Simple mathematical problem (?)

[DanishDynamite]

with this one


When k tend to infinity
the result tends to a / (1 - x)

There are equal signs all the way through. Try to understand that the expression "0.999..." is simply another way of writing the expression "0.9 + 0.09 + 0.009 + ..." These expressions are EQUAL. And the latter expression is EQUAL to 1 as the sum of a geometric series shows. EQUAL. There is no "tends" involved.

 

[LW]

When k tend to infinity
the result tends to a / (1 - x)

And as I wrote, that particular equation is a theorem taken directly from university-level mathematics textbook.

Now ask this question from yourself:

Which one is more probable, that I am right and a university professor of mathematics is wrong and not a single soul realised that in the first six editions of a hugely popular textbook, or that I am wrong and the professor is right?

 

[LuxFerum]

There are equal signs all the way through. Try to understand that the expression "0.999..." is simply another way of writing the expression "0.9 + 0.09 + 0.009 + ..." These expressions are EQUAL. And the latter expression is EQUAL to 1 as the sum of a geometric series shows. EQUAL. There is no "tends" involved.

0,9999... is not 1
0,3333... is not 1/3
no matter how close you get.
The difference tend to zero.
The limit is the point where they tend to go, but they really never get there.
If you have a computer with and infinity precision and infinity power of calculation, and you put him to add "0.9 + 0.09 + 0.009 + ..." and only stop when the result is 1,he will never come out.
this sum will never reach the "1" point.
And here where the limit idea comes from.
you can see where this sum is going, you can make it as close as you want.
making a infinity sum is not the best way to get there, but it is a good way to see where the result shoul be.
It is like the runner and the turtle problem, if you stick with the infinty sum, the runner will never transpass the turtle, But with a different analyse you can see that he will.

 

[LuxFerum]

And as I wrote, that particular equation is a theorem taken directly from university-level mathematics textbook.

Now ask this question from yourself:

Which one is more probable, that I am right and a university professor of mathematics is wrong and not a single soul realised that in the first six editions of a hugely popular textbook, or that I am wrong and the professor is right?

the equation is right, your interpretation of it is wrong, imho.

 

[Mendor]

If you have a computer with and infinity precision and infinity power of calculation, and you put him to add "0.9 + 0.09 + 0.009 + ..." and only stop when the result is 1,he will never come out.

Yes, it will. When t = infinity

(sorry, couldn't resist)

 

[LuxFerum]

Yes, it will. When t = infinity

(sorry, couldn't resist)

as I said, never

 

[Mendor]

as I said, never

Don't bring the real world into this! This is a Maths thread!

tch...

 

[LuxFerum]

Don't bring the real world into this! This is a Maths thread!

tch...

If the two of us were immortal, we could wait for the machine to give us the result.
And I will be mocking you all of that time, while you wait for the result.

 

[LW]

If the two of us were immortal, we could wait for the machine to give us the result.
And I will be mocking you all of that time, while you wait for the result.

Tell me, if I shoot an arrow at you, will it reach you in a finite time? [Supposing that I first travel near to you].

 

[LuxFerum]

Tell me, if I shoot an arrow at you, will it reach you in a finite time? [Supposing that I first travel near to you].

yes it will. and that is where you missed the point.
that is when you change the analise from and endless sum to a linear movement.
The endless sum is a good tool but incomplete.
you need that jump from one analise to another to make it pratical.
When you can see where that sum is going, that is the time when you leave the loop.
And that is why the computer will never leave the loop.
The computer will not make that assumption.
And will never reach the end of that sum.

 

[LW]

The endless sum is a good tool but incomplete.
you need that jump from one analise to another to make it pratical.
When you can see where that sum is going, that is the time when you leave the loop.

Oooh, we are almost there.

So, you seem to agree that if we have the sum:

sum( (1/2)^k ) where 0 < k < infinity,

then we don't have to use infinite amount of time to sum the terms individually, but we can instead use the above-mentioned formula to get that the value of the infinite sum is 1.

But then, that particular sum is of the form:

sum (a * x^k)

where a = 1 and x = 1/2.

Why can' t we then use the same kind of thinking when a = 0.9 and x = (1/10)? to conclude that the value of that sum is also 1 without having to sum the terms individually?

 

[Walter Wayne]

The people who still believe 1 != 0.999... seem to still extrapolate from finite to infinite.

0.9 != 1
0.99 != 1
0.999 != 1
...
When does the difference get to zero? Never. But at what time will you have added an infinity of 9's? Never.

Any when will you get there questions assume finite. When will you get to the end of a line (mathmatical definition). As a line is infinite, never. When will you get to the end of the 9's ...

As for limites note the language of limits

As x approaches infinity, y approaches ...
We are not talking about approaching infinity when we write 0.9..., we are talking about already being there. When you right it as the end of a sequence (0.9, 0.99, 0.999, ...) we are talking about approaching it.

Walt

 

[LuxFerum]

Oooh, we are almost there.

So, you seem to agree that if we have the sum:

sum( (1/2)^k ) where 0 < k < infinity,

then we don't have to use infinite amount of time to sum the terms individually, but we can instead use the above-mentioned formula to get that the value of the infinite sum is 1.

That formula already have that "jump" from an endless sum to another analise.

