.999... = 1

[InkL0sed]

But the difference between 1 and 0.99recurring is not a really small number. It is zero with an infinite number of zeros after the decimal place. Which is identical to zero with infinite degrees of accuracy.

If it is zero, then you wouldn't be writing 9's after the decimal place, and 1 before it. Besides, that wasn't the point - proof by ignoring the other persons argument is not, I assure you, a recognised logical method.

Archimedes said that infinitesimals do not exist due to the Archimedean property. But the Archimedean property does not hold. Therefore infinitesimals can exist, meaning that an immeasurable distance between 0.999... and 1 can be defined, given the series (1-x) where x = 10^-n, for n an element of the natural numbers.

I don't see how the Planck length has anything to do whether our abstracted number system can have an infinitesimally small number or not.

[e_i_pi]

But the difference between 1 and 0.99recurring is not a really small number. It is zero with an infinite number of zeros after the decimal place. Which is identical to zero with infinite degrees of accuracy.

If it is zero, then you wouldn't be writing 9's after the decimal place, and 1 before it. Besides, that wasn't the point - proof by ignoring the other persons argument is not, I assure you, a recognised logical method.

Archimedes said that infinitesimals do not exist due to the Archimedean property. But the Archimedean property does not hold. Therefore infinitesimals can exist, meaning that an immeasurable distance between 0.999... and 1 can be defined, given the series (1-x) where x = 10^-n, for n an element of the natural numbers.

I don't see how the Planck length has anything to do whether our abstracted number system can have an infinitesimally small number or not.

Because the Planck length example transforms the argument from theory to the real world. Unless you can think of some other place where human-invented numbers are used?

[InkL0sed]

But the difference between 1 and 0.99recurring is not a really small number. It is zero with an infinite number of zeros after the decimal place. Which is identical to zero with infinite degrees of accuracy.

If it is zero, then you wouldn't be writing 9's after the decimal place, and 1 before it. Besides, that wasn't the point - proof by ignoring the other persons argument is not, I assure you, a recognised logical method.

Archimedes said that infinitesimals do not exist due to the Archimedean property. But the Archimedean property does not hold. Therefore infinitesimals can exist, meaning that an immeasurable distance between 0.999... and 1 can be defined, given the series (1-x) where x = 10^-n, for n an element of the natural numbers.

I don't see how the Planck length has anything to do whether our abstracted number system can have an infinitesimally small number or not.

Because the Planck length example transforms the argument from theory to the real world. Unless you can think of some other place where human-invented numbers are used?

Can you think of some other place where abstractions are used?

An abstraction is an abstraction, not a string or a stone, despite the etymology of the word calculus.

[john9blue]

His "proof" that the two numbers were unequal relied on his initial assertion/definition that they are unequal. My statements were merely to show that it is no proof at all to say the numbers are in different domains, because you cannot say that before determining whether or not they are equal. I did not ignore his argument, I brushed it away as rubbish

This is all very silly though, its clear that neither side will convince the other, because we have been restating the same arguments for many pages. I will stick with the commonly accepted mathematical philosophy, if you continue to disagree with the mathematicians that is fine.

Dammit Sully, you're no fun. This is the off-topics forum, stop respecting other people's opinions!

[jonesthecurl]

I was gonna say something, but I was so wrong in the Monty Hall thread that this would only be ammunition for the people that are wrong...

[TheProwler]

Sorry what?

Replace infinity with 100.

Does 2/100 approach anything faster than 1/100? Of course not. These are equations with specific answers. The same applies if you replace 100 with infinity. 1/infinity and 2/infinity are not limits and they do not approach anything. They would be limits if we said f(x)=1/x and then we could say that as x approaches infinity f(x) approaches 0. However that's obviously not what we're talking about!

Ahhh, the dangers of mixing real numbers with infinity. Does 1/0 equal infinity? It's really the same question as "Does 0.999recurring equal 1?".

[TheProwler]

I liked your post, AAFitz.

0.999... does not approach anything. It is a single value which does not change.

I'm still kinda hoping someone can come up with an equation that will result in an answer of 0.999recurring without using recursion or a recursive number or some declared value that is "infinitely small". If you can't, then I would like someone to explain how 0.999recurring can even exist. Obviously, if something can't exist, it can't equal 1.

