Why Not Force Dividing by Zero?Date: 10/22/2002 at 04:09:07 From: Rindert Bolt Subject: Divide by zero Hello, I wonder if there has ever been an attempt to force dividing by zero just as it has been done so successfully by forcing the square root of -1 to be i. Of course it has been attempted, but why is it not used? Best regards, Rindert Date: 10/22/2002 at 09:46:18 From: Doctor Mitteldorf Subject: Re: Divide by zero Dear Rindert, It's a good question, and there's a good answer. When we define the square root of -1 as i, we expand the real number system to a complex number system. Every number has two parts - the real part and the part that is a multiple of i. We get an interesting and useful system. Numbers like 3+2i have meaning, and can be added, subtracted, multiplied, and divided with self-consistent results. Problems such as sqrt(i) and sin(1+i) turn out to have meaning, and the answers are consistent and obey the same algebraic laws as square roots, and the same trig identities that sines and cosines of real numbers obey. What's more, we don't have to expand the system any more or define any new levels of imaginary numbers, because all algebraic equations now have solutions, even if the coefficients in those equations are complex numbers. But suppose we tried the same trick for 1/0. We define 1/0 to be z, and we say from now on, all numbers will be expressed as two parts, as, for example, 2+3z. When we try to calculate in this system, we run into strange results, leading to contradictions. For example, you can prove that z+1=z. Just divide both sides by z. 1 + 1/z = 1 Since 1/z=0, this must be true. But if it is true that z+1=z, then we can subtract z from both sides and we get 1=0. But if 1=0, then it is certainly true that 2=1, or any real number is equal to any other real number. So all real numbers are the same, and we shouldn't confuse things by calling 7 a different name from 100. The moral of this story is that we take it for granted that if we solve a problem in two different ways, we will get the same answer both ways. Mathematics is consistent. But this is not a fact about "the world"; rather this is a design feature of the rules of mathematics. If we design our mathematics in the "wrong" way, we will get inconsistencies. (If you follow this kind of thinking deeper into the philosophy of mathematics, you will come across the program of Bertrand Russell in the early part of the 20th century. He tried to prove that the mathematical system that we all know so well had the simple, desirable properties that 1. Every statement is either true or false. 2. Every true statement has a proof, and every false statement has a disproof. It seems that these things are almost obvious. We are assuming something like this every time we set out to solve a math problem. But Russell encountered frustration. He couldn't seem to convince himself that these simple critera were met. A few years later, along came Kurt Godel, and at last we knew why. This is a fascinating story for another day.) - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ |
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