0.999... and Infinity
Date: 09/17/2003 at 17:23:33 From: Shane Subject: paradox? What's wrong with this? x = 0.99999999...forever so 10x = 9.99999999...forever so 10x - x = 9.9999999 - 0.9999999 9x = 9.0 x = 1 How does x = 0.9999...forever turn into x = 1?
Date: 09/18/2003 at 01:04:25 From: Doctor Mike Subject: Re: paradox? Shane, I'll tell you what's wrong with what you have written. Nothing. Nothing is wrong with it. It's fine. I can see, though, why you might be surprised about what you have found. Infinity is a pretty powerful concept. Why am I talking about infinity? Because whenever you use a "forever" repeating decimal, it doesn't just go on for a "really long time", it goes on forever. And whenever you have an infinite number of parts of something (an infinite number of places in a number, in this case) you might well expect something amazing to happen. OK, let's get beyond the philosophy and on to exactly why this happens. An infinite decimal "officially" is considered to be a sequence of numbers, by mathematicians who think about the origins and foundations of our number systems. For instance: 0.9 0.99 0.999 0.9999 0.99999 0.999999 0.9999999 and so on forever. Another familiar example you may have encountered is 3. 3.1 3.14 3.141 3.1415 3.14159 3.141592 3.1415926 and so on with the rest of the digits of pi, forever. In each case, the final and official value of the number is the limit of the sequence of these numbers that get "closer and closer" to that limiting value. In the case of 0.99999... the limiting value is 1. If you go out far enough in the terms of the sequence, you can get "arbitrarily close" to one. You might ask, "Can you go out far enough so that the terms in the sequence are closer to 1.0 than one trillionth = 1/1000000000000 ?" Yes, anything past the term 0.9999999999999 is closer to one than one trillionth. "Can you go out far enough so that the terms in the sequence are closer to 1.0 that one octillionth?" Yes. "What about 1/google?" Sure, no problem. As close as you want to specify, you can go out far enough in the sequence so that every term in the sequence, from there on, is closer to 1.0 than the degree of closeness you specified. That's what limits are all about. Now, the bottom line is: If this number 0.99999999... is closer to one than anything that can be measured... if it is closer to one than anything that you can even think about measuring... if it is closer to one that any possible number you could think of... if it is closer to one than any number that anybody could think of, no matter how small... THEN essentially 0.9999999... is indistinguishable from one. Said another way, if your number x = 0.999999999... is such that the difference "x-1" is smaller than any positive number (however small), then we consider x - 1 = 0, so that x = 1. I hope this helps you to understand your paradox. Thanks for writing to Dr. Math. - Doctor Mike, The Math Forum http://mathforum.org/dr.math/
Date: 09/18/2003 at 03:02:03 From: Shane Subject: Thank you (paradox?) Dr. Math is the most impressive thing I've found on the internet to this day - and I've been on it a very long time. I'm amazed at the quality and promptness your answers, and you do it without asking for anything in return. You are a true credit to society, thank you!
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