Actually, it is the exact reason that it cant "exist" that it does equal one

Can something that does not exist equal something that does exist?

The only thing more infinite or pointless about this conversation, really is its uselessness.

I agree. For a few reasons.

One, it will eliminate the obvious possibility that people will think 1/3 times 3 equals 0.999recurring (and other similar mistakes). This will occur when people don't recognize the limitations and inaccuracies of trying to represent fractions with recurring decimals.

Two, differences this small never actually exist. Quantum physics and all that stuff. Heck, one little electron floating around an atom will cause your measurements to constantly change if you are trying to measure so precisely.

This is kinda why I asked if someone could show me an equation to generate an answer of 0.999recurring without using, etc..

Right or wrong, the difference if there was one between 1 and .99... would be infinitely small. As an infintely small number, it would be = to 0.

Same argument. I think an infinitely small number is not equal to zero.

Some would argue that zero doesn't even exist. One sentence, but very relevant to this entire thread. So I'll say it again.

Some would argue that zero doesn't even exist. And they might be right.

since .999... in every practical and useful meaningful way, will always and always has =1

I agree with that.

, then .999...=1.

But, I disagree with that.

[owheelj]

Sorry what?

Replace infinity with 100.

Does 2/100 approach anything faster than 1/100? Of course not. These are equations with specific answers. The same applies if you replace 100 with infinity. 1/infinity and 2/infinity are not limits and they do not approach anything. They would be limits if we said f(x)=1/x and then we could say that as x approaches infinity f(x) approaches 0. However that's obviously not what we're talking about!

Ahhh, the dangers of mixing real numbers with infinity. Does 1/0 equal infinity? It's really the same question as "Does 0.999recurring equal 1?".

No it's not, because you could hypothetically divide something in to an infinite number of pieces, but you can't divide it into 0 pieces. Hence there is no answer to 1/0. It's a nonsensical question. If I have a 200 g cake and I divide it in to 0 pieces, what is the weight of each piece of cake? If I divide it in to an infinite number of pieces then obviously the weight of each piece is 0g. It does not "approach" 0 grams because limits are only something you use with a variable, and we don't have one.

[SultanOfSurreal]

Because the Planck length example transforms the argument from theory to the real world. Unless you can think of some other place where human-invented numbers are used?

i dunno

there are zero places i guess. so that would be a negative

[owheelj]

Human invented numbers are used all the time. Here is a bunch of examples of the application for using complex (imaginary) numbers - ie. the square root of negative numbers.

http://en.wikipedia.org/wiki/Complex_nu ... plications

[xelabale]

Some would argue that zero doesn't even exist. One sentence, but very relevant to this entire thread. So I'll say it again.

Some would argue that zero doesn't even exist. And they might be right.

I just checked my bank balance - 0 definitely exists.

[jonesthecurl]

Of course zero exists - they even wrote an opera about it...

Figure 0

[Snorri1234]

But the difference between 1 and 0.99recurring is not a really small number. It is zero with an infinite number of zeros after the decimal place. Which is identical to zero with infinite degrees of accuracy.

If it is zero, then you wouldn't be writing 9's after the decimal place, and 1 before it. Besides, that wasn't the point - proof by ignoring the other persons argument is not, I assure you, a recognised logical method.

Archimedes said that infinitesimals do not exist due to the Archimedean property. But the Archimedean property does not hold. Therefore infinitesimals can exist, meaning that an immeasurable distance between 0.999... and 1 can be defined, given the series (1-x) where x = 10^-n, for n an element of the natural numbers.

I don't see how the Planck length has anything to do whether our abstracted number system can have an infinitesimally small number or not.

Because the Planck length example transforms the argument from theory to the real world. Unless you can think of some other place where human-invented numbers are used?

But doesn't the fact that it's our abstract system mean that it is correct. I mean, math only has to be internally consistent.

[TheProwler]

Some would argue that zero doesn't even exist. One sentence, but very relevant to this entire thread. So I'll say it again.

Some would argue that zero doesn't even exist. And they might be right.

I just checked my bank balance - 0 definitely exists.

Not really. Show me zero money. Not a visual representation of zero money. Show me zero money. Or zero anything.

