1. PerfectHacker:

Albeit this is an older thread, would you be kind enough to expound upon your post? It has always been my experience that any repeating decimal can be proved rational via fundamental algebraic manipulation, i.e., application of "logical laws" within an established framework of undefined terms, axioms, definitions and previously proved theorems. It seems to me that declaring an infinite sum of natural powers equal to a particular rational number by definition is akin to writing into law that no child shall be delivered if the weight of such child exceeds that of the natural mother. Laws and definitions aside, the aforementioned mathematical consequence will occur without permission just as surely as maternal weight in the pre-birth shall not exceed twice that of the post-term mom. Both instances can be assured by proof which (by unintended irony) translates to theorem according to definition of the same. It is the theorem that separates truth from assumption...mathematics from wizardry.

...just a thought

Regards,
Rich B.

Originally Posted by ThePerfectHacker
The answer is basic, DEFINITON: THE REPETION OF A DECIMAL EQUALS THE VALUE OF ITS CONVERGENT. Thus, what most people do not relized is this is not paradoxical (nothing in math is paradoxical) but rather it is a DEFINITION and not a theorem. It is proven by looking at it geometric sum.

2. Originally Posted by ThePerfectHacker
The answer is basic, DEFINITON: THE REPETION OF A DECIMAL EQUALS THE VALUE OF ITS CONVERGENT. Thus, what most people do not relized is this is not paradoxical (nothing in math is paradoxical) but rather it is a DEFINITION and not a theorem. It is proven by looking at it geometric sum.
I find that odd. A definition assigns a label to an object with certain properties, not additional properties to an object with certain properties. Therefore, stating that "THE REPETION OF A DECIMAL EQUALS THE VALUE OF ITS CONVERGENT" is not a definition. Furthermore, you said paradoxically that "It is proven by looking at it geometric sum"; a definition would not need to be proven.

3. When a decimals repeats or is infinite we DEFINE its value as its convergent (it is reasonable). Thus, to show that .999.... is equal to one. We show that the convergent is equal to one, to show the convergent is equal to one we use the concept of an infinite geometric sum. Thus, by the geometric sum its convergent is one thus .99999.... must be one.
Same thing happens with infinite continued fractions.

Rich B, you cannot just use algebraic manipulation whenever you wish. You first need to show if the numbers you are using are real. For example, let us say you are using Grassman numbers then your algebra will go insane. Because all your manipulations would result from undefined terms. An example in some Structure a+b need not be b+a this is called commutative property. Thus, sometimes you get unusual results because you are using a series which is divergent thus it is not a real number thus the Laws for the reals no longer apply. Thus, sometimes using these divergents you get an unreasonable answer.

4. I tried, but I just can't convince myself that .99999 repeating equals 1.

consider an asymptote or a limit, they get closer and closer to a value, but never actually arrive at that value, and It would be wrong to say that they equal that value. To me this is what is happening in this case.

5. Originally Posted by angel.white
I tried, but I just can't convince myself that .99999 repeating equals 1.
Try harder

consider an asymptote or a limit, they get closer and closer to a value, but never actually arrive at that value, and It would be wrong to say that they equal that value. To me this is what is happening in this case.
Consider what 0.999.. means. (its the sum of a geometric series, which
can be summed and surprise surprise its sum is 1).

In summary 0.9999.. means:

$\lim_{N \to \infty} \sum_{n=1}^{N} \frac{9}{10^n} = 1$

nothing more; nothing less.

RonL

6. Originally Posted by angel.white
I tried, but I just can't convince myself that .99999 repeating equals 1.

consider an asymptote or a limit, they get closer and closer to a value, but never actually arrive at that value, and It would be wrong to say that they equal that value. To me this is what is happening in this case.
Yes, consider the limit or the asymptote. Also consider the language. The limit IS. The asymptote IS. We do not use language such as the limit is NOT QUITE, or the limit is ALMOST, or the limit APPROACHES. (Well, sometimes we hear that last one, but it is wrong.) The idea of "never actually" is not appropriate. "never actually" is a finite concept. Discard it. What you mean by "never actually" is: Stop the decimal expansion at some finite point and compare to the desired value. That is not what a limit is about. Do not think this.

hoeltgman suggested my favorite convincing argument. If 0.999999..... is NOT equal to 1, then please tell me how far from 1 it is. You cannot do it. No matter what you say, other than zero, it can be shown to be closer that that. You MUST, therefore, come to terms with equality.

