# 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?

## Discussions

5 years ago

Wow. This isn't even wrong. Your lack of understanding of any of the manthematical or physical points you made, what the concepts mean, how they are derived, what "units" represent, is truly staggering in its universality. To end on a more positive note, you do appear to have a terrific grasp of grammar, syntax, and spelling.

Reply 5 years ago

It is not a lack of understanding, it is a lack of being able to explain the concepts in terms that make sense for everyone. Yes, my Ampere example was clearly wrong and for that I apologize; misinformation is not a good thing.

The point that I was trying to make was that numbers, and everything that they represent, relies very heavily in the system in which you are using. As an example, on a construction project, if you measured 15/16ths of an inch, for all purposes it is generally considered to be 16/16ths or 1 inch. Again, tolerences, criteria for the project and so on an so forth, will dictate this.

This example makes much more sense then the ampere, which I was mistaken on. The point of using 6.24 was to show how numbers can be exact, but in real world applications sometimes they do not make sense. If you cut an apple into thirds, we can say one piece is 1/3rd, which we can express as a ratio in decimal form of 0.333 repeating to represent one third. However, if we add it all up, we get 0.999, which is not equal to exactly 1, however, for the purposes of dividing an apple up into 3 equal pieces, then combine them back together, you end up with in fact, one apple.

Forgive my inability to take thought and express it in text. Sometimes the thought and point can get lost in translation. Thank you though for compliment on the grammar, spelling, and syntax. As my wife is an author, if she caught me with bad grammar, well, it would be safe to assume that disappointment would ensue.

Apologies if this example does not clarify my original post, I only aim to help and share what knowledge I have with people; which is the basis for this entire site. Again, I hope this helps anyone in the future with the concepts of 0.999 and the, "Is equivalent to 1," problem. =]

Reply 5 years ago

Your example with pi was equally wrong as your example of charge, and for exactly the same reason. Pi is a ratio of two non-commensurate numbers, and is therefore guaranteed to have a non-repeating expansion in _any_ number base (decimal, binary, sexigesimal, continued fractions, whatever). If you do not understand that, then you are missing some fundamental properties of real numbers.

Your example in this instance of "equating" 15/16 with 1 because of tolerances is also wrong. Engineering and construction drawings specify the

central valuedesired for a measurement, together with the allowed tolerance (usually as +/- 100%). If a part is specified to be 15/16" +/- 1/16, then cutting it to 1" is not correct, and a machinist or woodworker who does that will very quickly fail. Not knowing this indicates a fair lack of understanding of both engineering and drafting.Your claim that an infinitely repeating sequence is not equal to the corresponding fractional representation is wrong. It reflects the common error of conflating a terminated value with a non-terminating one, and demonstrates a basic misunderstanding of limits. The equality of the

infinitely repeatingvalue 0.9~ with 1.0 is nearly trivial to demonstrate, provided the person doing so understands the concepts involved.You have definitely clarified your original post. You've demonstrated that you really do misunderstand the issues you were trying to "explain."

Reply 5 years ago

Do you know how to read properly? When I said, "As an example, on a construction project, if you measured 15/16ths of an inch, for all purposes it is generally considered to be 16/16ths or 1 inch. Again, tolerences, criteria for the project and so on an so forth, will dictate this."

Did you not read that criteria for the project will dictates what is allowable and what is not?

It is not a lack of engineering it's a fact that on projects if you need a measurement and it is close, you regard it as the higher value. Have you never been on a construction project & measured anything...at all? You will be saddly mistaken if you think everything is exact and equal on any project, no matter the size or scale of it. Because we live in a real world where your 0.9999 forever and ever IS equal to one. Go to any drafter, go to any engineer and find me a perfect design or project. One that has absolutely no room for anything but EXACT measurements. Again, please read what is wrote. Because you can clearly use laws and axioms and jargon, does not in anyway reflect what is actually going on.

Misunderstanding of limits has nothing to do with tolerances and why we consider values that are very close to a certain value, rounded up. Go grab a piece of lumber, cut it anywhere you wish. If you think that the cut is perfect and straight, you're seriously wrong, and reading anything about construction would do you some good. That cut is going to have parts where it is cut more on one side or the other, however we consider the cut to be straight and correct. Because such tiny inaccuracies in things can be negligible. As is the case with 0.99999999999999999999999. We live in a very real world where things do not come out perfect, so we compensate for this by establishing certain tolerances. Any engineer worth their salt knows this.

Go ahead, say something clever.

Reply 5 years ago

1/16" used to be a good tolerance 0.0625". When Sir Joseph Whitwoth, the inventor of the standard screw was an apprentice, a "bare" 16" was the height of precision. When he retired, he could measure 1/1000000"

Reply 5 years ago

Oh Kelsey. I have very little mathematical understanding, however I am very interested in the magic of it all and often read/look at stuff that is way over my head. I only got as far as fractions at school yet I have worked in research statistics and analysis. I freely admit my "ignorance" but hate being pointed out because of it. My brain just doesn't get it. Your's does.

Reply 5 years ago

Don't worry about it! The software for statistics and analysis handles a lot of the arithmetic for you, but it can also be helpful to know what's happening behind the scenes.

My comment was in response to the long screed from "justinmcg67", not you. His commentary was a mishmash of scientific-sounding jargon which did not reflect reality.

5 years ago

Much wrongness. Your 6.24 x 10^18 means that one couloumb of charge, when composed of electrons, is comprised of 6240000000000000000 electrons. It isn't a fraction of a charge at all.

Your comment about PI is equally specious.

8 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.

Reply 8 years ago

Careful :-) .9 repeating for any

finitenumber 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

infiniterepeat, 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.Reply 8 years ago

That's what I was saying.

Reply 5 years ago

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...

Reply 8 years ago

. 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"?

Reply 8 years ago

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 notterminate, 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.

Reply 8 years ago

. 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.

Reply 8 years ago

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

allthe steps, then N=infinity, and 1/infinity=0. The sum is exactly 1.Reply 8 years ago

. 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 x

^{0}always equals 1 (except when x=0). heehee (that one took a while to sink in back in school)Reply 8 years ago

In fact, the two statements are equivalent: 1/

∞= 0 and 1/0 =∞. The latter requires a bit of care, as I discuss belowYou 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.

Reply 8 years ago

. 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.

Reply 8 years ago

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).

Reply 6 years ago

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.

8 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

showyou.Or put another way, it's a question of rounding-up / approximating decimals.

L

Reply 5 years ago

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.

6 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.

Reply 6 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.

Reply 6 years ago

Oh snap.

Reply 6 years ago

crackle and pop. .9... + .9... = 2

6 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

8 years ago

It's close enough for government work.

Reply 8 years ago

I like this answer the best.

8 years ago

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:

https://www.instructables.com/id/Decimal-to-Fraction-2/

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