Poll: Does 0.999.. equal 1 ?

[mps4li3n]

You will literally never encounter that number in real life, it's a nonsense number that you'd round eventually anyway.

And even if it didn't you still wouldn't because it would take infinity to finish seeing how it's different from 1 and we have better things to do.

 

[SlasherX]

Technically no, in practice yes

 

[Darth Crater]

snipped lots of stuff

The "Definition of a number set" argument is circular, and the last argument appears to be disproved immediately below. As for my statement, I don't see how it could be used to set two arbitrary numbers equal to one another; could you explain further?

I don't see how in standard mathematics they would be considered equal. In proper math, everything is as is, and as such 0.999... != 1.0.

The problem here is that "everything is as is" is an English statement that does not apply to the actual math. Nearly every page of the thread contains a legitimate proof that the two are equal.

If you paid attention i have been saying that actually... 0.(9) = 1 because you never get to the difference is the only way to make sense of the proof in real life terms imo.

Either that or reality is an illusion...

My apologies for the misunderstanding, then.

 

[Skratt]

It goes 0.999... on to infinity. You'd think logically, you would just add a number that went 0.000...1. However, you can't have a 1 at the end of an infinite number of zeros, for that assumes there is an end to infinity, which there isn't...

2. The "if you do understand that .999~ will never quite get that last little bit..." statement is wrong, because there is no "last little bit". There is no "bit" or "number" etc. that separates the two real numbers, therefore they are the same. It's very important to realize that this is NOT a matter of opinion. This is a factual, direct conclusion from the basic axioms which define what a real number is.

Now, if you DO decide to change what a real number is, then some ambiguity may arise. But this is equivalent to saying "I won this track meet because I redefined what "running" is, and showed up in a jet plane. Doesn't my argument make sense?"

I don't fully understand the concept, but these explanations coupled with the wiki article get me pretty darn close.

 

[maninahat]

I'm not sure what you are saying as I am not familiar with most of these terms. Could you please explain in layman's terms?

Layman's terms: X times 10 is X0. If the last recurring digit isn't a zero, then it's not following the basic rules of maths - it's following the nearest possible because you're trying to use a finite operation on infinity.

Though I lack the knowledge to properly contend what you are saying, I think you're wrong and that you are somehow misapplying your knowledge). Recurring numbers are workable, despite being infinite in length. x can represent any number, and for the purpose of the proof, it represents 0.(9). It is perfectly possible to use operatives on infinitely long decimal numbers both on paper and practically. In fact, you could just do it counting beans. You have three beans. One of them is one third the total number of beans. In otherwords, 0.(3). I can take another bean along with that one. That makes two thirds of the total, or 0.(6). I add the final bean to the pile. 100% the beans, despite each individual bean representing 0.(3). How could this be possible? How could I be able to do this to the beans, when the fractions they represent are infinitely recurring numbers? How do you explain that 3/3rds makes a 1, when 1/3 makes 0.(3)?

Equally, 10 times infinity is still infinity. 10 divided by infinity ≡ 0.

I'm not asking anyone to multiply infinity. I am asking them to multiply a real number that can be displayed to an infinite number of decimal places. That is perfectly straightfoward, and it is something that mathematicians do even with normal numbers like 5 or 17. That is because it these numbers aren't just "5" or "17", they are "5.(0)" and "17.(0)". For convenience sake, we terminate them at the first zero, but that doesn't change the fact that there are an infinite number of zeros behind any of these numbers and that these infinite zeros don't make a blind bit of difference when I want to apply operatives to them.
If I want to double 3.333... I can. The answer is 6.666. I'm not doubling infinity itself.

Any number can be represented by x, or any symbol I choose to give. And I can perform whatever operatives I like on the number. I recall there being a particulary famous one that represents an infinitely long irrational number. Pi, I think they call it.

As someone so apparently savvy with mathematics, I don't understand why you are having this much trouble dealing with an issue that has long been accepted by mathematicians for decades. [http://en.wikipedia.org/wiki/0.999...]

 

[The_root_of_all_evil]

As someone so apparently savvy with mathematics, I don't understand why you are having this much trouble dealing with an issue that has long been accepted by mathematicians for decades. [http://en.wikipedia.org/wiki/0.999...]

Because I did a Computing/Maths Degree with Honours, and recognize the difference between an accepted truth and an actual truth. As does the other Maths Degree person I asked.

If you can't take note that decimals are an approximation of fractions, then I wave my hands of the argument. The level of approximation, however small, leads to errors - and that's why they're equivalent.

 

[maninahat]

i'm pretty sure that's what it says in your description, no? the turtle gets a headstart of 10, moves at a speed of 1, while the man starts at 0, but moves 10 times as fast.

