Negative Ratios and Dividing by ZeroDate: 8 Jun 1995 12:53:31 -0400 From: Andrea Berni Subject: Question Question which has been hovering in my mind for a while... 1 : -1 = -1 : 1 Then how can a bigger number be to a smaller number as a smaller number is to a bigger one ? Another little question to which I never had a satisfying answer... X divided by 0 = infinity Why not ? Thanks for wasting your time with me. Ciao ! - - - - - - - ---> Andrea ! Date: 8 Jun 1995 13:27:08 -0400 From: Dr. Ken Subject: Re: question Hello there! Well, hmm. The concept of a ratio can of course be dealt with by dividing the two numbers, i.e. 1:-1 = 1/-1 = -1, and -1:1 = -1/1 = -1, so they're the same thing. But what does it mean to have a ratio of a positive number to a negative number? I'm not sure I have a good answer, but I'll try. Look at this: Let's say Bob is in debt $5 and Yolanda is in debt $10. What's the ratio of Yolanda's money to Bob's money? Well, -10/-5 = 2. So does Yolanda have 2 times as much money as Bob? Not really, she's in more debt than he is. So the ratio of two numbers only gives you a good idea of how big they are (in the positive direction) when you can be sure both of the numbers are positive. I think that's the best I can do. >X divided by 0 = infinite >Why not ? I've got a better answer to this one. You see, it's kind of a gray area. The way most people think about it, x/0 is kind of infinite, as long as x isn't also 0. But nobody can really say that, because there are a few problems with it that you'd better be prepared to deal with. 1) When you divide one number by another, you expect to get yet another number, not some abstract concept like infinity. Infinity isn't a real number because it doesn't act like one. For instance, you can have infinity + infinity = infinity, not 2(infinity) like you'd expect. 2) If you treated infinity as a number, and you said that 3/0 was infinity, then what would 6/0 be? Would it be twice as big? The same size? 3) Would 3/0 be positive infinity or negative infinity? The sequence 1/(1/2), 1/(1/3), 1/(1/4), ... goes to 1/0, and you can simplify the sequence to be 2,3,4,..., which goes to positive infinity. But the sequence 1/(-1/2), 1/(-1/3), 1/(-1/4), ... goes to 1/0 too, and it's better known as the sequence -2,-3,-4,..., which goes to negative infinity. But yeah, go on thinking that in an abstract sense, x/0 really does go infinite. -K Date: 9 Jun 1995 09:02:19 -0400 From: Andrea Berni Subject: Re: question Ke> 1) When you divide one number by another, you expect to get yet Ke> another number, not some abstract concept like infinity. Infinity Ke> isn't a real number because it doesn't act like one. For instance, you Ke> can have infinity + infinity = infinity, not 2(infinity) like you'd Ke> expect. Ke> 2) If you treated infinity as a number, and you said that 3/0 was Ke> infinity, then what would 6/0 be? Would it be twice as big? The same Ke> size? Well, Cantor suggested there are different infinites (i.e. infinites of different sizes), and according to what I understood from reading his explanation, 6/0 and 3/0 would give the same sized infinite, however, in this case maths is mainly an opinion... Ke> 3) Would 3/0 be positive infinity or negative infinity? Well, the best I can think of is that it behaves as a square root: 3/0= +- infinity Ke> But yeah, go on thinking that in an abstract sense, x/0 really does go Ke> infinite. I guess this is the only way out. However the positive or negative infinity you mentioned is really a good argument against this. Thanks again ! Ciao Date: 9 Jun 1995 09:34:48 -0400 From: Dr. Ken Subject: Re: question Hello there! Yes, you're right, Cantor suggested different orders of infinity, but the sense he was talking about was in the sense of _counting_ elements of a set, i.e. Cardinal numbers. For instance, the size of the set {1,2,3,...} is the same as the size of the set {2,4,6,...} and they're both the same size as the set of all rational numbers, but none of them is as big as the set of all real numbers. It's important to make the distinction between numbers that are used to count elements of a set and numbers that are used in equations. In fact, infinity is the only context in which this really makes a difference. The infinity that you get when you talk about 3/0 is just kind of a vague, general infinity, whereas the infinity you get when you talk about the number of elements of the set {1,2,3,...} is a very specific kind of infinity, called "countable" infinity or "Aleph 0." For what that's worth. -K Date: Wed, 25 Feb 1998 16:02:58 -0500 From: Ken Shoemake Subject: Ask Dr. Math: Negative Ratios ... Howdy. I was browsing through the old "Ask Dr. Math" stuff to find something interesting for my young niece, and came across the "Negative Ratios and Dividing by Zero" item. May I suggest another kind of answer? Would you agree that if both numbers in a ratio, like 2:3, are multiplied by the same number then the meaning of the ratio should be the same? We will avoid zero for now. Then we can proceed. We can multiply both parts of 1:-1 by -1, which gives -1:1. So that's an easy answer. But do you know why -1 times -1 gives 1? Think about (2-1) times (2-1), and look at each of the four pieces it makes: 1 = (2-1)(2-1) = 4 - 2 - 2 + (-1 * -1) = -1 * -1 So -1 times -1 must equal 1. As for X/0 and infinity, we can use a similar approach. Let us say that two fractions, a/b and p/q, are equal whenever a*q equals b*p. We get this from multiplying both fractions by b and by q. Now suppose X/0 is some definite fraction, let's say a/b. If X/0 equals a/b, then we expect X*b equals 0*a, which is 0. It doesn't matter what a is, and b must be 0 since X isn't. Is this really what we want? If we try to make X/0 be some kind of number then anything over zero equals anything else over zero. So we usually don't do it. We say that X/0 is undefined; it is not any number at all. Otherwise it has to be some special new kind of number, with its own rules. But what about dividing by 1/2, 1/3, ..., approaching zero? We can get as close as we like to zero without a problem. One clever trick is to say "I have a number that acts like an integer, except that it is bigger than any integer." We should not call this number "infinity", expecting that it is the only "number" that will be that big. But we can now say that one over that big "number" is "infinitesimal"--closer to zero than any ordinary fraction, but still not zero. This is the idea used in calculus, as described by A. Robinson. There are many different ideas about what the word infinity can mean. Don't let that bother you. Have you heard the joke that "Noses run and feet smell"? We use all kinds of words in different ways! -- Ken |
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