Is 0.999… = 1 ?
4:58 am on November 23rd, 2006 | Tech
What would you say if I said that 0.999… (Also written 
or
) is equal to one? NO! Obviously… since we think that let there be any number of 9′s after the decimal, its still infinitesimally smaller than 1. That’s what I thought out loud when I first saw this article. But the weird thing is that there exists not one but numerous proofs of varying difficulties proving the above fact that indeed they are equal if we take into account Real Number System i.e., the normal numbers we use for the mathematically challenged.
Proof by Fractions:
Algebraic Proof:
The proof which is so obvious is based on a few basic assumptions of Real Numbers but they are assumptions anyways. Any one of those assumptions have to be undermined to make this equality void.
For A more lucid account see Wikipedia’s Featured Article on 0.999…
23 Responses to “Is 0.999… = 1 ?”
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Your first proof is obviously flawed.
The second one would convince any engineer. But how many nines can a finite universe contain? Still flawed.
This is completely and utterly wrong.
First of all, .3333… does not equal 1/3, it is simply the decimal approximation. this can be proven by multiplying each my 3:
.333*3=.999; (1/3)*3=1
Furthermore, a variable cannot change when you want it to. It must always be the same. in this algebraic proof, the author adds a 9 at will:
If c=.999, 10c= 9.99, not 9.999.
Therefore, 10c-c=8.991, not 9.
You cannot assume an infinite number of 9′s when you change the variable. If c has infinite 9′s, then 10c has infinity-1 nines.
Any idiot can tear this apart. Also, Wikipedia is never a valid source for anything. If you disagree, just look at their slogan :”the free encyclopedia that ANYONE CAN EDIT.”
Wow…..these guys posting before me don’t seem to understand the meaning of repeating. .999 is just his way of writing .9 repeating. But then again, weaker minds tend to have trouble grasping infinite. And yes, .333 does not equal 1/3, but when .333 is a quick way of annotating .3 repeating, then it is equal to 1 third.
Nick, you forgot that infinite-1 is still infinite. So the proof is perfectly valid.
http://en.wikipedia.org/wiki/0.999
there, do you agree now?
Oh my God. John and Nick may want to get some basic Math lessons.
0.333… * 3 is exactly equal to 1. The three dots (ellipsis) following the last ’3′ indicate that the number is not a decimal *approximation*, but is in fact the exact decimal expansion. The ellipsis indicates that the expansion repeats without end.
I think Harsha’s basic statement should probably include some information about terminology, to avoid confusion by people such as Nick and John. Nick is correct that 0.999 is not equal to 0.99. He did not understand the mathematical convention that an ellipsis indicates a continuing, repeated expansion. 0.99… is exactly equal to 0.999….
The first proof could use a little work, in my opinion, to make it easier for non-mathematicians to follow. Something like:
Let k = 1/3 = 0.3….
3k = 3 x 1/3, = 1.
But, since k also = 0.3…, 3k = 0.9….
Thus, 0.9… = 1.
Something along those lines…
If we could not have trusted these basic assumptions, we shouldn’t have built the limiting idea , differentiation and integration years ago.So it seems useless to dig this issue.
Those who cannot accept the fact that 0.999… is = 1
would by their own logic deem motion impossible via Zeno’s Paradox where you must get 1/2 way to a distance before getting there. If you keep doing that you reach into infinity where it would seem that you can never reach your destination because that small distance between you and your destination. However many people can’t grasp the concept of infinity that if you keep going for an infinite amount of time you will reach that point because it will add up to the distance from your original position to your destination and because all it requires is infinite time, it is satisfied because time in itself is infinite. You can divide one minute into an infinite amount of minute “time” segments and therefore have that infinite time to get from point A to B. Else motion would be impossible.
@Nick:
You’ve seem to have missed the point of repeating 3′s as Prospectus has mentioned. Moreover don’t slander wikipedia. Its not like its a graffiti wall where anyone can paint. Try editing some good page with whatever you like … It will be reverted to its original content within seconds. ( BTW have you ever even tried editing/contributing wikipedia? )
Hope this clears up ur thing: http://en.wikipedia.org/wiki/Talk:0.999…/Arguments#This_is_Ridiculous.21
The way I tend to look at this with no mathematics is;
Any two different numbers will have at least 1(see note) number between them.
There is no number between .999…(repeating 9s) and 1. Since there is no number between them, they are equal.
note = I think there are, technically, infinite numbers between any 2 numbers, but I just leave it at 1 to make it simpler, and I’m not positive about that.