This formula is what a computer will never assume.
If you keep only adding, you will never reach that point.
k=0;
sum=0;
while sum!=2 do
(
sum=sum+(1/2)^k;
k=k++;
)
end



"but we will get closer and closer to the final answer" <----This is the magical point.

This is when you stop adding and use your imagination to get to the result by another way, and not by an endless sum.

The representation of a problem in an endless sum need that jump to make it viable.

and everytime that you say that 0,999...=1 you are making that jump.

but if you dont have that jump that sum will [echo]NEVER , never ...[/echo] reach that point.

 

[Cabbage]

It's clear from reading the responses that noone in this thread that is claiming that .999... is not 1 is even slightly familiar with the standard construction of the real numbers.

What does it mean when we have a nonterminating decimal expansion (such as pi, or even .999..., such as this thread is about)? Such a decimal expansion by definition denotes the real number that is the limit of the "partial decimal expansions", for lack of a better term.

In other words, take pi for example. It's often said that pi = 3.14159...., but what does that mean? By definition that means that pi is the limit of the rational numbers

3
3.1
3.14
3.141
3.1415
...

and so on.

This is how the real numbers are constructed. This is how the decimal notation works. Any nonterminating (or terminating, for that matter) decimal is equal to the limit of that expansion, by definition!

Any argument along the lines, "Well, .999... approaches 1 as a limiting value, but .999... never reaches 1, so .999... and 1 are not the same" is complete nonsense, because (one more time): .999... is defined to be equal to its limit, in the first place! Since that limit is 1, we have .999...=1, exactly.

Re: The surreal numbers. While there are such things as surreal numbers, and there are surreal numbers that are different from one, yet closer to one than any real number, I have never seen anyone use decimal notation to represent a surreal, nonreal number (with, of course, the exception of a few people in this thread, and that's simply because they don't know what they're talking about).

Anyone giving the surreal number argument that .999... and 1 are different: I challenge you to give me one verifiable example of a professional mathematician using the notation .999... to describe a surreal, nonreal number. Or is that just something you made up?

(Edited because I screwed up my first sentence).

 

[LuxFerum]

Any argument along the lines, "Well, .999... approaches 1 as a limiting value, but .999... never reaches 1, so .999... and 1 are not the same" is complete nonsense, because (one more time): .999... is defined to be equal to its limit, in the first place! Since that limit is 1, we have .999...=1, exactly.

1/x with x tending to 0, is infinity or Undetermined?
when x tend to 0, it gets pretty close to infinity, but when it is 0 is it infinity?

 

[Martin]

But this proof does not work.

X=0.999(rec)
10X=9.999(rec)
10X-X=9.999(rec)-0.999(rec)
9X=9
X=1

So then 10X=9.999(rec) should be true if X=1 is true and it is not.

The only way this can work is by using circular reasoning

Hrmm...

We can use proof X to derive statement Y.
Therefore proof X only works if statement Y is true.
Therefore using proof X to derive statement Y is circular logic.

Well, it's certainly an interesting take on things

 

[Cabbage]

1/x with x tending to 0, is infinity or Undetermined?

Well, actually the limit doesn't exist, but it's not uncommon to say such things as "the limit from the left is -infinity, while the limit from the right is +infinity" (though neither one sided limit actually exists, either--saying the limit is +/- infinity means the limit doesn't exist--it's just a special way of saying the limit doesn't exist).

when x tend to 0, it gets pretty close to infinity, but when it is 0 is it infinity?

It's undefined when x=0 (division by zero).

 

[LW]

That formula already have that "jump" from an endless sum to another analise.

I don't have any idea why you keep on bringing that "jump" forward or even what you really mean by it.

This formula is what a computer will never assume.

And a computer doesn't have anything to do with how real numbers are defined. Actually, computers don't even work on real numbers. There is absolutely no way to store an irrational number in a binary computer. I'm not certain who was the first person to analyse infinite serieses but they have been used at least since the 18th century and now I'm playing it safe, that is several hundreds of years before any computers.


[quoteThis is when you stop adding and use your imagination to get to the result by another way, and not by an endless sum.{/quote]

No, you use the properties of the bloody infinite series. Not your bloody imagination.

Now. let me make it clear. If you still hold that 0.99... is not precisely equal to 1, then you are contradicting every single mathematician who has written about infinite series since after they have been invented.

Now there are exactly two choices:

1. Every single mathematician who has ever worked with real numbers and infinite series is wrong.

2. You are wrong.

There is no third alternative.

 

[DanishDynamite]

0,9999... is not 1
0,3333... is not 1/3
no matter how close you get.
The difference tend to zero.
The limit is the point where they tend to go, but they really never get there.
If you have a computer with and infinity precision and infinity power of calculation, and you put him to add "0.9 + 0.09 + 0.009 + ..." and only stop when the result is 1,he will never come out.
this sum will never reach the "1" point.
And here where the limit idea comes from.
you can see where this sum is going, you can make it as close as you want.
making a infinity sum is not the best way to get there, but it is a good way to see where the result shoul be.
It is like the runner and the turtle problem, if you stick with the infinty sum, the runner will never transpass the turtle, But with a different analyse you can see that he will.

I give up. Mathematically, when considering real numbers, 1 and 0.999... are equal as has been proven. You can wax philosophical about whether this seems reasonable or not, but that is the way it is.

"You can lead a horse to water, but you can't make it drink".

 

[LW]

"You can lead a horse to water, but you can't make it drink".

And after this argumentation I'd need a drink.