Understanding Imaginary Numbers
Date: 09/18/2002 at 20:03:39 From: Anonymous Subject: Imaginary number Dr. Math, I have asked and looked many places trying to understand the imaginary number. I don't have it in school yet, but I have to do a report on it. Why is the square root of i negative, and what about when it is cubed? Why is i times i equal to -i? Also, could i be considered real? Not real as in real number, but real as in real life. If so, how can we apply it today? I just want to know why it has its properties. Please help me. Thanks
Date: 09/18/2002 at 22:04:51 From: Doctor Ian Subject: Re: Imaginary number Hi, The following item from our archive should get you started: Complex Numbers http://mathforum.org/library/drmath/view/53877.html Complex numbers follow the same rules of exponents as real numbers: i^3 = i * i * i = (i * i) * i = -1 * i = -i >Why is i times i equal to -i? Recall the definition of i. We _define_ i such that i * i = -1 This is a little bit like defining the product of two negatives: -1 * -1 = 1 With negatives, you can pair up the signs, and each pair disappears, because it's equivalent to 1: -2 * -3 * -4 * -5 * -6 = -----(2 * 3 * 4 * 5 * 6) = ---(2 * 3 * 4 * 5 * 6) = -(2 * 3 * 4 * 5 * 6) With i, you get a similar effect, but you have to get _four_ i's if you want to cancel: i * i * i * i = (i * i) * (i * i) = -1 * -1 = 1 so 2i * 3i * 4i * 5i * 6i = iiiii(2 * 3 * 4 * 5 * 6) = i(2 * 3 * 4 * 5 * 6) You can think of it this way. Multiplying by -1 is like making a 180-degree turn on the number line. But multiplying by i is like making a left turn on the coordinate plane. It takes only two 180-degree turns to cancel out. But you need to make four left turns to get back to the direction you were going in originally. Does that make sense? >Also, could i be considered real? When you get right down to it, no numbers are 'real'. Numbers are a device that we make up in order to keep track of things. Counting numbers are good for keeping track of of sets of distinct objects (sheep, voters, pennies). Rational and irrational numbers are good for keeping track of quantities that can take values in continuous ranges, (lengths, intervals of time, areas, volumes). There's an old Calvin and Hobbes cartoon that I really like. The family drives over a bridge with a sign that says the bridge has a weight limit of, say, 20 tons. Calvin asks his father how they know what the limit is. His father says that they build the bridge, and then they drive heavier and heavier trucks over it until it collapses. Then they build another one just like it, and put up the sign with the weight of the last truck that crossed safely. Now obviously, this isn't the way it's done! But it illustrates why we find it so convenient to work with numbers. We can use numbers to model the behavior of real things, as a way of predicting how those things will interact in the real world. The things are real, but the models aren't, and the numbers that make up the models aren't. What are complex numbers good for? One thing that complex numbers are really good for is keeping track of things that behave like waves. The most important kinds of waves that we deal with are light, sound, and electric and magnetic fields. (Complex numbers might also be used in the design of a bridge, if the bridge is long enough; since one thing that can make long bridges fall is that they can start vibrating in resonance with winds, so when you build a bridge, you need to know what its preferred frequencies of vibration are. Vibrations are modeled with waves, and waves are modeled with complex numbers.) It's important to realize that what each new kind of number gives you is a notation that's easier to work with. We could get along with nothing but counting numbers, but it would be very cumbersome. Using negative numbers gives us a very compact notation for talking about quantities that have opposites (east vs. west, up vs. down, credit vs. debt), but that's all it does. Using rational numbers allows us to break larger units into partial units, so we can measure things in feet and fractions of feet instead of some smallest unit (like the radius of an electron). Similarly, there is nothing that we can do with complex numbers that we couldn't do without them. It would just be more cumbersome to do calculations, that's all. It doesn't make them 'real'. But it sure makes them useful! Some of the stuff I'm telling you I've only figured out within the last few weeks, while trying to explain to someone else why the product of two negatives has to be positive. In case you've ever wondered what the volunteer math doctors get out of doing this, this is a big part of it: We're forced to re-examine things that we already think we 'know', and find out that maybe we don't know them as well as we think we did. Believe it or not, that's fun. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ See also these answers from the Dr. Math archives: Imaginary Numbers in Real Life http://mathforum.org/library/drmath/sets/shortcuts/dm_imaginary.html - Doctor Sarah, The Math Forum http://mathforum.org/dr.math/
Date: 01/17/2003 at 16:28:48 From: Anonymous Subject: Imaginary number Well, it's been a while since I asked all this about imaginary numbers, and I don't think that I ever thanked you for replying so wonderfully. This really helped me understand it and I'm so grateful. I know where to come to if I ever need more help. Thanks!
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