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0.999...= 1

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Re: 0.999...= 1

#21

Post by onion2k »

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…

I think the source of confusion is that people can't cope with recurring numbers. When someone says .999...=1 the listener assumes that the 0.999... stops at some point, and if that happens it'll always be below 1 because they can imagine adding another 9. Essentially, people actually ignore the "..." because that's the hard part.

Re: 0.999...= 1

#23

Post by ant6n »

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..

Well, here you reduced 1=.9999... to 1/3=0.333... What if I don’t believe that second equation.

Re: 0.999...= 1

#24

Post by notRobot »

iso1631 wrote: post #12
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

or,

      x = 0.9999...
    10x = 9.999...
    10x = 9 + 0.999...
    10x = 9 + x
     9x = 9
      x = 1
Presented slightly more clearly

https://en.wikipedia.org/wiki/0.999...#Algebraic_arguments

Re: 0.999...= 1

#25

Post by larschdk »

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..

I don't mean to troll you, but if you were doubtful that 0.999... = 1, then you should also be doubtful that 0.333.. = 1/3. Any argument that 0.999... is not quite 1 can also be used to argue that 0.333... is not quite 1/3.

I think it's mostly a matter of definition, since mathematicians consider sums of infinite series equal to their limit (if it's finite), i guess for many practical reasons. If you accept this, then 0.999... = 1. If you don't, then 0.999... can't be assigned a value (but converges to 1), which may be the intuitive understanding of infinite series for some.

Re: 0.999...= 1

#26

Post by voxl »

rini17 wrote: post #8
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?

Diagnolization is a pretty deep argument about fixpoints, Godels incompleteness argument is essentially a diagnolization. So why wouldn't there be fascination?

Re: 0.999...= 1

#27

Post by steerablesafe »

virgilp wrote: post #19
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.

Arguably 1 is just as an infinite number of decimals as 0.(9) . 1 is just short for 1.(0) .

Re: 0.999...= 1

#28

Post by ginko »

ant6n wrote: post #23
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..

Well, here you reduced 1=.9999... to 1/3=0.333... What if I don’t believe that second equation.

As in the 0.333... will stop at some point? That would still mean that 3 time 0.333... with a LOT of 3s end up being being 1.

I also figure it's a bit more intuitive for pupils to just try out calculating the decimal representation of 1/3 and seeing that it'll just keep going forever.

Re: 0.999...= 1

#29

Post by notRobot »

larschdk wrote: post #25
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..

I don't mean to troll you, but if you were doubtful that 0.999... = 1, then you should also be doubtful that 0.333.. = 1/3. Any argument that 0.999... is not quite 1 can also be used to argue that 0.333... is not quite 1/3. I think it's mostly a matter of definition, since mathematicians consider sums of infinite series equal to their limit (if it's finite), i guess for many practical reasons. If you accept this, the…

> if you were doubtful that 0.999... = 1, then you should also be doubtful that 0.333.. = 1/3

I disagree. Any middle school student can calculate 1/3 to be 0.33333... using long division, but there's no immediately obvious way to go from 1 (or 1/1) to 0.9999...

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