Question about infinity

[Justin Horne]

Hi. I'm a freshman in High school, and my algebra teacher gave me a problem. She said that .9 repeating = 1 Now apart from all the ways this didn't make sense to me, I thought of a way that might make it not work. If you were to put something like .0 [infinity of 0's] 1, wouldn't that be able to be added to .9 repeating, making 1? Correct me if I am wrong here, please. Thanks,
Justin.

[chroot]

http://www.physicsforums.com/showthread.php?s=&threadid=9684

http://www.mathforum.com/dr.math/faq/faq.0.9999.html

- Warren

[sridhar_n]

There is a limit to how much we can measure. In fixing this limit we also fix our approximations. When there are infinites of nine we say that it is very close to one and hence approximate it to 1. We could approximate 0.999999999999999999999999999999999999999999999... to 0.9, but 0.9999999999999999999999999999999999999999.... is more closer to 1 than 0.9. Similarly, 0.0000000000000000000000000000000000000000001 is more closer to 0 than 1. Thus it is basically because there is a limit to our measurements and approximations that we approximate to te closest digit possible.

Sridhar

[chroot]

Originally posted by sridhar_n
we approximate to te closest digit possible.
Sorry, this is not correct. 0.\overline{9} is not approximately 1, it is absolutely 1. Remember, I'm not talking about a lot of nines, I'm talking about an infinite number of nines. That's a whole different bucket o' spaghetti.

- Warren

[HallsofIvy]

I know it is hard to accept but even teachers are right ocassionally! The problem with you idea is that you can't have an "infinite number of 0s" and then put a 1 on the end. With an infinite number of 0s there is NO end.

A non-terminating decimal such as 0.99999... is DEFINED as the limit of the sequence .9, .99, .999, .9999, etc. and that limit IS 1.0.

[HallsofIvy]

There is a limit to how much we can measure. In fixing this limit we also fix our approximations. When there are infinites of nine we say that it is very close to one and hence approximate it to 1. We could approximate 0.999999999999999999999999999999999999999999999... to 0.9, but 0.9999999999999999999999999999999999999999.... is more closer to 1 than 0.9. Similarly, 0.0000000000000000000000000000000000000000001 is more closer to 0 than 1. Thus it is basically because there is a limit to our measurements and approximations that we approximate to te closest digit possible.

Sridhar

I really have to object to this. When we say that 0.999.... is 1, we are not talking about "measurement" and we are not talking about approximations.

0.99999.... is not close to 1 it is 1: exactly equal to 1. As I said in another post, 0.99999... is defined as the limit of the sequence 0.9, 0.99, 0.999, ... and it is easy to show that that limit is 1.0.

[Organic]

When 1.000... and 0.999... are two representations of the same number then:

1.00... = 0.999...

0.100... = 0.0999...

0.0100... = 0.00999...

0.00100... = 0.000999...

0.000100... = 0.0000999...

0.0000100... = 0.00000999...

Therefore we can write:

0.100... + 0.0100... = 0.0999... + 0.00999...

0.0100... + 0.00100... = 0.00999... + 0.000999...

But this is not true because:

0.1100... not= 0.0999... + 0.00999... = 0.10999...8

0.01100... not= 0.00999... + 0.000999... = 0.010999...8

and so on ...


Digit 8 is not the last digit but the limit digit or the unreachable digit of 0.999...

Therefore 1.000... is not the limit of 0.999...

[HallsofIvy]

I sincerly hope that that was your idea of a joke.

[suyver]

Check out the other thread with the identical post to convince yourself that it wasn't a joke... [:(]

[chroot]

[moderator hat]

This thread should be locked, since it just parallels the other one.

[/moderator hat]

- Warren