Psyguy22
I accept that .999...=1, but what is the importance in that definition? What does it help us accomplish?

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It's not so much a definition as a consequence of other definitions. If it were not the case we'd be in something of a mess, like 1 = 3*(1/3) + 3*(.3333...) = .999... not being 1.

Psyguy22
So is that all its there for? Just so we can have those fractions make complete sense?

Vorde
Nothing 'is there for -> this'. People developed fractions and decimals and someone noted that by the rules they had created, .99999... = 1.

Edgardo
I accept that .999...=1, but what is the importance in that definition? What does it help us accomplish?

0.999... = 1 is not a definition. It is a result as mentioned by haruspex.

0.999... is defined as $\sum_{k=1}^{\infty}9/10^k$, i.e. it is the limit of the sequence $s_n = \sum_{k=1}^{n}9/10^k$.

s1 = 0.9
s2 = 0.99
s3 = 0.999
...
One can show that this sequence converges to 1, i.e. if I give you a small number $\epsilon$, then you could find an index m such that |sm - 1|< $\epsilon$.
For instance, I give you $\epsilon$=0.0001. Can you find an m?

Intuitively this means that $s_n$ moves arbitrarily close to 1.

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We use decimal notation because it is convenient notation for real numbers. If you follow through how decimal numerals correspond to real numbers, you find that the numerals 1.000... and 0.999... both correspond to the same real number.

Psyguy22
0.999... = 1 is not a definition. It is a result as mentioned by haruspex.

0.999... is defined as $\sum_{k=1}^{\infty}9/10^k$, i.e. it is the limit of the sequence $s_n = \sum_{k=1}^{n}9/10^k$.

s1 = 0.9
s2 = 0.99
s3 = 0.999
...
One can show that this sequence converges to 1, i.e. if I give you a small number $\epsilon$, then you could find an index m such that |sm - 1|< $\epsilon$.
For instance, I give you $\epsilon$=0.0001. Can you find an m?

Intuitively this means that $s_n$ moves arbitrarily close to 1.
Ok. I understand how to prove that .999..=1, but my question is what is the importance behind it?

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Ok. I understand how to prove that .999..=1, but my question is what is the importance behind it?

It's simply true. Why should it have importance?? What is the importance of 1+1=2? Or what is the importance that a cat has (usually) 4 legs??

Psyguy22
It's simply true. Why should it have importance?? What is the importance of 1+1=2? Or what is the importance that a cat has (usually) 4 legs??
So its nothing more than that? Just that its true?

Staff Emeritus
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So its nothing more than that? Just that its true?

I'm confused. What more do you expect?? What would you think is a good importance??

Psyguy22
I thought that it would resemble some kind of importance like e^(pi*i)=-1

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I thought that it would resemble some kind of importance like e^(pi*i)=-1

How exactly is that important in the first place??

Gold Member
I accept that .999...=1, but what is the importance in that definition? What does it help us accomplish?

The sequence of decimals .9 .99 .999 ... is a Cauchy sequence. That means that even though the sequence is infinite,the numbers in it cluster together and the further out the sequence the more closely clustered they get. To say that the infinite sequence equals one, says that it actually converges to the number 1. A prioi one might imagine that such an infinite sequence might end up nowhere, But in fact any infinite decimal sequence will converge to some number. This means that the numbers have no holes, that there is nothing missing in them. That to me is the importance of that expression.