Imaginary Numbers, Division By Zero
Date: 07/03/2000 at 19:22:27 From: Marc Subject: Imaginary Numbers/Division By Zero Dr. Math, Here is my question. If we can create a new set of numbers for something that doesn't exist in our real number system for the square roots of negative numbers (imaginary numbers), then why can't we do the same thing for division by zero? The number i was created to equal the square root of -1, so why can't we define another variable, for example, x, as equal to 1/0, 2x as equal to 2/0, 3x as equal to 3/0 and so on? Why is there a set of numbers that uses the square root of negative numbers, something that is impossible in our "real" number system, but NOT one that uses division by zero, something that is also impossible in our real number system? I have one more question. Are there any real life uses for imaginary numbers? And if not, why were they created? How can you use a number that has no "real" value for any useful purpose? Thank you for your time.
Date: 07/03/2000 at 21:08:42 From: Doctor Peterson Subject: Re: Imaginary Numbers/Division By Zero Hi, Marc. In creating the complex numbers, what we are doing is extending the set of real numbers by adding one new number i, defined so that i^2 = -1, and then applying the properties of the operations on real numbers to complete the set; for instance, we can multiply i by any real number, giving the imaginary numbers, and then we can add any of these to any real number to produce a complex number. If at any point the properties of operations led to a contradiction (for instance, if it turned out that multiplication as defined were not commutative), we would have to give up the idea of complex numbers, or at least recognize that they are not an extension of the real numbers as we know them. If we tried to do the same thing by adding an "infinite" number (a number, not a variable!), say I = 1/0, we would find lots of contradictions. For example, I would be defined more specifically as the number for which I * 0 = 1, but since (I + 1) * 0 = I * 0 + 1 * 0 = 1 + 0 = 1 so that (I + 1) * 0 = 1, we would have to say that I + 1 = I By subtracting I from both sides, we would find that 1 = 0 Now, any system of numbers for which this is true is not very useful, so we just can't add this "I" to the system and still follow the rules of the real numbers. You can read more about this in our FAQs on Dividing by 0 and Large Numbers and Infinity: http://mathforum.org/dr.math/faq/faq.divideby0.html http://mathforum.org/dr.math/faq/faq.large.numbers.html To answer your other question, imaginary numbers are indeed used in many ways, particularly in physics and engineering. The first use was in solving cubic equations, where it was found that by passing temporarily into the complex realm, problems whose ultimate solutions were real could be solved! But many variables in physics turn out to be complex; physical quantities can often be thought of as just the real part of a complex number. Here are a couple answers we've given with more details: Applications of Imaginary Numbers http://mathforum.org/dr.math/problems/zakrzewski10.14.97.html Imaginary Numbers http://mathforum.org/dr.math/problems/matt02.28.99.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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