Is 0.999... Really Equal to 1? Exploring the Mathematical Proofs

[chroot]

So how many 9s are after the decimal point in 0.99...? There is no real number for them, so the number can't exist?

Infinity exists, it's just not a real number -- specifically, I mean [itex]\infty \nin \mathcal{R}[/tex].

1 2 3 4 5 6 ... infinity
2 3 4 5 6 7 ... infinity + 1 = ?

The notation "infinity + 1" is nonsense.

Why can't we add an 8 after the 9s?

Because there are an infinite number of nines. No matter where you stuck the eight, you'd by definition no longer have an infinite number of nines before it.

- Warren

 

[musky_ox]

1) How do you get from 1.00... to 1.10...? Do you not have to first pass through 1+infinitesimal to get there?

2) In base 12, 1/3 is a terminating number... so you don't get the small rounding error when you use it for calculations. If 0.99... truly equals 1, then it wouldn't be confined to just our base 10 (decimal) system.

 

[chroot]

1) How do you get from 1.00... to 1.10...? Do you not have to first pass through 1+infinitesimal to get there?

"1 + infinitesimal" doesn't exist, so the question is meaningless.

2) In base 12, 1/3 is a terminating number... so you don't get the small rounding error when you use it for calculations. If 0.99... truly equals 1, then it wouldn't be confined to just our base 10 (decimal) system.

Numbers are numbers, no matter what base you put them in. Numbers exist independently of representation, as has been shown with numbers like 6/4 and 1.5. There is no "rounding error" involved anywhere.

The fact that 1/3 is a terminating number is base 12 actually helps our case, not yours. If 0.333... did not truly equal 1/3, then it would truly not equal 1/3 in any base, not just decimal.

- Warren

 

[chroot]

Besides, musky_ox, you have still not answered my question:

Can you find a number between 0.999... and 1?

- Warren

 

[musky_ox]

1 - 0.99... = infinitesimal

As you say that a nine occupies every space in 0.99..., in infinitesimal, a 0 occupies every space except the last one. I don't see how infinity is any more real than infinitesimal.

 

[chroot]

1 - 0.99... = infinitesimal

No, it equals zero, because 1 and 0.999... are the same number.

- Warren

 

[Hurkyl]

in infinitesimal, a 0 occupies every space except the last one.

Why do you think there is a "last space"?

I don't see how infinity is any more real than infinitesimal.

I repeat, the real number system has neither infinite numbers nor infinitessimal numbers.


Incidentally, I get the feeling the terminology is misleading you; the "real" in "the real numbers" is unrelated to the English word "real".

 

[musky_ox]

asymptote

\As"ymp*tote\ (?; 215), n. [Gr. ? not falling together; 'a priv. + ? to fall together; ? with + ? to fall. Cf. Symptom.] (Math.) A line which approaches nearer to some curve than assignable distance, but, though infinitely extended, would never meet it

So is there no such thing as an asymptote?

Incidentally, I get the feeling the terminology is misleading you; the "real" in "the real numbers" is unrelated to the English word "real".

So 0.99... is not a real number in the first place? Math must deal with more than just real numbers then.

 

[chroot]

So is there no such thing as an asymptote?

What? Why are you bringing another concept into this muddled discussion? Please answer the questions that have already been asked of you, rather than trying to complicate it further.

- Warren

 

[chroot]

So 0.99... is not a real number in the first place? Math must deal with more than just real numbers then.

0.999... is a real number in the sense that it is a member of the field [itex]\mathbb{R}[/tex]. Infinity is not a real number in the sense that it is a not a member of the field $\mathbb{R}$. That is what mathematicians mean by the term "real." That is what we in this thread mean by the term "real."

- Warren

 

[musky_ox]

I see no questions to answer... And no, I am not bringing a totally unrelated topic in here. Read the definition of an asymtote. By your reasoning, there is no such thing as one. Id like to know so next time someone starts talking about an asymptote i can tell them that it doesn't actually exist because there is no such thing as infinitely approaching a number without being defined at it.

 

[chroot]

I see no questions to answer... And no, I am not bringing a totally unrelated topic in here. Read the definition of an asymtote. By your reasoning, there is no such thing as one. Id like to know so next time someone starts talking about an asymptote i can tell them that it doesn't actually exist because there is no such thing as infinitely approaching a number without being defined at it.

From what I can tell, this has nothing to do with the discussion to this point. I fear you are misreading what people are saying.

The only question I have to ask you is this one:

Can you find a number between 0.999... and 1?

- Warren

 

[musky_ox]

Sorry i looked for a graph of this but couldn't find one.

Say you graph a function that has a horizontal asymptote of 1. By definition, as the x value goes to infinity, the y value is infinitely approaching 1, but will never be 1. I see no reason that just because i cannot identify a number between 0.99... and 1 that you can say it equals 1.

 

[chroot]

By definition, as the x value goes to infinity, the y value is infinitely approaching 1, but will never be 1.

Except at infinity, where it (might) be 1.

The notation 0.999... does not mean "a lot of nines," it means "an infinite number of nines," which is completely different.

- Warren

 

[chroot]

BTW, I must mention that Webster's dictionary is an abhorrent place to find the definitions of mathematical or scientific words. Rarely are the definitions provided in a common English dictionary adequately precise for technical communication.

