Is It Possible That x/0 is Not Really Undefined?Date: 06/14/2004 at 04:16:42 From: Stephane Subject: Can we not define x/0 in the example of the complex number i As we are all told, n = 1/0 is undefined, since no number n exists that, when multiplied by 0, gives the result 1 (i.e., n*0 = 1). However, drawing an example from the theory of complex numbers, there also exists no obvious number i that, when squared, gives the result -1. The square root of -1 would, in past centuries, have been described as nonsense or undefined. Mathematicians nonetheless define just such a number, enlarging the known numbers from the real to the complex, and use the number i successfully in many real-world calculations. Can we be certain that, for any nonzero x, x/0 is actually undefined (i.e. unable to be ascribed any actual value or meaning), or is it possible that an important number and new number class has so far escaped discovery? Regrettably, I do not have the required mathematical training to flesh this proposition out substantially, except to suggest that if, by some novel number definition (say, j = 1/0), an additional "dimension" of numbers can be defined, perhaps this new, larger, set also encompasses the complex numbers, as these encompass the real numbers. To put it another way, if we can think of the "real number line" being extended into a "complex number plane" by the definition of i = sqrt(-1), perhaps the complex plane can be extended into an even larger, "ulterior volume" by the definition of a number j = 1/0. Recognizing that all this is probably just idle speculation, I appreciate the time and effort anyone is willing to devote to providing an answer. Date: 06/14/2004 at 08:26:44 From: Doctor Mitteldorf Subject: Re: Can we not define x/0 in the example of the complex number i Dear Stephane, This is a good, thoughtful question, and there's a satisfying answer to it. You don't have to have an extensive math background to appreciate the answer, just some basic algebra and the confidence to apply it. The definition i = sqrt(-1) is such a rich mathematical construct because of two facts. First, the definition is self-consistent. It does not introduce any contradictions into the system we already have in place. Second, it doesn't lead to more numbers that need defining, but provides solutions (in the form of complex numbers) to all polynomial equations. For example, you don't need to define a new number for sqrt(-i), because the square root of -i can be calculated in terms of the complex numbers (using 1 and i) that we have already defined. (You can verify that sqrt(-i) = (1-i)/sqrt(2) by multiplying this number by itself and seeing that you get -i as an answer.) The problem with defining j = 1/0 is that it doesn't pass the first test I mentioned. The mathematics that results is no longer self-consistent, and it's therefore not very interesting. Here's an example of an inconsistency: By definition 1/0 = j. Multiply both sides by 2, and you find that 2/0 = 2j. The problem comes when we multiply both sides through by 0. We find in the first case that 0 * j = 1 and in the second case that 0 * 2j = 2. So far, so good. But multiplication is supposed to be "associative". If you multiply (a*b) * c you should get the same answer as a * (b*c). For example, we can write 2*3*4 without specifying which multiplication is done first, because (2*3)*4 = 6*4 = 24, and 2*(3*4) = 2*12 = 24. We get the same answer both ways. Now try this with 0 * 2 * j, and you'll see what I mean about introducing inconsistency. - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ Date: 06/14/2004 at 20:44:27 From: Stephane Subject: Thank you (Can we not define x/0 in the example of the complex number i) I see your point, and why we can't just arbitrarily define such a number. Thank you for taking the time to answer my question. |
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