- #1
Psyguy22
- 62
- 0
I accept that .999...=1, but what is the importance in that definition? What does it help us accomplish?
Psyguy22 said:I accept that .999...=1, but what is the importance in that definition? What does it help us accomplish?
Ok. I understand how to prove that .999..=1, but my question is what is the importance behind it?Edgardo said:0.999... = 1 is not a definition. It is a result as mentioned by haruspex.
0.999... is defined as [itex]\sum_{k=1}^{\infty}9/10^k[/itex], i.e. it is the limit of the sequence [itex]s_n = \sum_{k=1}^{n}9/10^k[/itex].
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 [itex]\epsilon[/itex], then you could find an index m such that |sm - 1|< [itex]\epsilon[/itex].
For instance, I give you [itex]\epsilon[/itex]=0.0001. Can you find an m?
Intuitively this means that [itex]s_n[/itex] moves arbitrarily close to 1.
Psyguy22 said:Ok. I understand how to prove that .999..=1, but my question is what is the importance behind it?
So its nothing more than that? Just that its true?micromass said: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 said:So its nothing more than that? Just that its true?
Psyguy22 said:I thought that it would resemble some kind of importance like e^(pi*i)=-1
Psyguy22 said:I accept that .999...=1, but what is the importance in that definition? What does it help us accomplish?
.9999 is a decimal representation of a number, specifically 0.9999. This number is often used to denote a value that is very close to but not exactly equal to 1.
While .9999 is very close to 1, it is not the same as 1. .9999 is a repeating decimal, meaning it goes on infinitely. However, 1 is a whole number and does not have any decimal places.
No, .9999 is not exactly equal to 1. However, in many cases, it is considered equivalent due to rounding and approximation in calculations.
.9999 is often used in mathematics to represent a limit or approximation. It is also used in various mathematical proofs and equations.
.9999 can also be written as 99.99%, 0.9999, or as a fraction 9999/10000. These are all different ways to represent the same value.