A More Formal Definition of the Imaginary Unit i
Date: 01/12/2004 at 05:29:39 From: Daniel Russell Subject: Contradiction in Simple Imaginary Number Algebraic Equation Although I have read the few pages on your site that deal with problematic imaginary number proofs, I feel that I am still not satisfied why the following is not valid: 1. 1/i = sqrt(1)/sqrt(-1) 2. 1/i * i/i = sqrt(1/-1) 3. i / -1 = sqrt(-1) 4. -i = i I read somewhere on this site that we are not allowed to combine roots unless the arguments of the roots are both positive. But this rule is blatantly ignored all the time! For instance, it is common practice to do all the steps on the left side of the equality sign in my proof. Another example is the idea that an imaginary number such as sqrt(-4) = sqrt(4) * sqrt(-1) = 2i. Why is this allowed, yet the contradictory proof above is not? What exactly are the rules of imaginary numbers? When can we do operations with imaginary numbers, like multiply, divide, etc, and with what kinds of other operands can we do these operations?
Date: 01/12/2004 at 06:14:43 From: Doctor Pete Subject: Re: Contradiction in Simple Imaginary Number Algebraic Equation Hi Daniel, You bring up a very important point when dealing with complex numbers, and the only way to give you a satisfactory answer is to say that the "usual" definition of i = Sqrt[-1] is really not a formal one, and does lead to a great deal of confusion. The actual definition is much more clear on this issue, and is well-understood by mathematicians. A complex number is an ordered pair (a,b) of real numbers, with the following rule for addition: (a,b) + (c,d) = (a + c, b + d) and the following rule for multiplication: (a,b) * (c,d) = (ac - bd, ad + bc), where addition and multiplication on the left-hand side are performed in the domain of the complex numbers, and addition and multiplication on the right-hand side is naturally done on reals. Thus defined, it is obvious that the complex numbers contain the real numbers as a proper subset, since for all real numbers a, a = (a,0). (To verify this, all one needs to do is show that the rules for addition and multiplication in the complex numbers correspond to regular real addition and multiplication for numbers of the form (a,0).) Furthermore, let us define i = (0,1). Then the computation i^2 = (0,1) * (0,1) = (0*0 - 1*1, 0*1 + 0*1) = (-1,0) = -1 shows that i has the property that its square is equal to -1. We can then proceed to observe that if a and b are real numbers, we have a + bi = (a,0) + (b,0)(0,1) = (a,b) so our definition of complex numbers as ordered pairs is consistent with our "usual" notation. Set up in this manner, there is absolutely no ambiguity whatsoever with the imaginary unit i, since we have not talked about square roots at all! The reason for the "paradox" in your original question is that there are two complex numbers, (0,1) and (0,-1), whose squares are equal to -1. In fact, there are exactly two other complex numbers whose squares equal to any given complex number, with the exception of zero. Thus, we call the square root function a "one-to-two" function, just like the square function is a "two-to-one function." Formally speaking, the function f(z) = z^2 maps two values in the domain to a single value in the range, and the inverse g(z) = z^(1/2) maps every value in the domain to two values in the range. You may also be interested to know that there are certain complex functions which are "many-to-one" and "one-to-many," such as f(z) = e^z g(z) = log(z), the complex exponential and logarithm functions. So we must be very careful when making a statement like log(-1) = Pi*i, because in truth, the logarithm of -1 is actually an infinite set of values, each differing from the others by a multiple of 2*Pi*i. I know my discussion does not pinpoint the fallacy in your question, and so sidesteps the issue entirely with a formalized definition, but the reason for this is that I want to re-conceptualize your notion of complex numbers as having anything to do with square roots of negative numbers, and instead encourage you to think of them as ordered pairs of reals, which obey particular rules for multiplication and addition. Metaphorically speaking, I don't want to patch the leaky boat you've found yourself sailing in, but rather give you a new vessel--a sturdy, streamlined one. Undoubtedly you will have more questions about our discussion, as I know you'll do your own exploration on this fascinating subject--feel free to write back! - Doctor Pete, The Math Forum http://mathforum.org/dr.math/
Date: 01/12/2004 at 07:04:22 From: Daniel Russell Subject: Thank you (Contradiction in Simple Imaginary Number Algebraic Equation) Thank you very much! That answer was far better than I expected, and exactly what I needed. I hope it will make its way onto the site because just about every student I know has been taught incorrectly that i = sqrt(-1).
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