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0.999...=1 and why people believe it is false.


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Since you cant go an infinite amount of places to the right you will only be able to read 0.00000000000.... forever; therefore, .999... is 1

 

 

 

1.00000000->

 

-0.99999999->

 

--------------

 

0.<-00000001

 

 

 

The infinity goes in the direction of the decimal point forever, but there is still always going to be that 1 on the far right.

 

 

 

 

 

Infinity does not end, there is no far right for that 1 to be on, as such it doest exist.

 

 

 

I never understood this concept, even as my last course being AP Calc. If I walk precisely 99.999...% am I touching the wall? I assumed that I would begin walking infinitely closer to the wall, without ever touching it. Assuming that based on your proof that 99.999 also equals a value equal to that of 100. I understand the proof, don't get me wrong, I don't understand the application.

 

 

 

 

 

99.999 isnt equal to 100, 99.9... is. The difference is that in the latter case the 9's after the decimal point go on forever

there are no stupid questions

just way too many inquisitive idiots

balance is scary to people who like things easy for them

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As I said, 0.9999....Exists only in theory, as infinity exists only in theory. For example:

 

 

 

No matter how big a number is, you can always add one to it.

 

 

 

So, In theory, you're right. However, 0.999... Cannot really exsist, so I think that would make this entire situation impossible...?

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So, would you not agree that 0.999...9 + 0.000...1 = 1?

 

 

I'm going to use this as an example, as it seems my point was missed. I'm saying Infinity cannot be compared to a regular decimal number as infinity itself is not a real number; it cannot be treated in the same fashon. Just as the number is infinietly increasing, it's increasing by an infintly small amount. An example would be this:

 

 

 

X = Real # (I.E. 10)

 

 

 

Loop

 

X = X + (X * .5)

 

End Loop

 

 

 

X will increase an infinite amount, but it will never reach, say, one million, as it is increasing by an infinitely smaller amount.

 

 

 

 

Whoa... Madman and Compfreak agree with me on this topic? I wasn't that expecting at all.

 

You know, aside from those two debates, I generally tend to agree with you. Before that, I used to think of you as someone who mirrored my ideas\beliefs fairly closely. Then again, if I was right, your stubborn and abrasive when arguing (just like me) so it makes sense that the tiniest difference would spark an explosive argument :lol:

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As I said, 0.9999....Exists only in theory, as infinity exists only in theory. For example:

 

 

 

No matter how big a number is, you can always add one to it.

 

 

 

So, In theory, you're right. However, 0.999... Cannot really exsist, so I think that would make this entire situation impossible...?

 

 

 

 

 

The square root of -1 can not really exist either, but you would be amazed at the vast number of things its used in and the mathematics thats built up around it.

there are no stupid questions

just way too many inquisitive idiots

balance is scary to people who like things easy for them

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99.999 isnt equal to 100, 99.9... is. The difference is that in the latter case the 9's after the decimal point go on forever
It was a flaw on my part, but almost implied given the subject. In any case, thank you, I will correct that.
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0.999... != 1

 

 

 

The reason they're not equal is that 0.999... is not a number. 0.333... is not a number. It's an approximation of 1/3. If you were to divide 1 by 3, it would never end, there are an infinite amount of digits. The same goes for 0.999... How many 9s are there? Infinite. Infinity is not a number, and a number can not have an infinite number of digits. You could, and rightly, assume that it would infinitely remain 3, or 9, or whatever your number of choice. However, like pi and e[]/i], it is not truly a number, it is a limit.

 

 

 

A limit is best described as a theoretical number, in my opinion. In some cases, the limit is also the true value of a function. For example, the limit of f(x) = x + 1, as x approaches 1, is 2. This means that as x gets infinitely close to 1 from either side, the answer becomes infinitely close to 2. In thise case, f(1) is also equal to 2. This holds true for some limits, but not all limits.

 

 

 

For example, the limit of f(x) = x / x, as x approaches 0, is 1. As the value you enter gets infinitely close to x, the value remains 1. However, anyone with even the most basic mathematical knowledge can tell you that you can't divide by 0. 0 / 0 is not 1, it simply does not exist. x / x as x approaches 0, on the other hand, exists as the limit of 1.

