Why Isn't Infinity a Number?
Date: 15 Feb 1995 15:08:54 -0500 From: Steven Bahrij Subject: infinity Dear Dr. Math, I realize that you cannot divide by zero; however, the limit of 1/x as x approaches 0 is either positive or negative infinity. The concept of infinity is not considered a number. Why? Frances Shefl Grade 11 St. Mary's high School O'Neill Nebraska math instructor: email@example.com
Date: 15 Feb 1995 20:25:22 GMT From: Dr. Math Subject: Re: infinity Hello there! One of the main reasons we don't call infinity a number is that it doesn't act like one. Let's say we call infinity the number A (some people actually do use the hebrew letter Aleph to name one kind of infinity). Then I can show you that the equation A + A = A. Consider the number of terms in the sequence 1,3,5,7,9,11,13,... , and consider the number of terms in the sequence 2,4,6,8,10,12,14,.... They have the same number of terms, because I can match them up in a one-to-one way, right? Match them up like this: 1 3 5 7 9 11 13 15 ... 2 4 6 8 10 12 14 16 ... So the number of terms in these sequence is the same, and we'll say that they each have A terms. Now what happens if I take both of these sequences together? I get the sequence 1,2,3,4,5,6,7,8,.... And here's the funky part: this sequence, which looks like it should have twice as many terms as either of the first two sequences, actually turns out to have the same number of terms as each of the first two sequences, as I can demonstrate by matching them up like this: 1 3 5 7 9 11 13 15 ... 2 4 6 8 10 12 14 16 ... 1 2 3 4 5 6 7 8 ... So if you believe this stuff, we've just shown that infinity is a number that you can add to itself and get itself. That's a pretty bizarre characteristic to have in a number, unless it's zero (and infinity certainly isn't zero!). So this is one of the properties of numbers that infinity wouldn't stand up to. I hope you keep thinking of good questions like these! -Ken "Dr." Math
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