InfinitiesDate: Mon, 21 Nov 94 13:08:00 EST From: Binford M.S. Gifted Program Subject: problem Dear Dr. Math, Can one infinity be larger than another? Sincerely, Gina Miller Date: 21 Nov 1994 19:27:52 GMT From: Ken "Dr." Math Organization: The Math Doctors Subject: Re: Gina's problem Hello there, Gina! Great question. Yes, one infinity can be larger than another, a weird kind of concept. One standard way of dealing with the different sizes of infinities is to take one kind of infinity as a "normal" infinity or a "base" infinity, and then compare its size with other infinities. What we do in practice is take the number of counting numbers 1,2,3,4,5,... as our normal infinity, and we call it "countably infinite." The way we compare the sizes of two infinite sets is to see whether we can pair the elements of the two sets up in a one-to-one correspondence. For instance, if we match up the even numbers and the odd numbers like this: 2,4,6,8,10,12,14,16,18,... 1,3,5,7, 9,11,13,15,17,... We can see that there are exactly the same number of even numbers as odd numbers. What if we try to match up the even numbers and the counting numbers? 2,4,6,8,10,12,14,16,18,... 1,2,3,4, 5, 6, 7, 8, 9,... Again, we can do it, so there are exactly the same number of even numbers as counting numbers, so we say there are a "countably infinite" number of even numbers. Notice that all I've done above is write the even numbers in a list, making sure that I list them all. This is the usual way to show that there are countably many (a synonym for countably infinite) numbers in some set: try to find a way to list them all. What about the rational numbers? Remember that the rational numbers are numbers that can be written as the quotient of two integers. I'll write a list of all the positive rational numbers; can you find the pattern? If you're really ambitious, you could try to find a formula for the n'th number in the sequence! 1/1, 2/1, 1/2, 3/1, 2/2, 1/3, 4/1, 3/2, 2/3, 1/4, 5/1, 4/2, 3/3, 2/4, 1/5, ... Here's a hint about how to look at the numbers in that sequence: look at the numerators seperately, then the denominators. Do you believe that I'll get all the positive rational numbers this way? If so, I've just shown that there are countably many positive rational numbers. How could you use a similar sequence to show that there are countably many rational numbers (both positive and negative)? It is an interesting fact that you cannot write such a listing of all the real numbers (which includes both the rationals and the irrationals). If you're interested, write back and I'll help you show that. If that's true, what does that say about how many real numbers there are? There's a lot of them, more than the number of counting numbers. So there are two infinite sets that really do have two different sizes. I hope this helps you. Ken "Dr." Math ________________ Date: Mon, 22 Nov 1993 10:12:04 -0500 From: Phil Spector Subject: Re: problem (Gina Miller's) Gina - Great question! Infinity is a difficult thing to understand, but if it weren't there, then there probably wouldn't be many classes left for math majors to take in grad school. In any case, depending upon what your definition of larger is, the answer is yes. There are a bunch of ways of looking at this. If there's one thing that people are sure of, it's that infinity clearly doesn't always = infinity. infinity - infinity doesn't equal zero, and infinity/infinity doesn't equal one. Rather, they are both undefined, because it is recognized that one of the infinities could be much larger than the other. How? Well, let's start counting from 1. You could go on for ever. In fact, I think it's safe to assume that since you could count forever, there are an infinite number of numbers in that set. In other words, {1, 2, 3, 4, 5, 6....} and so on has an infinite number of numbers in it. But then what about {1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5....} and so on? This seems to have twice as many members (can you see why?), but it still of course has an infinite number of numbers in it. Finally, what if we were to look at every single number you can think of. That includes all the above numbers, plus a lot more. And of course there are an infinite number of numbers you can think of. Therefore we see that in each case there were an infinite number of numbers, but in each case there were more numbers than in the previous one. It seems that indeed, one infinity can be bigger than another. This is, of course, very silly. Even though there are twice as many numbers in the second set as in the first, mathematicians have never gone off the deep end and said that 2 * infinity = infinity. Rather, they make efforts to differentiate between different kinds of infinity. For example, they call the first set (where you can count the infinity) a "countable set" of infinite numbers, while they call the third example (all the numbers you could ever think of, including all sorts of wacky decimals) an "uncountable set". I'm sure some other Dr. Math's will jump in with other ways of looking at it, but I like the above one, if for no other reason than it's the only one I could think of. Hope this helps a little! Again, great question! You asked about something that has been plaguing mathematicians for a long, long time... Phil, Dr. Math |
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