 

 

 

So how is this relevant? 0.999... is NOT a number. It entails infinity, so it is the limit as the number of digits approaches infinity. That is the key misconception. The limit, as x approaches infinity, if x is the number of 9s, is one. However, .999... by definition can not equate to 1. The moment you perform any mathematical function on it, be it dividing it, multiplying it, adding to it, or whatever else you used in your proofs, your answer is no longer a number. Your answer is a limit.

 

 

 

Tl;dr version: The limit of 0.9 repeating is 1. 0.9 repeating is not 1.

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Infinite means there is no limit. .999... means that there will always be more until it equals one. Which will not happen, but if we go on FOREVER and keep adding more it will go to one.

 

But it's also decreasing infinitely at the same rate, keeping it from ever reaching 1. It's like standing 10 feet from a wall and halving the distance away from it each second. You can do that forever, but you will NEVER reach the wall.

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Soma, your terminology is wrong as is your conclusion.

 

 

 

0.9... isnt an approximation of anything, it is a representation. The elipsis simply means that the preceding characters are repeated forever.

 

 

 

In particular 0.9... is a representation of the sum capture5t.png, (-1)^2 and -e^(i*pi), they are all exactly the same number. Just because they look different doesnt mean that they are.

 

 

 

Also Pi, and e are numbers, transcendant numbers, yes, but numbers nethertheless.

there are no stupid questions

just way too many inquisitive idiots

balance is scary to people who like things easy for them

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I agree with soma's post me. It would indeed be a limit. The fractional proof does not work because decimals are never as precise as fractions. (.333... does not actually exist, it only represents something which does.)

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Infinite means there is no limit. .999... means that there will always be more until it equals one. Which will not happen, but if we go on FOREVER and keep adding more it will go to one.

 

But it's also decreasing infinitely at the same rate, keeping it from ever reaching 1. It's like standing 10 feet from a wall and halving the distance away from it each second. You can do that forever, but you will NEVER reach the wall.

 

Its hard to fully understand the concept of infinity.

 

 

 

Its like dividing by zero. 5/0 what do you get?

 

 

 

PS: I remember a math trick I used in 7th grade for fractions. 1/9 = .111... 2/3 = .222... 3/9 = .333... And then 9/9...

 

 

 

Its a weird concept.

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Infinite means there is no limit. .999... means that there will always be more until it equals one. Which will not happen, but if we go on FOREVER and keep adding more it will go to one.

 

But it's also decreasing infinitely at the same rate, keeping it from ever reaching 1. It's like standing 10 feet from a wall and halving the distance away from it each second. You can do that forever, but you will NEVER reach the wall.

 

Its hard to fully understand the concept of infinity.

 

 

 

Its like dividing by zero. 5/0 what do you get?

 

 

That's understood perfectly, but it takes a math genius to explain it. This page does a great job: http://www.math.utah.edu/~pa/math/0by0.html

 

 

 

PS: I remember a math trick I used in 7th grade for fractions. 1/9 = .111... 2/3 = .222... 3/9 = .333... And then 9/9...

 

 

 

Its a weird concept.

 

That's not a weird concept, it's perfectly understandable. .111111111111 is an approximation, .111... is not a number as it relies on infinity, and as such cannot be compared to regular numbers.

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NEW INFO: I just re-read that page, and discovered it has a great example to back up what I'm saying. The link is http://www.math.utah.edu/~pa/math/0by0.html

 

 

 

What is infinity? It is not a number! Why not? Because if we treated it like a number we'd run into contradictions. Ask for example what we obtain when adding a number to infinity. The common perception is that infinity plus any number is still infinity. If that's so, then

 

infinity = infinity+1 = infinity + 2

 

 

 

which would imply that 1 equals 2 if infinity was a number. That in turn would imply that all integers are equal, for example, and our whole number system would collapse.

 

 

 

It's written by a professor of math at a prestigious university, credentials that I daresay outstrip anyone else here, so I think it can be considered much more reliable :P

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Since I'm terrible at math, my favorite way to approach this one is actually pretty simple, and thankfully in layman's terms.

 

Find a number that is larger than 0.999... but smaller than 1.0.

 

You can't do it. They're the same number.

 

 

 

.9999

 

He's using the "..." at the end to represent infinity, but since infinity is NOT a number, any equation associated with it CANNOT be compared using standard algebraic expressions. See above for a math professor's much better explanation of it. Basically, comparing infinity to real numbers is saying Apple + Orange = Apple.

