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Why or why not does .9 repeating equal 1?

I already posted this as a regular question, but someone told me to make it as a burning question, so here it is:

HI! I am wondering if .999...=1? I mean if 1/3=.333 and 1/3+1/3+1/3=3/3 and 3/3=1. So wouldn't .333...+.333...+.333...=1? But it actually equals .999... not 1. I googled this question and I got many different answers
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For example...(someone wrote):
I'm not really into math, but a friend brought something up to me today that really seemed very strange. (For the duration of this post, .999 will mean .9 repeating unless otherwise specified- just for the sake of ease)
.999=x
10x=9.999
10x - x = 9x
9x=9
1x=1.
.999 = 1.
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Someone else wrote:
Numbers are fake. They are a manifestation of our minds to describe something, similar to words. Just because we say "red" doesn't mean something is red. what is red? Languages and math are very similar. Math is universal...at least for our planet though. .999~ does not = 1. But what .999~ repeating represents, does in fact equal what 1 represents.

No-one will ever be able to comprehend infinity, os its time to stop trying. Think of space and the universe. IT IS GROWING. how can it continue to grow with no stop? what is there to contain it? WE need something to contain it in order for us to understand it. We need a stopping point, but there is none.
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Another person wrote:

In your proof that .333...*3=.999... you forgot to include the fact that .3333... is NOT 1/3. 1/3 if not a number that can be turned into a decimal in any way. I thought someone might like to know this fact.

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And the last person wrote:

Just a pedantic point the equation for differentiation is given by lim x->0 [f(x+h)-f(x)]/h, it has the minus sign. It is after all just telling you the slope and is no different really from doing simple trig using the tan function. Here you just take a really small triangle.

As to the 0.99~ thing, this is really just writing the supremum (spelling might be off) of the numbers less than 1. Just think of it as taking the smallest number 'n-word' than 0 away from 1. They are not identical for if they were we would not have a continuous number line, but rather a dashed one with lots of wholes in it. I could simply argue that 0.99~8 is just as close to 0.99~9 as 0.99~ is to 1. For those who really want to understand go and look up supremum numbers and the axioms of the real number line.
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So who is correct? Do you agree that .999...=1? Or .999... does not equal 1?

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NachoMahma5 years ago
> I mean if 1/3=.333 and 1/3+1/3+1/3=3/3 and 3/3=1. So wouldn't .333...+.333...+.333...=1?
.  1/3 does NOT equal .333 - it's .3 repeating. You are using an approximation. 1/3 + 1/3 +1/3 = 3/3 = 1
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.  .9 repeating does NOT equal 1. As caitlinsdad says, it's real close (past a few decimal places doesn't really matter for most DIY projects), but not quite there.

Careful :-)  .9 repeating for any finite number of places is less than 1 (in fact, less than by exactly 10-n, where n is the finite number of places. 

If you specify the infinite repeat, then it's a freshman calculus exercise to show that the infinite sum converges to identically 1 (the equivalent of saying that 10-infinity = 0.

That's what I was saying.
Just go to KhanAcademy.org and watch the presentation on the theory of limits. kelseymh has explained it to the point where, if that doesn't break it down enough for you, then reading up on math might behoove some of you as trying to understand rocket science without understanding how gravity works, is about as an appropriate analogy as I can think of. Like running before walking, or something else...
.  So 0.8rep = 0.9? 1.9rep = 2? 1/3 = 0.4?
.  Sorry, but this Calculus-ignorant Ibler just doesn't get it. No matter how many decimal places you carry it out to, 0.9rep just never quite reaches 1. If you carry it out to an infinite number of places, you get "infinitely close" but no cigar.
.  From a Calculus POV, where am I going wrong? Do numbers have a "quantum"?
You're mistaking "infinity" with "very very large."  They're different.  Any finite but terminating string of 0.99999.....9 is less than 1 by exactly 1e-N, where N is the number of repetitions.  Now, 1/infinity = 0 exactly, hence 1e-(infinity) = 0 exactly, so if the repetition does not terminate, then the "difference" from 1 becomes exactly zero.

For your two examples, 0.888888.... = 8/9, which is not 0.9.  1.999... = 2.

