What is i?Date: 9/24/95 at 14:31:58 From: Anonymous Subject: SQR(-1) ?? Hi Dr. Math, I want to know if it is correct to say: i = SQR(-1) ? Or am I only allowed to say: i^2 = -1 ? Because: -1 = SQR(-1) * SQR(-1) = SQR((-1)*(-1)) = SQR(+1) = 1 and -1 is not equal 1!! I really don't know what is right, but I've read both in separate math books. Thanks for your help (hope so). Bye TOETI Date: 9/27/95 at 11:30:30 From: Doctor Ken Subject: Re: SQR(-1) ?? Hello! Actually, you're right. You just proved that -1 = 1. Just kidding. Well, the problem is that i isn't defined to be the square root of -1. You see, any number has two square roots. For instance, 4 has the square roots 2 and -2. For convenience, we almost always define the square root function to give us the positive square root of a real number (we pick 2 instead of -2). Well, it's similar with imaginaries. There are actually two square roots of -1, there's i and there's -i. So we say "i is defined to be a square root of -1 and that makes -i the other one," not "i is _the_ square root of -1." With that in mind, it's more correct to say that i^2 = -1, although people will usually know what you mean if you say i = Sqr{-1}, just like people will usually know what you mean if you say 2 = Sqr{4}, although the concept of "taking the positive square root" is a little weird when the answer is imaginary. So the flawed step in that chain of equations you wrote is right at the end; it's because 1 and -1 are both square roots of 1. It's similar to saying -2 = Sqr{4} since (-2)^2 = 4 = 2. This is pretty fun stuff to think about. Hope you get something good out of it. - Doctor Ken, The Geometry Forum |
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