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many people are reluctant to believe that 1/3 = .333... or 2/3 .666... because they do not believe that decimals can represent fractions. but it can. like someone said earlier, 0.5 is 5/10, 0.125 is 125/1000 and people have no problem accepting that. Consider that .5 is a "infinitesimal" in a way because there is an infinite amount of 0s trailing the number 5. That must be true in order for 0.5 to equal 5/10 exactly. If people can accept .500... = 5/10 exactly, then why can't they accept that .333... = 1/3 exactly??


.333... or .3 with bar on top means EXACTLY 1/3. Because there is no way to write out the whole thing as a decimal, mathematicians use .333... or .3 with bar on top as a representation for it. Representations of a fraction do indeed equal the fraction they represent. In other words, .333... and 1/3 mean exactly the same thing just as "O-N-E" and "U-N-O"(one in Spanish) mean the same thing except they are spelled differently.

Niranjan Srinivas

It's surprising how difficult it is to convince people about this. As a mathematics major at university, i've had to do it quite often. the trouble is partly that people do not know what decimal expansion really is, and what 0.99...is defined to be, or why it needs to be defined.


I was bored, so I wrote a little dialogue! Hope you enjoy.


"I'd like to prove to you that the mathematical object ##@@## equals 1."

"But what do you mean by ##@@## ? Please define it for me."

"Ok, don't worry, ##@@## is just notational shorthand for the limit, whatever it is, of the following sequence of real numbers (written in base-10):

0.9, 0.99, 0.999, 0.9999, ...

If you don't know what a limit is in this context, I'll briefly describe it to you: it's a number. Just one real number."

"OK, I think I follow you; basically, you've defined this thing ##@@## to be the limit of that sequence. But the limit of that sequence is defined to be a number, right? So basically, by a round-about sort of way, you've defined #@@## to be a number."

"That's right."

"Cool! I wonder which number? Is it 12? 23.4543?"

"Nope. It's 1. That's right, ##@@## = 1."


"Calculate the limit of the sequence I gave you (if you don't know how to do this, check out http://en.wikipedia.org/wiki/Limit_of_a_sequence)."


"Gosh, the limit of the sequence is 1. But you defined ##@@## to be the limit of the sequence. So ##@@## is 1."


"But wait a minute. The sequence never actually reaches 1, it just gets closer and closer to 1."

"That's right. What's your point?"

"Well, if the sequence never reaches 1, ##@@## can only be an approximation to 1, because it will never reach 1".

"No. You're wrong there, pal. I didn't mention anything about the terms of the sequence when I defined ##@@##. I merely mentioned the limit, which is just one singe number. I don't care about what the terms of the sequence do, as long as they give me the right limit. If you like, I could redefine ##@@## as the limit of this sequence instead:

1, 1, 1, 1, 1, 1, 1, 1, 1, ...

Since this also has limit 1, it doesn't make the tiniest difference whether I redefine ##@@## to be the limit of this sequence instead of the previous one. In fact, I might just do that - it makes no difference at all."

"But this new sequence has only 1's in it!"

"That's right."

"Hmm, I guess ##@@## must be 1, then."

"Yay, you got it."


Now find-and-replace "##@@##" with "0.999..."



I'm not disagreeing with anybody, but I'm confused by your logic.

"Let x = .9999999..."

Wait...I thought an Infinite number couldn't equal anything...because it goes on forever. (Wait that doesn't sound right let me explain) So you couldn't multiply Infinite by ten because because you still don't know what Infinite is. Same thing is 1/3 = .333... you can multiply 1/3 by 3 to get 1, but you can't multiply .333... * 1 because you cant define .333... without rounding somewhere. All the calculations I see that get .333... * 3 = 1 Round .333... at some point.


Polymath wrote:
"If you claim that "point 9 repeating" doesn't equal 1, then the burden falls on you to find me the number halfway between the two."

Why halfway? Why not just in between the two? Surely if it can be shown that B exists in between A and C, then A cannot equal C, right?

.99 is in between .9 and 1
.999 is in between .99 and 1

This goes on infinitely! Therefore .9 repeating does not equal 1!



i agree that any number in between will do. but i looked through your infinite list of numbers between numbers, and i'm afraid i didn't see .999... anywhere. every entry in that list has finitely many 9's.

read my arguments carefully...trying to find a number between .999... and 1 leads to some bizarre contradictions.


I cannot argue about 1=.99... but I can at least prove that 1=2. If this is possible, anything can happen...

1 = 1 | *x
x = x | x^2
x^2 = x^2 | -x^2
x^2 - x^2 = x^2 - x^2 | 3. binom to the right
x(x-x) = (x+x)(x-x) | /(x-x)
x = (x+x)
x = 2x | /x
1 = 2

Next time someone will tell us that the world is NOT flatt and I'm hanging top down.

PS: feel free to reuse any kind of joke or sarkasm you can find in this comment ;-)


Hi Polymath,

I have spent a lot of time reading many of these posts, but forgive me if the following has already been asked:

is it correct to say that 1-.999...=.000...(1)?

or could it be said to be more acurately represented as 1/oo?

if the latter, can it be shown that there does exist a distance b/w 1 & .999...?

here's what I'm thinking:
(1/oo)oo=1 where as 0*oo=0

Be gentle, i haven't been in a math class for almost a decade.



How can you nerds understand this stuff???
I am 54 and have been studying math everyday for the past 48 years.


I agree that 0.999...=1 and that 1-0.999..=0 but what is 1-0.888...= ?


And also.. if
2/3 = .666...
1/3 = .333...
1 = .999...
Therefore .444...=.333... because .666...+.444..=1 ?



.6666...>.6, right?
.4444...>.4, right?

so if you add them, surely you get something greater than 1.

6/9 + 4/9 = 10/9 = 1.1111....


Oh, my mistake. I'm sorry :)
but what about 1-.888...= ?


Denis, let's actually do the subtraction then:


You know how to carry the one, right? I will use "T" to represent "10" as a single digit:




Couple more repeats:


Well, it's obvious that we would be carrying the one for all eternity (i.e. the 9s will go on forever), so let's reflect that in our equation:


Wait, did you notice something? The 1 just transformed into 0.999...! And since no subtraction has been performed yet (and carrying the one DOES NOT change the value either), 0.999... is equal in value to 1. Let's proceed with the actual subtraction:


Now, what's the fractional representation of 0.111...? It's 1/9. And since the fractional representation of 0.888... is 8/9:

1-0.888... = 0.999...-0.888... = 0.111... = 1/9 = 9/9-8/9 = 1-8/9

Amazing how this all works out, huh?


i am getting a lot of spam on this post. i am closing it to comments for a while to see if that helps.

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