# Does .999999~ = 1?

no this isnt another one of my joke topics.

Try to answer the question and say why you think it does or doesnt.

Yes, actually, it does.

A few weeks ago (coincidentally) I read a pretty massive topic on another forum about this exact dilema, which finally concluded (after extensive debate) that 0.999… really does equal 1.

The most classic demonstration of this uses fractions.

We know that 1/3 = 0.333… (I’m using “” because we can’t draw the appropriate line above the repeating characters on this forum).

And, 2/3 = 0.666…

Thus, 1/3 (0.333…) + 2/3 (0.666…) = 3/3 (0.999…) = 1.

The problem lies with decimal math, and it’s inability to express infinity.

very good theory. My answer is differant, but if anybody else has a reason go ahead and post it.

actually, ill jsut post it now because it doesnt seem like this will be a popular topic. LOL, i googled it but it makes sence to me. I dont think that a basic knowledge of geometry could actually figure it out this way.

mathforum.org/dr.math/faq/faq.0.9999.html

does it make sence to you?

I like Athiest’s idea better…less confusing. I normally could decipher yours but I’ve been really tired recently.

lol, im in calculus and this stuff sounds pretty familiar.

I recall a similar problem concerning the cumulative area under the curve of y=1/x. This was done by adding up rectangles defined by the curve. With change in x representing the base of each rectangle and the resulting y being the height, all rectangles added up. Theres two ways to go about this, with part of the rectangle going above the slope, and the rectangle stopping at the slope leaving unaccounted for area. Okay, so naturally with the curve sloping downward as x increased, one would think that it would never reach infinity and stop at like 1.75 total area or something. Wrong, with the x approaching infinity and part of the rectangle going above the curve in the second way to determine area, the area of the space under the curve 1/x, as x approached infinity, was actually infinity! (its surprising, cuz you look at the downward slope and think ‘wtf?!?’)

In other words summation from 1 to infinity of 1/n.

This is later proven through integration. For all of you who know the lingo, it is integral from 1 to infinity of 1/x*dx. (dx being change in x, i think lol). The integral of 1/x is ln(x). This then means ln(x) from 1 to infinity, or ln(infinity) - ln(1). The ln of infinity is infinity, and infinity minus 1 is still infinity.

I thought this was one of the actually interesting things in calculus. The rest of it i have repressed. Oh wait, another memory is coming up from the darkness, yeah, Squeeze Theorem. Anyone hear about it? I think that thats even more insane.

Nice theory Sentient Haven’t heard about the Squeeze Theorem though.
Atheist’s explanation and dnLMicky’s url are also perfect examples to prove 0.999=1. There are probably many other proofs of this… but it really comes down to the fact that decimals are insufficient to express infinity, so you have to work with limits. Hence the possible confusion.
There’s nothing much to add I guess…

Bizarrely enough, I was thinking of this exact topic before I came onto the forum today. I was even thinking of posting about it.

Yes it’s true, somehow 0.9999… = 1.

The 1/3 thing is a good way to work it, and the 1/9 one works as well:

1/9 = 0.111…
2/9 = 0.222…
3/9 = 0.333…
4/9 = 0.444…
5/9 = 0.555…
6/9 = 0.666…
7/9 = 0.777…
8/9 = 0.888…
9/9 = 0.999… or 1.

hmm… This is confusing.

Surely 0.9999 = 0.9999, and 1 = 1 ?

How can 0.9999 = 1, if its 0.9999? Doesnt make sense…

because it is .999999999999999999999999999999999999999999999999

repeating forever.

Since it repeats forever, it is really one, somehow…

i guess i kind of get it and kind of don’t…

like .9 could equal one fraction and .99 another and .99999999999999 another, and they keep getting closer and closer and closer to equalling one, so at infinity, then, they actually do reach one?

is that right?

think about it, say .9 to the millionth power… well i can’t comprehend how large that would be… but you would only need to add just a tiny little .1 to the milionth power

(a decimal followed by a milion or so zeros followed by a one) to get it to make one.

but since .9 repeating goes on forever, you actually don’t need to add anything to it to make it equal one, because at infinity, that value would in fact be 0?

I don’t know anything at all about math, could someone back up my theorizing or shoot it down? If we mapped infinity and therefore oculd not go beyond infinity, where .9 repeating ends, at infinity, nothing else could be added to it to make it turn into one, becuase it would already be there?

would the same then hold true for anything that inifnitely repeats?

Another way to approach the question is to subtract.

``````  1.0000000....
- .9999999....
--------------
0.0000000....
``````

Sure looks equal to me. What about the “1” at the end, I hear you ask?
Well, I’ll write it as soon as I finish writing infinitely many 0s.
Any decimal place you name (say, the four billion three hundred
twenty-eight million two hundred seven thousand four hundred
ninety-fifth) has a 0 in it. A number with a 0 in every decimal place
is certainly 0.

that makes sense to me.

o_O

Oh man a math related thread AND

Well, I would have to say that infinity and, well, recurring numbers, are beyond our comprehension, but the subtraction argument is probably the most convincing (IMO) that I’ve seen.
I mean, 3/3 does equal one, .3333 is more of an approximation, so multiplying it by three will still only give you an approximation.
And saying it’s like a geometric progression - well that tends towards a number, rather than being equal to a number
The subtraction, however, is less disputable.

I know at least two ways to prove that 0.999… = 1 holds.

The easiest is to study this equation:

x = 0,999…
10x = 9,999…
10x - x = 9,999… - 0,999…
9x = 9
x = 1

You have actually half way proved it there. Just use the same kind of theory and apply it to infinite sums! For example 0.999 can be seen as the infinite sum 0.9 + 0.09 + 0.009…

This is a geometrical sum with the quotient 10. There is a theorem that says that any infinite geometrical sum that has a quotient > 1 exists as a finite number, which is equal to the limit of the sum (I don’t know the proof of that theorem by heart). 10 > 1 → 0.999… exists. The limit of 0.999… is of course 1. (If you don’t believe that, you can rewrite it as: 1-1/(10^n) which clearly goes towards 1 when n->infinity!)

This is one of the major bases… if .999… doesn’t =1 calculus is all wrong, because limits don’t work.

I like Atheist’s version of the topic. It is simple easy to understand and to the point.

The mathforum.org/dr.math/faq/faq.0.9999.html sure has a long drawn out way of doing it.

Ok, me and my Grade 10 math skills is finding this a bit confusing, I like Athiest’s idea, but for WHY it does this here is my imaginary theory of why this is true

(humor me)

So you know parallel lines? Now they say that they will touch because if they go on forever somehow the galaxy will bend them. It would make sense that .99999 if you made it into a three dimentional figure would do the same, slowly bend into 1.

Ok, I’m done

Also for ppl who aren’t familiar with fractions…

1/1, 2/2, 3/3, 4/4, etc. (any numbers that are same above and below) always are equal to 1 in math’s theory, but since they have showed to be .9999~ also… that means .999~ = 1

Hope that makes everything a bit clearer.

I have to agree that .9999~ = 1… math formulas are there to prove that.

Actually in my opinion… in real world… NONE is pure solution in whole number… everything has fractions or decimials. We always round everything up. .9999~ is 1 in real world’s eye. At least it is in my eye. Hahaha.

Interesting one I must say!

if .99------------- = 1 then how come .333------------ doesn’t = 1?

1.0
-.3333333333333333333333333333333333333

well, that would have to be .000000000000000000000000000000000 forever, so… assuming there were a last number it would have to be a 7… but…
am I missing something very obvious?

I dont understand still. so calculus is wrong.