If you want a headache at 12am

[nordberg]

er, um....i thought the answer was 42

[wouter999]

Gerbick did you read his bit at the bottom? 0.99... doesn't itself get closer to 1

Also, something that is infinately small IS zero. Eg take the sum of a geometric progression: Sn=a(1 - r^n)/(1-r) If you take this series to infinity you get S∞=a/(1-r) because the - r^n part becomes infinitely small, tending to 0.

Something that is infinitely small IS NOT zero. zero means a non-existence. Not an infinitely small amount.

Take a piece of bread. You break this apart into tiny little bread crumbs. You break these apart into cells and fibers. All these little cells and fibers are broken to organells and liquids. You break this down into tiny little molecules, which are so incredibly small you can't even begin to visualise what they look like. But you do accept they're there, right? Let's now break down these molecules into atoms. Again, atoms do exist right? Now we break down the atons and we have a load of protons, neutrons and electrons. Then you break those down and what you'll find is quarks, neutrino's and all that mumbo jumbo. If we go any deeper we'll find ourselves in superstring regions, but this is quite far enough.

My point is that no matter how often you cut this piece of bread, you'll still be left with a bunch of stuff, however small it may be. If you cut the bread an infinite amount of times, you'll still have a teensy weensy tiny little bit of matter, infinitely small, but it's still there. It doesn't magically disappear--which would be implied if we accept that infinitely small equals zero.

Assume you bought a loaf of sliced bread. Perhaps you have an unkempt household, perhaps you were away for a couple of days, either way, when you return home you find out that one of your slices is reduced to 0.9 of it's original size. A mouse the size of a golf ball took a chunk of your bread and converted it into little mouse poop. Suppose another mouse, ten times smaller, the size of an ordinary green pea, took a bite out of another loaf of bread. This slice is 0.99 of it's original size. Another slice of bread has been ravaged by an even smaller mouse than that, the size of a grain of sand. That slice is 0.999 it's original size. An even smaller mouse took an even smaller bite out of another slice, et cetera, et cetera.

Now imagine an infinitely small mouse, not even the size of a lone neutron, took a bite out of one of the slices of bread. Does that mean he didn't actually take a bite out of the bread? Of course he did, only the bite is so incredibly small, mathmaticians made up rules with which they can usually neglect this tiny little difference.

There are only a handful of occasions when this infinitely small number actually makes a difference, and this example is one of them. This example deals with two of the most difficult mathmatical concepts, 'infinity' and 'zero'. Both concepts cannot be defined, 'infinity' for its immense magnitude (or in this case, smallness), and 'zero' because of its nothingness. You cannot specify 'zero', because whatever way you try to define it, it will always be an abstract way of formulating 'nothing', and that purpose is defeated the moment you try to visualise or define it. 'Nothing' becomes 'something'. 'Infinity' becomes 'finity'.

[creativeinsomnia]

whoa.

[silverx2]

wouter needs to stop masturbating, and get a girlfriend.

[yasunobu13]

For those of you nonbelievers left out there
No, I'm Sorry, It Does
Refutations

[random25]

i think this quote says it all:
'Today, a young man on acid realized that all matter is merely energy condensed to a slow vibration - that we are all One Consciousness, experiencing itself subjectively. There's no such thing as death, life is only a dream and we're the imagination of ourselves. Here's Tom with the weather....' - Bill Hicks

Thats on the tool album i've been listening to this week.

[MechaPiano]

For those of you nonbelievers left out there
No, I'm Sorry, It Does
Refutations

are you for real? that man should NEVER be allowed to teach! and he's clearly wrong! (and I think it was even in comments on Digg about this)

if you do believe that 0.999... = 1, then you believe that ALL numbers are equal why do you think he leavs out integers? because it's too obvious in that case

[JWin]

The people arguing against this only seem to be using number theory whereas the people for it are using actual numbers. Makes it hard to side with the opposition.

[MechaPiano]

The people arguing against this only seem to be using number theory whereas the people for it are using actual numbers. Makes it hard to side with the opposition.

fine, you think you can handle actual numbers? let's see:

if 0.999... ...999 = 1 then
0.999... ...998 = 0.999... ...999 = 1
0.999... ...997 = 0.999... ...998 = 0.999... ...999 = 1
do you see where I'm going?

urgh, just take any of those "arguments" he has and I'll prove he's wrong

[JWin]

no mecha, cause you are defining finite numbers. .999... ...999 does not = 1 .999... does. according to the theory. Those arent the same and certainly .999... ...998 does not equal .999...

I'm not arguing for either side just pointing out the inaccuracies in your solution. But anyway, the nerds on digg argued it enough for the whole interent
http://digg.com/links/.9999999%3D1

[yasunobu13]

fine, you think you can handle actual numbers? let's see:

if 0.999... ...999 = 1 then
0.999... ...998 = 0.999... ...999 = 1
0.999... ...997 = 0.999... ...998 = 0.999... ...999 = 1
do you see where I'm going?

urgh, just take any of those "arguments" he has and I'll prove he's wrong

Your numbers like "0.999... ...998" make no sense, so that entire argument is nonsense.

[MechaPiano]

Your numbers like "0.999... ...998" make no sense, so that entire argument is nonsense.

please explain why

[MechaPiano]

no mecha, cause you are defining finite numbers. .999... ...999 does not = 1 .999... does. according to the theory. Those arent the same and certainly .999... ...998 does not equal .999...

ok, I see where's the problem now it's all about the concept of "infinity"... I'm afraid I'll not be able to explain that clear enough as I'm not a math teacher.

