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0.999...= 1

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Re: 0.999...= 1

#11

Post by username90 »

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.

The intuition that there is something in between isn't really wrong, it make sense and they work, otherwise physicists wouldn't be able to work with them. So that intuition is correct, it is mathematicians who just don't understand it fully yet. Maybe fully formalizing this is what unlocks the final piece keeping us from creating a unified theory in physics?

Re: 0.999...= 1

#12

Post by iso1631 »

ginko wrote: post #9

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..

Another secondary school 'proof'

  x = 0.9999.....
  10x = 9.9999.....
  (10x -x) = 9x = (9.9999.... - 0.9999....) = 9
  x = 9/9 = 1

Re: 0.999...= 1

#13

Post by saagarjha »

rs23296008n1 wrote: post #6

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?

Arithmetic breaks, as multiplication is no longer the inverse of division. (For example, 1/3 * 3 = 0.999… would no longer work.)

Re: 0.999...= 1

#15

Post by saagarjha »

ltbarcly3 wrote: post #7

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?

0.999…5 obviously.

Re: 0.999...= 1

#16

Post by steerablesafe »

Maybe the major source of confusion is that our decimal representation for whole numbers is supposed to be unique. Then when we extend it to rationals and reals this property fails at rationals in the form of a/10^n.

Arguably the sign symbol ruins it for whole numbers as well, as +0 and -0 could be equally valid representations of the number 0. We just conventionally don't allow -0 as a representation. There are other number representations that don't have this problem.

Re: 0.999...= 1

#17

Post by edflsafoiewq »

ltbarcly3 wrote: post #7

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?

Or: if they're different, what is their difference?

Re: 0.999...= 1

#18

Post by ltbarcly3 »

rs23296008n1 wrote: post #6

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?

So, lets take .9, .99, .999 and so on. If a sequence of rational numbers converges, it converges to a real number. What number does .9, .99, .999, .9999 (and so on) converge to? Which is to say, is there a number that it gets closer and closer to at every step? Clearly it gets closer and closer to 1 at every step, so the sequence converges to 1.

This is one of the many (equivalent) ways the real numbers are defined to begin with, https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...

There are lots of other ways to define sets with operations, but they won't be anything at all like the normal numbers you are used to.

Re: 0.999...= 1

#19

Post by virgilp »

steerablesafe wrote: post #16

Maybe the major source of confusion is that our decimal representation for whole numbers is supposed to be unique. Then when we extend it to rationals and reals this property fails at rationals in the form of a/10^n. Arguably the sign symbol ruins it for whole numbers as well, as +0 and -0 could be equally valid representations of the number 0. We just conventionally don't allow -0 as a representation. There are othe…

Right - I also find it easier to say that really, "1" is just a different/shorthand notation for 0.(9) It's not "two different, but equal numbers" - it's two different notations for the same number. Like how you can write same number in different ways in different bases - this is just writing the same number, in the "infinite number of decimals" vs "natural" way.
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