The entire problem with Zeno's Paradox is that it takes the point where the man passes the turtle as an assymptote and approaches it at exponentially decreasing speed; therefore never reaching it. If you change the measurements to simply "What time does the man pass the turtle", then it's easy.

The man doesn't slow down. It is just that the distance between the man and the tortoise gets infinitely smaller. Basically, the paradox assumes that a man cannot possibly accomplish an infinite number of tasks (the tasks being run 10 meters, then run 1 meter, then 0.1, then 0.01 etc.). But this doesn't make sense: If a man cannot accomplish an infinite number of tasks, then neither can the a tortoise create an infinite number of tasks by moving on ahead. If we assume the tortoise can keep infinitely keep creating these gaps, we can assume the man can infinitely cross them.

 

[maninahat]

As someone so apparently savvy with mathematics, I don't understand why you are having this much trouble dealing with an issue that has long been accepted by mathematicians for decades. [http://en.wikipedia.org/wiki/0.999...]

Because I did a Computing/Maths Degree with Honours, and recognize the difference between an accepted truth and an actual truth. As does the other Maths Degree person I asked.

If you can't take note that decimals are an approximation of fractions, then I wave my hands of the argument. The level of approximation, however small, leads to errors - and that's why they're equivalent.

Okay, then why do none of the world's leading mathematicians follow your reasoning when confronted with the same issue? They categorically claim that 0.999... equals 1 (see the wikipedia link I sent). I don't like making this appeal to authority, but you did after all mention you and your friend's math degree.

I understand that decimals are approximations of some fractions, but I contend that it doesn't matter in this case. Mathematical realism states that "in the mathematics of computation, which includes calculus, we need be concerned only with what we can observe. Actual infinities therefore have no place and are not necessary. They have no practical effect on calculations in arithmetic or calculus."

 

[The_root_of_all_evil]

pfft. Believe what you wish.

 

[Torrasque]

No, they are not the same.
Yes, math is flawed.
You can see this especially in fractions.

disclaimer: fractions are a poor representation of non-fractions
1/2 = 0.5
1/4 = 0.25
1/3 = 0.333...
2/3 = 0/666...

The simple fact that it repeats forever and ever, means one thing: that it repeats forever.

Here's a fun fact: If you add up 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ..... it would equal 1.

disclaimer: fractions are a poor representation of non-fractions
I take it, that you're "....." means infinity? 1/64 + 1/128 + ... ?
Ironically enough, no, it would not equal 1. It would equal 0.999... and keep getting closer to 1, but never actually get there.

Infinity has a funny way of making numbers approach stuff sometimes. And sometimes the difference between the numbers and the stuff it approaches gets really, really small. So small, so infinitely small, in fact, that mathematicians give this gap a name: zero.

Actually no, they just say zero for the sake of argument, when in reality, it is infinity.
Just like how space has an end, but until scientists see it, they are calling it infinity.

Just like Pi does not exactly equal 3.14, it equals 3.14159265358979323846264338327950288419716939937510582097494459...

As it turns out, Pi is irrational, so it can't really be represented by either fractions or non-fractions.

π = C/d
C: circumference
d: diameter
Sorry, what did you say about it not being able to be represented by a fraction?

To put this into perspective, lets say you live on Earth, and I live 3 light years away.
That is an extremely long ways away, and only an idiot would say that I am touching you.
One day, I decide to teleport back to Earth to visit you, and teleport within 1 centimeter of you.
Compared to the ridiculous distance that separated us before, I am practically touching you. But I am not.
I move to within 1 nanometre of you, just because I am creepy that way, but do not touch you.

Think of 0.999... as that. The difference between 0.999... and 1 is so insignificantly small that depending on the case, you'd just ignore it. But there is still a difference between touching you, and not touching you, whether that is 3 light years, or 1 nanometre.

And what if the distance between us were infinitely small? That's not small; that's not even existent. It's zero.

Depending on the application and current technology, nanometres can be comparable to kilometres, or kilometres comparable to nanometres. Whether you want to agree or not, the fact of the matter is, if I am 1 nanometre away from you, I am still 1 nanometre away from you. Not touching.

Would have posted this yesterday, but Escapist apparently had downs for a day /shrug

 

[TiefBlau]

Ironically enough, no, it would not equal 1. It would equal 0.999... and keep getting closer to 1, but never actually get there.

Actually no, they just say zero for the sake of argument, when in reality, it is infinity.
Just like how space has an end, but until scientists see it, they are calling it infinity.

You're arguing semantics. And not even correctly.

