Why are there Imaginary Numbers? They have no purpose, they cannot be measured, and yet sometimes I find them in a problem that has to do with measuring something. Therefore creating a no answer situation, which can be maddening. I understand that we need negitive numbers, so if someone is in debt they can see how deep they are in. But Imaginary Numbers? I mean come one! The whole idea of it is pure stupidity. Someone give me a logical reason for Imaginary Numbers, please!
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Anonymous2008-05-01 1:34
"Complete knowledge of the nature of an analytic function must also include insight into its behavior for imaginary values of the arguments. Often the latter is indispensable even for a proper appreciation of the behavior of the function for real arguments. It is therefore essential that the original determination of the function concept be broadened to a domain of magnitudes which includes both the real and the imaginary quantities, on an equal footing under the single designation complex numbers."
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Anonymous2008-05-01 2:57
>>appreciation
Rarely is this stressed in mathematics
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4tran2008-05-01 4:10
They are quite useful, actually.
They make solving linear systems of differential equations much easier. RLC circuits are generally solved using complex analysis (complex impedances, etc).
Have fun integrating cos(ax)/(x^2 + 1) from 0 -> infinity without complex numbers. If you know about residues, this is easy.
They also makes fourier transforms more compact, and less messy.
They also allows all polynomials to have at least 1 root.
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Anonymous2008-05-01 7:47
>>1
Ever consider the fact that numbers are abstract things, and don't have to correspond to the real world at all?
Not to mention the fact that imaginary numbers play a big role in quantum mechanics and electrical engineering...
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OP2008-05-01 15:48
>>5 Explian their purpose in quantum mechanics and electrical engineering. Eventhough I will not understand it, please do so.
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Anonymous2008-05-01 15:57
>>6
pauli spin matrices and as >>4 said, used to solve circuits.
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Anonymous2008-05-01 16:26
>>1 Do you understand why do we need rational, or real numbers?
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Anonymous2008-05-01 18:43
>>8
Do you understand how to properly structure a sentence?
Lots and lots of reason. As an algebraic extension to the real numbers, the Gaussian integers very useful in number theory, they are one of the three unique division algebras over the reals, the theory of complex functions is useful for solving real integrals and as a purely academic exercise quite interesting, Quantum mechanics would not have a neat mathematical basis without the complex numbers (Or at least it would have been longer in coming, as I'm sure without the complex numbers we'd just express them as ordered pairs to suit our purpose with the same meaning but different notation)... I could go on