0.999...= 1

#1

Post by yurisagalov » Tue, Apr 28, 2020, 6:31 AM UTC

0.999...= 1
en.wikipedia.org

Re: 0.999...= 1

#2

Post by bytedude » Tue, Apr 28, 2020, 7:19 AM UTC

Flame wars over this used to be common on the internet. People intuitively have the notion that the left side approaches 1, but never actually equals it. They see it as a process instead of a fixed value. Maybe the notation is to blame.

Re: 0.999...= 1

#3

Post by orthoxerox » Tue, Apr 28, 2020, 7:20 AM UTC

I remember WarCraft 3 official forums being torn apart by this, with probably thousands of comments in the thread. Blizzard even had to post their official stance on the issue, but that didn't calm those who insisted 0.999... was 1 minus epsilon and not exactly 1.

Re: 0.999...= 1

#4

Post by klodolph » Tue, Apr 28, 2020, 7:22 AM UTC

An interesting consequence of this in proofs.

You’ll see various proofs involving real numbers that must account for the fact that 0.999…=1.0. There are, of course, many different ways to construct real numbers, and often it’s very convenient to construct them as infinite sequences of digits after the decimal. For example, this construction makes the diagonalization argument easier. However, you must take care in your diagonalization argument not to construct a different decimal representation of a number already in your list!

Re: 0.999...= 1

#5

Post by cjfd » Tue, Apr 28, 2020, 7:29 AM UTC

Ultimately this is more the definition of R than that it is a theorem. One can also work with sets of numbers in which the completeness axiom does not hold. E.g., sets of numbers in which one also has infinitesimals.

Re: 0.999...= 1

#6

Post by rs23296008n1 » Tue, Apr 28, 2020, 7:29 AM UTC

I'm actually curious what impact it would have on various proofs if 0.999... wasn't accepted as 1.

What gets broken? What consequences do we hit?

Re: 0.999...= 1

#7

Post by ltbarcly3 » Tue, Apr 28, 2020, 7:30 AM UTC

This is 'more intuitive' if you think about it this way:

If any two real numbers are not equal, then you can take the average and get a third number that is half way between them. Conversely, if the average of two numbers is equal to either of the numbers, then the two numbers are equal. (this isn't a proof, just a way to convince yourself of this)

What's the average of .9999... and 1?

Re: 0.999...= 1

#8

Post by rini17 » Tue, Apr 28, 2020, 7:32 AM UTC

klodolph wrote: post #4

An interesting consequence of this in proofs. You’ll see various proofs involving real numbers that must account for the fact that 0.999…=1.0. There are, of course, many different ways to construct real numbers, and often it’s very convenient to construct them as infinite sequences of digits after the decimal. For example, this construction makes the diagonalization argument easier. However, you must take care in you…

I never understood the fixation on diagonalization. Why can't ever exist another way for mapping any set to countables?

Re: 0.999...= 1

#9

Post by ginko » Tue, Apr 28, 2020, 7:33 AM UTC

I remember being doubtful when being presented with this in middle school, but after being shown this as fractions makes it obvious:

      1/3 =     0.333..
  3 * 1/3 = 3 * 0.333..
      3/3 =     0.999..
        1 =     0.999..

Re: 0.999...= 1

#10

Post by tgv » Tue, Apr 28, 2020, 7:34 AM UTC

bytedude wrote: post #2

Flame wars over this used to be common on the internet. People intuitively have the notion that the left side approaches 1, but never actually equals it. They see it as a process instead of a fixed value. Maybe the notation is to blame.

Repetition can easily be seen as a process, which would indeed approach 1. But I think the idea of infinite repetition is very hard to get.