If you understand the concept of a limit, you will see it. If you whack off the limit at some finite thought, you will not see it. So, yes, consider the limit and consider the asymptote, but do NOT chop them off when your concept reaches its finite capacity. Keep going.

7. Originally Posted by CaptainBlack
Try harder

Consider what 0.999.. means. (its the sum of a geometric series, which
can be summed and surprise surprise its sum is 1).

In summary 0.9999.. means:

$\lim_{N \to \infty} \sum_{n=1}^{N} \frac{9}{10^n} = 1$

nothing more; nothing less.

RonL
I agree that the limit of that geometric sequence equals one, I don't see how .9repeating actually equals that limit.

I would say the difference between 1 and .9repeating is $\frac{1}{\infty}$

8. Originally Posted by angel.white
I agree that the limit of that geometric sequence equals one, I don't see how .9repeating actually equals that limit.

I would say the difference between 1 and .9repeating is $\frac{1}{\infty}$
But $\infty$ is not a number. (Well, in the real number system, anyway.) Thus $\frac{1}{\infty}$ is not a number.

-Dan

9. Originally Posted by topsquark
But $\infty$ is not a number. (Well, in the real number system, anyway.) Thus $\frac{1}{\infty}$ is not a number.

-Dan
I have always thought of it as a decimal point, an infinite number of zeros, and then a 1.

I guess the way I think of it is sort of like the function f(x)=1/x

x can get infinitely close to zero, but it can never become zero.

Or for the case of .9r (r = repeating), f(x)=1/(1-x), x can get infinitely close to 1, it can be .9999999999999999 on to an infinite number of 9's, but it cannot become 1, because 1 is not in the domain, because that would cause the denominator to go to zero, which means you would be dividing by zero, which is not allowed.

10. Look at my post (#14) from 2005 you can clearly see that I still have the same hatred toward philosophers.

What is so hard?
$1+a+a^2+... = \frac{1}{1-a}, \ |a|<1$
So,
$10\times .999... = 9\left( 1+\frac{1}{10}+\frac{1}{10^2}+... \right) = \frac{9}{1-\frac{1}{10}} = 10$.
Thus,
$.999... = 1$.
Absolutely trivial.

11. Originally Posted by angel.white
I agree that the limit of that geometric sequence equals one, I don't see how .9repeating actually equals that limit.
But 0.999.. means no more than this geometric series. It does not matter what
you think it means, this is what it is defined to be, almost all other explanations
you will see are in fact nonsense (including most in this thread).

RonL

12. Originally Posted by CaptainBlack
this is what it is defined to be, almost all other explanations you will see are in fact nonsense (including most in this thread).
Amen to that!
Why does anyone waste time with this nonsense?

13. *shrug* no need for everyone to get upset, I can't force myself to believe something I don't believe (belief doesn't work that way), I thought about it several times today, and could not overcome my disbelief, so I guess it is what it is.

14. Originally Posted by angel.white
*shrug* no need for everyone to get upset, I can't force myself to believe something I don't believe (belief doesn't work that way), I thought about it several times today, and could not overcome my disbelief, so I guess it is what it is.
Therein lies the nonsense!
This is not about belief. In fact, any good program in mathematics training teaches that the word “belief” is a no-no. A mathematician may say “I think…” but may not say “I belive…”. Mathematics is not a system of “beliefs”. Mathematics is a system of axioms. That is a matter of ‘acceptance’ and not a matter of belief.

15. Originally Posted by angel.white
*shrug* no need for everyone to get upset, I can't force myself to believe something I don't believe (belief doesn't work that way), I thought about it several times today, and could not overcome my disbelief, so I guess it is what it is.
Listen to what Plato and CaptainBlank said they know what they are talking about (unlike many in this thread).

In order to understand what this perfectly you need to understand, "what is a number". And many people who (I should say everyone) is not a mathematician does not have a strong enough understanding of what this seemigly simple (yet not so trivial) question really means. Once you understand that you will realize how pathetic your argument is.

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