It’s not a ‘Proof’. but I think it is a simpler way to view it.
The point here that everyone seems to be missing is that there is no such thing as .333…
.333… is called an approximation, and it represents the limits of man made measuring instruments. It is not, however, a number. ’1′ *is* a number, and actually, by the logic presented here, is the same as ’1.000…’, even though the intent of that approximation is to indicate the slightest augment above 1 itself.
And that’s kind of the point: numbers themselves are man made measuring instruments – simply components of the language intended to allow us to express quantity in writing. This is evidenced by fraudulent accounting – writing ‘$1,000,000′ on a ledger line somewhere does NOT magically create 22 pounds of hundred dollar bills.
So, the reality is that while mathematicians play with numbers and symbols, there really is no such thing as .3333…; mathematicians just use it and other abstract concepts as an excuse to work on something besides mowing the lawn.
.999… = 1 – (1/infinity) by definition
.999… is only exactly equal in finite equations.
in infinite equations such as:
(1 – .999…) x infinity. it does not hold up because (1-.999…) is exactly equal to (1/infinity) in any equation and when multiplied by infinity is equal to 1.
Cliff, you’re an idiot. Learn basic mathematical terminology, and learn it correctly.
.333… is not an aproximation. .333… = 1/3 EXACTLY. It is the EXACT decimal expansion of the result of one evenly divided into three parts. .333 (no elipsis) would be the approximation you are referring to.
Try this: use the approximation and the decimal expansion to figure out 2/3:
2 x Approximation of 1/3 = .666 (no elipsis)
2x Decimal = .666…
Best Aproximation to 3 decimal places = .667 (no elipsis)
Sleeper:
You’re dividing by 0. F(x) = 1/x. As x approaches infinite, F(x) approaches 0.
By the same token, I can prove that 1=0
A=B
A/B=1
With me so far?
Now, let A=0
Dave is right, but he should have said Multiply, not Divide.
1/infinite would be equivalent to 0.000…01 (an infinite quantity of zeros preceding a 1…) For the same reason that 0.999… = 1, 0.000…01 = 0
inifinites do strange things like that…
WOW
you girls r trying to convince people that :
0.999… = 1 or
0.333… = 1/3
am i glad i’m a programmer ! In my world :
“0.999…” = “0.999…”
“0.333…” = “0.333…” and oh ya
1 = 1
get it ?
for those of you who don’t understand… this is something you learn in high school geometry but use as fact in 90% of calculus problems. No mathematical applications past algebra would exists without this assumption: limit(n/(n+1)) n->infinity = 1
you can’t really argue with fact.
http://www.coolmath.com/limit1.htm
How about this?
Two numbers are either the same or different.
If they are different, one must be less than the other.
Then there must be another number between the two.
But no number exists between 0.999… and 1.
Does this proof work or is there a flaw?
zeno’s paradox has been dismanlted. although one may think that it is impossible to cover an infinite number of distances, in actual fact one is actual moving a finite distance. with a finite distance, amotion is thus possible. and .3333… is in fact an approximation of 1/3, since .333 is an infinite series. Simple mathematical functions such as multiplication and division do not work with simple numbers. infinitely many .3333s minus an infinitely .2222s is undefined, not an infinte string of .111s. The logic in accepting the notion of infinity as mathematically defined by this way is thus flawed.
@mike: the thing is, we’re using the concept of a limit. Its like an asymtope (sp?). it will tend towards the value, but never reach it. NEVER. .999 is not 1, it simply tends to one. or else its like saying .98888 tends to .9999, so on till i reach 0 = 1, 0= 2 so on. limits are limits, not the exact value.
Exactly! .999… never reaches 1 & that’s my point. Just like in the demonstration on the website. You can keep adding sides to a polygon… but you can add sides forever and it will NEVER truly be a circle. You can keep subdividing 1 forever but it will never be a whole number. This argument is the difference between number theory and poor arithmetic. .999… != 1
You’re right Dave… but I didn’t mean to make you mad. I’m really not an idiot – I’m saying that even if we infinitely write threes (and that’s what .333… really means), it still does not equal 1/3. It just approaches it. Maybe that’s the ‘appro-’ word I’m looking for – sue me.
If you get fired up enough about repeating digits to call someone you’ve never met an idiot, then perhaps it’s time to chill out.