I highly recommend Jack's insructable.  He goes through the summation exercise in detail.
.  I read the iBle before posting the prev msg and still can't say I grok it. I can follow your formulas (or at least they seem to make sense), but it just doesn't "feel" right. Still seems more like a parlor trick.
.  Luckily, this is not something I am ever likely to use in an important situation (I've gone almost 54 years without it coming up), so I'll just take your word for it.
The key is "limits".   The same methodology is used to resolve some of Zeno's paradoxes, like "you can never reach your destination, because first you have to get halfway there, then you have to get half of the rest of the way, and so on.  So you're always short." 

No, because the _infinite_ sum converges.  Yes, each step is half of the step before, 1/2, 1/4, 1/8, 1/16, etc.  If you add them up, you get closer and closer to 1.  If you stop the sum at any point, you're short by 1/2^N (e.g., after four steps, you're at 15/16).  But if you really take all the steps, then N=infinity, and 1/infinity=0.  The sum is exactly 1.
.  The "1/ = 0" part is what's messing me up. If division by zero is undefined, then why isn't division by everything undefined? Zero and ∞ are truly strange creatures.
.  But:
.  1) Your explanation makes a certain amount of sense
.  2) I believe you know what you are talking about
.  3) You like to kid around a bit, but I've never seen you actually mislead anyone
.  Ergo, I'm going to "take it on faith" that 0.9rep = 1


.  I don't think infinity had been invented when I studied Math. <snicker> And why would they name a mathematics concept after some high-end speakers?
. I suppose the next thing you are going to tell me is that x0 always equals 1 (except when x=0). heehee (that one took a while to sink in back in school)
In fact, the two statements are equivalent:  1/ = 0 and 1/0 = .  The latter requires a bit of care, as I discuss below

You can derive these again by taking the limit of 1/x, as you take x to either +/-infinity or to zero.  For the latter, the result is actually "undefined" the limit of 1/x for x<0 goes to -infinity, but the limit for x>0 goes to +infinity.

As you say, zero and infinity are strange.  You cannot trivially use results from finite numbers and expect them to "just work" with infinite limits.
.  In my math studies, I never made it past x/0 is undefined. I didn't make it to Riemann spheres, formal calculations, Calculus, &c. That's why they don't have me teaching college and working on colliders. ;)
.  Thanks for taking the time to explain. I can't say I grok it, but I think I understand as well as I do capacitors.
The idea of limits and infinite sums are basically the introduction to calculus.  If you can master those, it's pretty much all you need for anything in the real world (why the circle and sphere formulas work, why ballistic trajectories are parabolas, and so on).
It's the old: if a snail is in a race and moves 1/2 the distance to the end point each day, he will never reach the end point. In reality of course, that is just not physically true.
lemonie5 years ago
Basic errors:
0.333 is not equal to 1/3
10x 0.999 = 9.99 (not 9.999)

If you're going to pre-set a limit to decimal places displayed, you're effectively asking what a calculator will show you.
Or put another way, it's a question of rounding-up / approximating decimals.

L
OMG, when you say .999 repeating it means .999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999... and so on and so forth

so when you say that 10x.999=9.99 not 9.999, it doesn't make any difference, because the .9 is repeating.
lime3D3 years ago
No it does not equal 1. In your example, you state that 1/3 = .333, but that is not true. 1/3 = .3 repeating.


.3 repeating + .3 repeating + .3 repeating = .9 repeating.

1/3 is a fraction, not a decimal. So 1/3 + 1/3 + 1/3 truly = 1.
kelseymh lime3D3 years ago
Are you familiar with first-year calculus? In particular, the convergence properties and limits of infinite sums? Please evaluate Sum_n=1^\infty 10^(-n), then multiply that result by nine.
Oh snap.
crackle and pop. .9... + .9... = 2
zyzyqy3 years ago
Why not write it as either

<><> - 1/<><>

or

(<><> - 1)/<><>

Either one gives repeating 9's

The <><> is the best way I could write the infinity symbol

Hope this helps
caitlinsdad5 years ago
It's close enough for government work.
 I like this answer the best.
Zero point nine, with an infinity of repeating nines is, in fact, another way to write 1.

Any number with repeating decimal, i.e. a rational number, can be expressed as a fraction with integers in both the numerator and denominator, i.e. a ratio of two integers.

In fact, I already wrote an instructable explaining how to do this:
http://www.instructables.com/id/Decimal-to-Fraction-2/

Follow along, and you'll see that 0.999[repeating] simplifies to 1/1.