0.999... does not equals 1

[yasunobu13]

please explain why

0.999... means an unending series of nines, short for
9/10 + 9/100 + 9/1000 + ...

What does 0.999... ...998 mean? An unending series of nines followed by an eight? That doesn't make sense.

0.999... does not equals 1

Yes, it does.

ok, I see where's the problem now it's all about the concept of "infinity"... I'm afraid I'll not be able to explain that clear enough as I'm not a math teacher.

Then why don't you believe math teachers when they say it does?

[MechaPiano]

Then why don't you believe math teachers when they say it does?

because he's wrong, don't twist it! he didn't prove anything! he just didn't!

I can prove he's wrong, it's just that you don't fully understand the concept of infinity so there's no point for me to even try.

however,I have an idea and I will try
this will not prove anything, but it will show you the whole thing from the different angle

ok, here goes:

draw a line. let's say that the length of this line is 1. now, let's split this line in 2. you'll have 2 sections each is 0.5. now, take any section and split it in 2 - we have 3 sections: 0.5, 0.25 and 0.25. Now take the last section and split it in two, you'll have 0.5, 0.25, 0.125 and 0.125. now split the last section in 2... and so on. You can go on like this forever, you can always split a section in 2, no matter how small it is.

now, you have an infinity of numbers, but if you sum all of them, you'll always have 1 (is anything ringing at this point?)

ok, now understand this - when you sum those numbers going from left to right (0.5 + 0.25 + 0.125 + ...) your final number will get closer and closer to 1 (0.999...), but will never reach it, unless you'll add all sections. the last section (that does not really exists, since you're always spliting) is 0.000... ...001 big

"Infinite" does not mean it "has no end" (and at th same time it does). You have infinite numbers, but if you sum them - they'll give you 1 It's like i (-1^1/2), it doesn't really exists, but it is.

I hope this helped just a little bit...

[phires]

Aren't both answers correct? .99999... = 1 and .99999... != 1

To clear up what I mean I'll use a short theoretical example. Lets say I take my index finger and my thumb and I touch the two tips together. I can see when looking at the tips of my index finger and thumb that there is no space between them; lets say this is = 1.

Now if I was some infinitely small creature then there would be an infinite amount of space between my two fingertips; we'll set this = .99999...

From my own perspective I know that if I was infinitely small then there would be an infinite amount of space between my two fingertips (.99999...) but since I can see my two fingertips touching each other I can say .99999... = 1.

The infinitely small creature however, could never see 1 in the way that I see it. That creature could never reach a fingertip, nor would it even know that they exist.

To you and I, infinity is just a concept that we can't really experience and so saying .99999... = 1 is easy to do and makes sense because we experience the limit and only conceptualize of the infinity. To the small creature however, it lives the infinity and trying to place a limit on it would be unimaginable and from that perspective .99999... != 1.

So .99999... = 1 and yet .99999 != 1 just depends on how you're viewing the problem. Since we live in a world of limits, saying .99999... = 1 works for the vast majority of other problems that we need to tackle but that doesn't necessarily mean that it's true in all cases.

[yasunobu13]

ok, now understand this - when you sum those numbers going from left to right (0.5 + 0.25 + 0.125 + ...) your final number will get closer and closer to 1 (0.999...), but will never reach it, unless you'll add all sections. the last section (that does not really exists, since you're always spliting) is 0.000... ...001 big

We're talking about a number 0.999...
We aren't getting closer or farther from anything, simply discussing a single number on the line of real numbers. In your example (0.5 + 0.25 + 0.125 + ...), we also have a number. Adding them up one by one, yes you will get closer to something, but we aren't talking about that. We're talking about the sum of all the numbers.

You're example is called an infinite geometric series. The sum of the series is well defined (and has been for quite some time). It equals 1. It doesn't get closer because it is a single number. I ask that you bring this up with someone who teaches math. This was settled a long time ago.

[gerbick]

I seriously never understood how zero factorial (0!) equalled one either.

[MechaPiano]

We're talking about a number 0.999...
We aren't getting closer or farther from anything, simply discussing a single number on the line of real numbers. In your example (0.5 + 0.25 + 0.125 + ...), we also have a number. Adding them up one by one, yes you will get closer to something, but we aren't talking about that. We're talking about the sum of all the numbers.

You're example is called an infinite geometric series. The sum of the series is well defined (and has been for quite some time). It equals 1. It doesn't get closer because it is a single number. I ask that you bring this up with someone who teaches math. This was settled a long time ago.

yes, this was settled a long time ago, and it's good that you know and understand this. don't leave out the last part of my explanation, that's important!

if you sum all those numbers, you'll get 1, but what if you'll not add the last "piece", the last number in the line, what will happen? what number will you have? do you know the answer?

[yasunobu13]

yes, this was settled a long time ago, and it's good that you know and understand this. don't leave out the last part of my explanation, that's important!

if you sum all those numbers, you'll get 1, but what if you'll not add the last "piece", the last number in the line, what will happen? what number will you have? do you know the answer?

What is the last piece? What does that mean?

I'm starting over.

The question is does 0.999... = 1?

Do you agree that 0.999... is a specific number on the real line?
If so, then all the proofs in those links I provided (plus refutations of the exact same arguments you and others here have tried) are all true.

Saying otherwise is to try and refute centuries of basic mathematics. What specific part of the proofs do you disagree with or do not understand?


Gerbs: http://mathworld.wolfram.com/Factorial.html

The special case 0! is defined to have value 0!==1, consistent with the combinatorial interpretation of there being exactly one way to arrange zero objects (i.e., there is a single permutation of zero elements, namely the empty set).