This was a debate on the other message board I go to, a debate that people had many opinions about:

Does 0.9999 = 1?
or,
is 0.9999 < 1 (less than 1)?

Some people seem to believe that because 0.9999 is so close to 1 that it is pretty much is 1. I have a different opinion about that. To me, 0.9999 repeating is just 0.9999. The decimal 0.9999 is that way because it isn't 1. It falls short of being a whole number. If it was to be 1 then I find it hard to believe that 0.9999 repeating would exist in the first place. I want to hear your opinions about the number.

the answer to the question is simple...

what do you add to 0.99999 repeating to make it into 1? Is there a number that will accomplish this goal? No. So if there is no number you can add to 0.9999 repeating to make it into 1 then there is no difference in their value.

Let's see... How detailed should I be?

.9999 is not equal to 1. People round it up to 1 because they're lazy.


Here's the Calculus response:

Limits are used fairly often when x is approaching either infinity or a value where it is undefined. If x=1 yields an undefined value for y, then there is a simple solution. Find the limit of y as x approaches 1. Basically, use values that are just above 1 and just below 1 to estimate what the value of y would be at x=1 if it was not undefined. .9999 (repeating) is used in almost every case when the limit as x approaches 1 needs to be found.

If the two values were equal, then the limit would not work. Calculus deals with infinite values, so this logic proves that .9999 (repeating) is not the same as 1.


P.S. Don't challenge me when it comes to math . I literally do math for fun .

Hmmm... this is... confusing, to say the least.

After contemplating both Cyro's and Geeo's theories, BOTH sound plausible. However, I'd have to side with Cyro. Math has several holes in it, and this is one of them. Since the number is repeating, it means that you can't add anything to make it = to 1. That said, 0.999999999999... doesn't LOOK like 1, so what would make it equal? It would be less than, wouldn't it? I don't know. No one knows.

BNuge's calculus stuff helps explain it though.

To me, .999999 repeating is .999999 repeating.

I can add .00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 to it to make it 1 (adding a 1 anywhere would do that)

So I guess I see it as basically 1.

No matter what, no matter how far the repeating stretches, you'll always have to add SOMETHING to make it "1". 0.99999~ is a number in itself. Just like 0.88888~ isn't 0.9. That 0.0000001 can be infinitely small, just as 0.999 can be infinitely long.

BNuge : I must whole-heartedly disagree, and therefore challenge you at math.

Using calculus, I believe the way to explain would be to use a geometric series. However, I am lazy this late at night and do not feel like thinking through calc (just took a test earlier today...).

So I will present you all with the very first explanation I was taught as to why .9999 (repeating) is, in fact, equal to 1:

Are you familiar with the fact that if you put anything over 9 (fraction), it repeats itself? 1/9=.111111 ; 8/9=.88888888 ; logic serves that 9/9=.99999999 and as a fraction, 9/9 equals 1.

Take, for example, the repeating decimals .6666666 and .33333. When you add them, in theory, you should get .9999 repeating, right? What about their fraction counterparts?

.666666=6/9=2/3
.333333=3/9=1/3
And what is 1/3 + 2/3 if not, in fact, 1?

In short, decimals, in my opinion, are flawed representations of numbers.


The above info completely blew my mind in precalculus in 11th grade. haha

That doesn't look like it's a good argument. Using fractions is a bit odd. Even more so is that you said 9/9 = .9999999. Open up the calculator on your computer. Put in 9 divided by 9. What do you get?









1, right? Not .999999. 9 pieces out of 9 total pieces is "1" whole. There is no decimal in that anywhere. I'm not sure if you thought that one through enough. When you add up fractions vs decimals you get different answers. 2/3 + 1/3 =1 and .6666 + .3333 = .9999. It doesn't work the way you suggested. When you convert fractions into decimals you play by the rules of decimals. You don't get outside help from the realm of fractions to make the solution of two or more decimals different.

danielbelitch :

6/9=.66666 (repeating)
3/9=.33333 (repeating)

Even though they are repeating, they are both estimates of a fraction. The fraction is the definitive number, while the decimals are just approximations. Adding the approximations is not the same as adding the fractions. Take this example.

