0.99... Repeating Can't Really Equal One, Can It?Date: 08/14/2008 at 01:49:03 From: John Subject: .999 repeating = 1 . . . . again Doesn't .999 repeating simply get infinitely close to 1, but never quite get there? Kind of as if it had an asymptote at 1? It always gets closer and closer, but will never quite get there, because if it did then all the 9s would turn to 0s, and then it would be 1. Each 9 that is added simply makes the number more precise, closer to 1. But the number will ONLY become more PRECISE, it will never become, well, precise, only MORE precise. Date: 08/14/2008 at 09:38:09 From: Doctor Peterson Subject: Re: .999 repeating = 1 . . . . again Hi, John. Sure, if you ever STOP somewhere in writing out 0.999..., then it will not be equal to 1. But that's not what a repeating decimal means. Think about 0.333... . That equals 1/3, right? But something like 0.33333333333333333333333333333333333333333333333333 does not equal 1/3, because I've left off all the digits after the 50th. The value of any non-terminating decimal like 0.333... is defined to be, not the value when you stop somewhere, but the "limit" that you approach closer and closer as you take more digits into account. So in fact what you say is true of ANY such number, even something like pi = 3.14159...; each digit you add gets closer to the actual value, giving you better and better approximations without ever actually reaching it. But that actual value is still out there, beyond all those approximations; it is not one of them. So the value of 0.999... is not the value of 0.99999999999999999999999999999999999999999999999999 but the value, namely 1, that you are approaching as you add more digits. (In fact, if you could stop somewhere and get a value of 1, that would prove that 0.999... was NOT 1, since the next digit would make it greater!) By the way, if 0.999... were not 1, how would you explain the fact that 1 = 3 * 1/3 = 3 * 0.333... = 0.999... ? If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 08/14/2008 at 15:51:17 From: John Subject: Thank you (.999 repeating = 1 . . . . again) Touche, sir/madame . . . . touche! |
Search the Dr. Math Library: |
[Privacy Policy] [Terms of Use]
Ask Dr. Math^{TM}
© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/dr.math/