Zero is the name given to a number that is infinitely small in our number system. Infinity is the name ascribed to a number that is infinitely large in our number system. Neither of which are considered "real numbers"; as such, they are umbrella terms for any number that is infinitely small or large. Not all zeros are equal and not all infinities are equal. For example, 0*Infinity is indeterminate; you wouldn't know what it is unless you had a bit more context.

As such, a number that is infinitely close to another number has a difference of zero in our number system, meaning they are one and the same. If you don't like this idea, blame our primitive number system, I guess.

Just like Pi does not exactly equal 3.14, it equals 3.14159265358979323846264338327950288419716939937510582097494459...

As it turns out, Pi is irrational, so it can't really be represented by either fractions or non-fractions.

π = C/d
C: circumference
d: diameter
Sorry, what did you say about it not being able to be represented by a fraction?

As it turns out, either circumference or diameter or both are irrational and therefore pi is irrational. If circumference is a rational number, it would be impossible to represent diameter as a fraction, and therefore pi would be impossible to represent as a fraction.

Your smugness only fuels my self-satisfaction in correcting you.

Depending on the application and current technology, nanometres can be comparable to kilometres, or kilometres comparable to nanometres. Whether you want to agree or not, the fact of the matter is, if I am 1 nanometre away from you, I am still 1 nanometre away from you. Not touching.

But you're not a nanometer away. You're a distance infinitely small. Infinite.

You might say that this would mean that the limit as x approaches k of f(x) would be equal to f(k). For the purposes of our number system, it does.

 

[mattsipple4000]

So basically the point made in this mess iv created here is that 0.999(r) isn't actually a number as it includes infinity!!
But also as the 9's go on forever and NEVER EVER EVER END there is nothing between the infinite 9's and 1.

infinite is hard to understand I'm guessing that's why most people are having problems here!

 

[Winthrop]

snip

I could be wrong on this, as I said I am rather confused, but for instance couldn't the same be said for 1.099.. and 1.1? or 1.0099.. and 1.01 and eventually for a large number of numbers? Sorry if I'm wrong. About the proofs for the second one that was my bad the disproof by Hypocrism was added this week and I had not noticed the change. For the first he does a poor job of explaining it. I believe that his logic is that .9 is in the set, .99 is in the set, .999 is in the set and so on so therefore every .9... is in the set as well while 1 is not in the set.

EDIT: for the first part I forgot to mention multiplication. If you take numbers to large exponents (which, assuming they are equal, does not violate any mathematical rules) the gap widens substantially (it is still negligible but much less negligible).

 

[Jack Skelhon]

I spent about thirty minutes with a friend on this one.

This friend happens to be in the top fifty in the UK for mathematics, so excuse me if I sound "smug" - intonation doesn't come across on the internet, so try reading this in a flat monotone and assume I'm a robot.

I won't go through the details but it was basically advanced algebra against my empirical thought process and U in AS Mathematics. We eventually came to a conclusion.

0.999 does not equal one.

0.999(r) DOES equal one IN THEORY.

Assuming infinity is applicable to the equation, 0.999(r) must equal one. If infinity is NOT applicable, the number is so close to one it is assumed to be equal to one.

So there you go. 0.999(r) equals one within the bounds of 'true' maths.

 

[mps4li3n]

Zero is the name given to a number that is infinitely small in our number system.

Eh... no. Zero wasn't even used in early math because it was always nothing... and until they got to math complex enough to need a number signifying nothing (like an object at rest when doing something with speed, thus it would have no speed) it wasn't needed.

So zero isn't infinitely small, it's not at all (and this doesn't change even if there is math where you use it instead of a infinitely small number, because that's just like 0.(9)=1, there might never be a practical difference, so why not just assume it as nothing, but that's the thing, you're taking the inf small number as if it where nothing).

If infinity is NOT applicable, the number is so close to one it is assumed to be equal to one.

If there's no infinity we might actually one day get to the end of any finite form of 0.(9)...

 

[TiefBlau]

Zero is the name given to a number that is infinitely small in our number system.

Eh... no. Zero wasn't even used in early math because it was always nothing... and until they got to math complex enough to need a number signifying nothing (like an object at rest when doing something with speed, thus it would have no speed) it wasn't needed.

So zero isn't infinitely small, it's not at all (and this doesn't change even if there is math where you use it instead of a infinitely small number, because that's just like 0.(9)=1, there might never be a practical difference, so why not just assume it as nothing, but that's the thing, you're taking the inf small number as if it where nothing).

In our number system, infinitely small numbers are equated to nothing. That's why 0*Infinity is an indeterminate number. All zeroes aren't created equal.