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GWD drops: 14,000x Bars, 1x Armadyl Hilt, 2x Armadyl Skirt, 4x Sara Sword, 1x Saradomin Hilt, 8x Bandos Hilt, 8x Bandos Platebody, 9x Bandos Tassets, 4x Bandos Boots, 43x Godsword Shard, 82x Dragon Boots

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Since I'm terrible at math, my favorite way to approach this one is actually pretty simple, and thankfully in layman's terms.

 

Find a number that is larger than 0.999... but smaller than 1.0.

 

You can't do it. They're the same number.

 

 

 

.9999

 

He's using the "..." at the end to represent infinity, but since infinity is NOT a number, any equation associated with it CANNOT be compared using standard algebraic expressions. See above for a math professor's much better explanation of it. Basically, comparing infinity to real numbers is saying Apple + Orange = Apple.

 

 

 

I know, I was just implying that you can't assume 2 different numbers to be equal just because humans can't divide to such tiny numbers.

 

 

 

1/3 is not equal to .333

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Since I'm terrible at math, my favorite way to approach this one is actually pretty simple, and thankfully in layman's terms.

 

Find a number that is larger than 0.999... but smaller than 1.0.

 

You can't do it. They're the same number.

 

 

 

.9999

 

He's using the "..." at the end to represent infinity, but since infinity is NOT a number, any equation associated with it CANNOT be compared using standard algebraic expressions. See above for a math professor's much better explanation of it. Basically, comparing infinity to real numbers is saying Apple + Orange = Apple.

 

apple-orange-cropped-7378931.jpg

 

 

 

OT: Its hard to get the vinculum (yes the line over the decimals) on the computer and tip.it'ers are lazy.

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I remember posting this on the last 0.999.. thread

 

[hide=]hardoeq.jpg[/hide]

 

 

 

everybody can shut up about this now. unless they have questions about my proof of course

 

 

 

I couldn't read part of your writing but one part that I could read I feel is incorrect.

 

 

 

No matter how big the fraction is, 1 over anything is not 0.

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I consider myself decent at math (Currently in an honors precalc class). But these proofs just seem off to me, and i refuse to believe that .9999... is equal to 1. .9repeating is as close to 1 as you can get without actually being 1. Just like you can't properly convert some fractions into decimals. (1/3) is understood to be represented as .3333...., so by adding (1/3)+(1/3)+(1/3), in decimal form is .9999... , but fractionally it equals (3/3) or, 1. So I say that fractions are more accurate and should be used. Decimals, although jsut another form of fractions, should be left to the obsolete computers that can't handle fractions. Just like how leaving the root symbol in an answer is more accurate than trying to calculate all the decimals. .999... can be ROUNDED to 1, fine. But it can never equal it.

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I remember posting this on the last 0.999.. thread

 

[hide=]hardoeq.jpg[/hide]

 

 

 

everybody can shut up about this now. unless they have questions about my proof of course

 

Why ignore my post? your comparing a number to a concept, something that CANNOT be done. Infinity cannot be used in an algebraic equation for reasons mentioned above. Any equation in which you try to include infinity (i.e. .999 infinity = 1) is completely invalid as infinity is not a number and does not follow rules that apply to our number system. Remember, it's not me saying it, it's a collage professor ;)

 

 

 

 

With proofs, you can basically make any number equal anything. At least from what I've seen.

 

You generally can, but everyone is still ignoring the fact that we AREN'T comparing numbers, we're trying to compare a number with a concept. It's like saying .999 X Shoe = 1. It cannot be done.

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Drops: 1x Draconic Visage, 56x Abyssal Whip, 5x Demon Head, D Drops: 37, Barrows Drops: 43, DK Drops: 29

GWD drops: 14,000x Bars, 1x Armadyl Hilt, 2x Armadyl Skirt, 4x Sara Sword, 1x Saradomin Hilt, 8x Bandos Hilt, 8x Bandos Platebody, 9x Bandos Tassets, 4x Bandos Boots, 43x Godsword Shard, 82x Dragon Boots

Dry streak records: Saradomin 412 kills Bandos 988 kills Spirit Mages 633 kills - Slayer Sucks

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