transformers, even though they dont really exist..

myheadasplode.

Son Goku

mawk wrote: »

im also led to believe that complex turns count in electrical transformers, even though they dont really exist..

myheadasplode.

Well, they exist just as much as any other number. In fact their existence is a lot less dubious than numbers like the "unnamables", which are real numbers. The complex numbers are just the 2D plane with multiplication tacked on. Nothing mysterious. Where as to go from the rationals to the reals you need to add in a bunch of numbers that can never be described.

MathsManiac

You don't have to have a difficult problem to demonstrate the usefulness of complex numbers. Most people should be able to appreciate this one:

e.g. Consider the sequence: 0, 1, 4, 11, 24, 41, 44, ...,
where the rule for getting each term (apart from the first two) is: "four times the previous term minus five times the one before that".

Suppose now that you want to get, say, the thousandth term without having to go through all of the terms to get there. You'd like to have a formula for the nth term of this sequence. It turns out that you basically can't get such a formula without complex numbers, but you can with complex numbers.

For the above sequence, by the way, the formula for the nth term is (i/2)*(2-i)^(n-1) - (i/2)*(2+i)^(n-1).

People often find it interesting that such a formula (with lots of complex numbers in it) always gives you a non-complex number as the answer, no matter what the (whole number) value of n is.

Sequences generated by rules like this are one of the very early things that people encounter in primary school maths, in number puzzles, etc., so it's interesting that to deal with them fully, you can't avoid irrational numbers (for, say, the Fibonacci sequence) or complex numbers for ones like the one above. Also, sequences like that do crop up in real applications.

kelly1

An Fear Aniar wrote: »

They can't be much use if they're only imaginary.

What practical use have they ever been?

When I studied electronic engineering, we dealt with complex/imaginary number all the time. The were used to simplify analysis of circuits which ordinarily would have involved the solution of differential equations so we used the Laplace transform to turn the analysis into a simple algebra problem. This was used to find the time and frequency domain response of a given circuit for a given input.

Co33iE

good example of complex numbers in real world @ Scots Guide to Complex Numbers, Remember j is used in electronics as imaginary number because i is current.