[jonesthecurl]

I've just sent you zero money in the post.
In zero envelopes.
With zero stamps.
Tell me when it gets there.

[e_i_pi]

Because the Planck length example transforms the argument from theory to the real world. Unless you can think of some other place where human-invented numbers are used?

i dunno

there are zero places i guess. so that would be a negative

Human invented numbers are used all the time. Here is a bunch of examples of the application for using complex (imaginary) numbers - ie. the square root of negative numbers.

http://en.wikipedia.org/wiki/Complex_nu ... plications

By "some other place", I meant some other place than the real world. So, thanks guys for pointing out the existence of complex numbers. With i being part of my username, you'd think I'd be aware of their existence and application. How stupid am I?

[e_i_pi]

Because the Planck length example transforms the argument from theory to the real world. Unless you can think of some other place where human-invented numbers are used?

But doesn't the fact that it's our abstract system mean that it is correct. I mean, math only has to be internally consistent.

...and once again I refer you to Kurt Gödel's incompleteness theorems

[InkL0sed]

Some would argue that zero doesn't even exist. One sentence, but very relevant to this entire thread. So I'll say it again.

Some would argue that zero doesn't even exist. And they might be right.

I just checked my bank balance - 0 definitely exists.

Not really. Show me zero money. Not a visual representation of zero money. Show me zero money. Or zero anything.

A vacuum.

[e_i_pi]

Some would argue that zero doesn't even exist. One sentence, but very relevant to this entire thread. So I'll say it again.

Some would argue that zero doesn't even exist. And they might be right.

I just checked my bank balance - 0 definitely exists.

Not really. Show me zero money. Not a visual representation of zero money. Show me zero money. Or zero anything.

A vacuum.

"A vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty," but in reality, no volume of space can ever be perfectly empty. A perfect vacuum with a gaseous pressure of absolute zero is a philosophical concept that is never observed in practice."

[InkL0sed]

Some would argue that zero doesn't even exist. One sentence, but very relevant to this entire thread. So I'll say it again.

Some would argue that zero doesn't even exist. And they might be right.

I just checked my bank balance - 0 definitely exists.

Not really. Show me zero money. Not a visual representation of zero money. Show me zero money. Or zero anything.

A vacuum.

"A vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty," but in reality, no volume of space can ever be perfectly empty. A perfect vacuum with a gaseous pressure of absolute zero is a philosophical concept that is never observed in practice."

Did you get that from wikipedia?

[sully800]

lol

It's somewhat but not entirely true. There are perfect vaccuums, especially within what we deem to be ordinary vaccums. The vaccuum of space is largely empty but not completely empty. A small sample of space is most likely completely empty. A small enough sample of solid matter is in fact completely empty, at least to our current knowledge. But none of that really matters does it

[TheProwler]

I've just sent you zero money in the post.
In zero envelopes.
With zero stamps.
Tell me when it gets there.

I'll wait an infinite amount of time. I will.

[TheProwler]

Some would argue that zero doesn't even exist. One sentence, but very relevant to this entire thread. So I'll say it again.

Some would argue that zero doesn't even exist. And they might be right.

I just checked my bank balance - 0 definitely exists.

Not really. Show me zero money. Not a visual representation of zero money. Show me zero money. Or zero anything.

A vacuum.

Upright, canister, or central?

Seriously, you can't think that proves anything. Are you trying to say that nothing is something? The absence of something is not something, it is nothing. That means it doesn't exist. i.e. Zero doesn't exist.

The debate over whether zero exists is a long and complicated affair. Don't be so smug to think that you can settle it with two words. Or were you just joking?

[owheelj]

I don't understand what you mean by "zero doesn't exist." I think you've misunderstood the debate about zero. I can easily show you "zero money." Just look somewhere where there isn't any money. The debate that many people have had over a long time is not "zero doesn't exist" but "zero isn't a number." These are obviously completely different statements. Jessica Alba exists, but she is not a number.

[TheProwler]

She is a solid 8.

When I talk about zero existing, consider it to mean existing as a number. The difference is just semantics. If zero only exists as a concept, I am saying this is equivalent to not existing.

A 17-headed sea creature with 1024 arms.

Now that I said it, does it exist?


Just like nothing. It is a word. But nothing does not exist. Or you wouldn't be here.