Can an Imaginary Number Be a Valid Answer?Date: 01/12/2006 at 15:06:55 From: Greg and the group from Neelin Subject: are imaginary numbers valid answers? We know that the answer to the equation x^2 + 1 = 0 is root negative 1 and the only answers to that are +/- i. But if "i" does not exist how can it be an answer? Date: 01/12/2006 at 22:16:40 From: Doctor Peterson Subject: Re: are imaginary numbers valid answers? Hi, Greg. Your question is just the one that bothered mathematicians when complex numbers were first discovered, and for centuries after. But the same question was asked about negative solutions; only positive solutions to equations were considered meaningful, because it was assumed that the solution had to represent some measurable length. Now, if your equation arises from a problem where only positive numbers make sense (such as a length), then the answer is that negative or imaginary solutions are invalid--but that's due to the problem, not the equation itself. Likewise, there are problems in which only a real number makes sense (such as a coordinate in space); then positive and negative solutions are valid, but an imaginary one is not. But an equation itself doesn't relate to any specific real-world problem; it only asks what numbers make it true, and complex numbers are numbers. So we can ask "what are all the real solutions to this equation?", and the answer will be "there are none", only because the problem was stated in terms of real numbers. Without that restriction, imaginary solutions are perfectly valid. After this idea was accepted, that complex numbers really are numbers (and DO exist, in the sense that any numbers, though abstract, can be said to exist), it was found that there ARE real problems in which complex solutions make sense. For example, in electronics there are problems in which the complex solutions of an equation tell you how a circuit will respond to certain inputs; imaginary numbers correspond to alternating current, and real numbers to direct current (more or less). In this setting, complex solutions not only are valid, they are what you're looking for! See this page: Using Imaginary Numbers http://mathforum.org/library/drmath/view/53879.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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