- Warren

 

[musky_ox]

I never said that it was "a lot of nines." I said as x-> infinity that y is infinitely close to 1. (infinitesimal away from 1)

Think of this analogy.
Lets assume that space is quantisized and just say that a quark is the smallest distance of space. Does length of 1 quark = 2 quarks just because there is no length between it?

 

[chroot]

I never said that it was "a lot of nines." I said as x-> infinity that y is infinitely close to 1. (infinitesimal away from 1)

And its behavior as it goes to infinity might have nothing at all in common with its behavior at infinity.

Think of this analogy.
Lets assume that space is quantisized and just say that a quark is the smallest distance of space. Does length of 1 quark = 2 quarks just because there is no length between it?

We're not talking about physics, for the last time. We're talking about pure math. The real number line is not quantized, and there is no "smallest number."

I'm going to ask you again, for the third time:

Can you find a number between 0.999... and 1?

- Warren

 

[musky_ox]

What is 2(.99...)? Isnt is 1.99...8? So theoretically if you could have this number, would it equal 2? I can think of a number in between it and 2, 1.99...

 

[chroot]

No, because once again numbers like 1.99...8 do not, and cannot exist! We've already covered this ground. 0.999... does not have a lot of nines, it has an infinite number of nines.

2(0.999...) = 2

I'm really beginning to believe you are just a troll. When things have been explained to you clearly, yet you continue to just repeat yourself and ignore what has been explained, you are trolling.

- Warren

 

[chroot]

I'm going to ask this question again for the fourth time. If you do not answer it directly, I will take action against you for trolling.

Can you find a number between 0.999... and 1?

- Warren

 

[musky_ox]

I guess i can't find any number between 0.99... and 1 then. I don't think it has any implication, but it does appear to be a glitch in the decimal system to me. The only reason i kept arguing is that in base 12 it seems to me that doing 1/3*3 gives you the same answer as 0.4*3.

BTW - What do you mean by "Troll?" I got a warning for saying someone had an IQ below 25 (an idiot) and i hope you weren't indirectly calling me stupid!

 

[chroot]

If you cannot find a number between 0.999... and 1, then 0.999... and 1 are the same number. There are literally dozens of proofs in this very thread, which I suggest you read in its entirety.

Your concept that numbers behave differently in different bases is simply wrong. There is no room to argue this. What is true in one base must be true in all other bases. The definition of the real numbers has nothing to do with what base you choose to represet them in. The definitions deal with the properties of the numbers themselves, independent of representation.

- Warren

 

[musky_ox]

Alright, i will admit defeat then, however 0.99... only equals 1 mathematically. Theoretically, it is still an infinitely small distance away from being 1. I have a question for you: How can a problem involving infinities be represented with equations?

 

[chroot]

Alright, i will admit defeat then, however 0.99... only equals 1 mathematically.

What else have we been talking about besides math? This is absurd. :rofl:

- Warren

 

[musky_ox]

Physics.

- musky ox

 

[BoulderHead]

I always appreciate it when [other] people can admit defeat. :rofl:

Three cheers for musky_ox not being a diehard !

Cheers - Cheers - Cheers

 

[Hurkyl]

I have a question for you: How can a problem involving infinities be represented with equations?

Generally by figuring out what you really mean to ask. (which usually turns out not to involve anything infinite)

For example, the statement "the limit of f(x) as x approaches infinity is L" is actually defined in a way that doesn't involve infinity at all. The thing it is "really" saying is:

You can make f(x) as close to L as you want simply by picking any sufficiently large x.

The precise definition is: for any positive number e, there is a number M such that:
if x > M, then |f(x) - L| < e.



And as I've already mentioned, when one says "0.999... has an infinite number of nines", what they "really" mean is

0.999... has a 0 in every position left of the decimal point, and a 9 in every position to the right of the decimal point.

 

[musky_ox]

Thanks Capt'n Obvious.

I was talking about an asymptote. It is never defined, doesn't matter if x=infinity, it is defined as NEVER being defined.

 

[arildno]

I was talking about an asymptote. It is never defined, doesn't matter if x=infinity, it is defined as NEVER being defined.

Worthless crap; learn some math.

 

[phoenixthoth]

Sorry if this has already been posted in some other form but I have to ask this question to anyone thinking that 0.9... does not equal 1:

Ok so .9... != (does not equal) 1. Then express the difference between the two different numbers in decimal form. Since they're different, their difference must be non-zero. Or, if you can't express the difference between them in decimal form, say what decimal that difference must be greater than. For example, if it was .9 and 1, you can say the difference is greater than .0001.

 

[Adam]

0.9 * 100 = 90

1.0 * 100 = 100

Big difference. Clearly not the same number. However, before you bother bombarding me with the proofs, I'll remind you that I am absolutely terrible at maths, don't know any of it, and this very thread has proofs aplenty of all manner of strange beast.

 

[Adam]

Does 0.98 = 0.99 then?

 

[Integral]

Adam,
You are missing a key symbol.

.999... Notice the 3 dots, known as an ellipsis, after the 9s. This is short hand for repeat this pattern (999) infinitely. So

.99 != .98

but it is true that:

.99 = .98999...

That infinite string of 9s makes a difference.

 

[Adam]

Oops, sorry. That was very silly of me. With the ellipses in use, I forgot that you meant the dot over the last digit instead. We don't use that here.

What is the Latex code for placing the dot over the digit?

Also, can someone show me some more about the reciprocal of that, and the value of that reciprocal?

 

[Adam]

For that matter, how do I write the reciprocal for an infinite series (0.9 with the dot over the 9)?