1/16+1/32=3/32 True

Let's approximate the decimals of the first two fractions to 3 places.

0.063+0.031=0.094 True

3/32 on Calculator = 0.09375

0.094 is not equal to 0.09375


This example illustrates that adding the fraction values does not yield the same result as approximating the fractions as decimals before adding. Adding 1/3 and 2/3 to get 3/3 is not the same as adding .3333 (repeating) and .6666 (repeating) to get .9999 (repeating). The fractions and the decimals may be close in value, but they are not identical. That disproves your theory.

Remember when I said not to challenge me lol
If you have any other theories, I'll be glad to disprove them

P.S. Cyro also replied to you, but didn't summon.

Disregard this.

BNuge :

Cyro Xero :

Alright. First, Cyro. I don't understand how you can say the fractions don't make sense. From what I'm reading, you nearly quoted my argument in order to disprove it. It's as if you agree, yet refuse to accept the facts. lol
It's like I said, decimals are flawed representations of numbers. Adding fractions vs decimals DOES yield the same result. They're 2 ways to express the same number.

Which leads me to you, BNuge. You agree with Cyro that decimals and fractions are not the same, but look at your very own example. Rounding IS WHY decimals are flawed. Besides, those are finite decimals, there's no need to round:

1/16=0.0625
1/32=0.03125

0.0625+0.03125=0.09375=3/32

So you're right, approximations are only that. However, approximating should only be used in more complex situations.


If that wasn't enough to prove what I said before is true, then I am now awake enough to use the geometric series explanation:


For the infinite sum from n=1 to infinity of any r<1 we have that the sum is equal to
ar/1-r where a is a constant multiple.

So, allowing E to be a sigma it looks like E 9(1/10)^n
9=a
1/10=r<1
where (1/10)^n for n is greater than or equal to 1 yields
1/10 + 1/100 + 1/1000 + 1/10000 +.....

which is .11111 (repeating)
so using our formula ar/1-r to find the sum of this series will yield the true number .11111111 (repeating) times 9 = .99999999 (repeating).
So let's do that:
ar/(1-r)=
9(1/10)/(1-(1/10))=
(9/10)/(9/10)=
1

So we should've gotten .9999 (repeating), yet our answer is just 1 after all.


To avoid further disputes of this information, here's a link I found that actually has both of my explanations, and much more, as to why .999 (repeating) = 1

http://en.wikipedia.org/wiki/0.999...

danielbelitch :

Regarding the beginning of your reply, approximations don't need to be saved for decimals that repeat or constant values like e and π. When counting money, you always round off to two decimal places. Just because a decimal is finite doesn't mean it shouldn't be approximated.

I know it makes me sound dumb, but I don't understand what you laid out. It's difficult to read when it's not written out in a mathematical form. The carats (^) always confuse me. I haven't done geometry in 5 years or so, so I wouldn't remember something like that.

I have Calc II tomorrow morning. I'll try to remember to ask my teacher what she thinks. It could be a nice way to kill time during an 8am class.

Wikipedia can be edited by anyone. It's never a good basis for an argument. The fact that all wiki articles can be edited makes the information questionable. That's why a lot of teachers won't accept it as a source for a project.


P.S.
Cyro Xero : This is the first debate thread that I've enjoyed coming back to. I may spend more time in this forum after this.

BNuge :

First of all, it was a Geometric Series, nothing having to do with geometry itself, really. You should have learned it by now in Calc2. My calc2 teacher is actually the one that told me a couple weeks ago that the fraction thing isn't 'formal.' And that the definition of a repeating decimal is a geometric series. But anyways, everything I did is laid out under that section in the wiki article. Wikipedia may be able to be edited by anyone, but it's constantly checked and updated to be accurate. They try to eradicate false info as soon as possible. Not to mention:

"Posted on 02-27-11 09:51 PM BNuge is Offline Post: 52 words - (ID: 340800) - Report Abuse | Link | Reply to Post
In Philosophy 101, we learned a lot about fallacies. My favorite that I didn't see in your list is Reductio ad Absurdum. It translates from Latin to mean Reduction to the absurd. It basically proves that an argument contradicts itself.

Further explanation is on Wikipedia:
http://en.wikipedia.org/wiki/Reductio_ad_absurdum


Also, Poisoning the Well doesn't have an explanation. "


Found that in the logic 101 thread. It appears that your taste in wikipedia isn't all bad. But anyways, the math symbols and whatnot are in the article if you want to take a look at the series. I love my math teacher. I doubt yours will disagree with him, but let me know she says.

*PS I don't mean to just bash you for downing my wiki article, I just liked how I got an ironic quote from you is all.

I still stand by my basic argument that if .999... is actually 1, why does ".9999..." still exist as a number? Why wouldn't a rule be made stating that ".9999...8, 1", instead of .9999...8, .9999..., 1"?

It seems to me that all these equations are tools to turn .9999 into 1, and don't really explain why it should be that way.

danielbelitch :

I never said I didn't like Wikipedia. I just said that the information isn't 100% reliable. In the context that you quoted, I wasn't using it for an argument. It was meant to give a bit more depth into my brief explanation. It's all good though

I had another thought earlier today and I generated a chart to prove my point. As x decreases on the graph below, y gets closer and closer to 1, but remains just below 1. As x increases on the graph, y gets closer and closer to 1, but remains just above 1. However, the domain of the graph is such that y can not be equal to 1. Regardless of what value you choose for x, y can never be equal to 1. .9999999999 will be extremely close to 1, but it can never reach 1 because that would make the function undefined.




Image upload: 669x371 totaling 64 KB's.

BNuge : Yes, I definitely understand what you're saying, limits weren't ever that bad to understand for me. I agree with your argument to a point. I don't like to believe it either, but the repeating decimal .999 is defined by an infinite sum. THAT'S what took me a while to overlook. How can an INFINITE sum have a finite solution? The concept just lags around in my mind, but I've accepted it overall.

8am for calc2 is rough, though. Sorry to hear that. Mine's at 5pm haha. Got calc3 in the fall at 3 o'clock--I'm not crazy for morning classes. But anyways, did you have time to ask your teacher what she thought?

I'm think I'm pretty much over the 'debate' part of this topic though. I can't do much more to support my theory other than direct you to youtube. lol At least that's more visual. I wouldn't have understood what I typed about the series either if I wasn't the one who did so.

BNuge, pretty much said exactly what I was going to say regarding maths, and he even had a graph almost exactly like mine to demonstrate that point

Here was mine. The asymptotes are not actually drawn in, but you can use the axes as them as y=0 and x=0 are them.

So yeah, like BNuge said, in the equation if the graphed line hits the asymptote then it is undefined, but as long as it is close it will never reach like in the graph below the line will go really close to the axis like .9999... to 1, but never actually reach it.






Image upload: 800x500 totaling 26 KB's.

septembern : I concur. That is EXACTLY what happens on a GRAPH for an EQUATION. Here's the bottom line that I'll apparently never get anyone to agree with though: (lol)

A repeating decimal is NOT an EQUATION. It doesn't have LIMITS. It is a number. The graphs you can generate have asymptotes because of a domain error: you can't divide by zero. And thus any EQUATION that is a fraction cannot have any circumstance where zero is in the denominator. The fact that I've stated before is that .9999 repeating is a Geometric Series and hence equal to one by definition. Clearly I can't explain well enough myself, but since septembern replied, here's my last attempt, someone actually explaining the simpler ways:

http://www.youtube.com/watch?v=79Q08UYknTY&feature=related

He does quite a few examples. I mean, it is fact in the math world. It's just killing me inside that I can't get anyone to agree on here... You know what I mean? haha oh well

danielbelitch : good post and good video. I agree with you.

0.9999(repeating) = 1

That's just the way it is.

geeogree : Thanks!

I'm satisfied now.