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# Converting Repeating Decimals to Fractions using Algebraic Method is Invalid

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dylancatlow
 Voting Style: Open Point System: 7 Point Started: 2/6/2013 Category: Science Updated: 5 years ago Status: Post Voting Period Viewed: 5,585 times Debate No: 29976
Debate Rounds (2)

37 comments have been posted on this debate. Showing 1 through 10 records.
Posted by KroneckerDelta 5 years ago
I wouldn't have really minded losing this debate if only it had been for the right reasons.

1) I only lost because somehow wolfman decided you had better conduct...I don't see how that vote is justified whatsoever.
2) The only way to vote for Con is by looking at my R2 where I explained that the algebraic method fails when you have a repeating number (no decimals), i.e. an infinite series that does NOT converge. But seeing as how you (Con) didn't really jump on that and, I think that's not what a repeating decimal is anyway, I still think I should have won. Even worse, the guy who voted against me clearly didn't understand the debate at all and basically had no business voting (yet he ended up being the deciding vote).
Posted by dylancatlow 5 years ago
Sorry I won, Kron.
Posted by dylancatlow 5 years ago
Sorry, I meant vote for Con.
Posted by dylancatlow 5 years ago
By the way, voters, I concede. So vote for Pro.

.999 repeating = 1 because:
1.1 to the power of infinity = infinity.
or 555555.... (infinity 5's) = infinity.
Posted by dylancatlow 5 years ago
Yeah, I know infinity doesn't exist in the real world, at least not in some regards. 0.99999 is a very useful substitute for 1, even without an infinity number of the 9's :) So I understand what you mean about real world application.
Posted by KroneckerDelta 5 years ago
You should realize that infinity is a mathematical concept, it doesn't exist in the real world, so trying to reason about it through the real world is kind of a futile effort. The entire study of calculus depends on the analysis of infinity.

Does this mean calculus has no real world applications? No of course not, we can use it to approximate real world things and it allows us to reason about things that we otherwise could not reason about (Quantum Mechanics comes to mind).
Posted by KroneckerDelta 5 years ago
"But wouldn't an infinite series have a part in it that is infinitely big of whatever you want an infinite number of times?"

No, not necessarily (if I understand your question).

Again, for instance 0.999... = 9/10 + 9/100 + 9/1000 + 9/10000 + ...

The numbers are getting smaller and smaller, so there nothing that is getting infinitely big. I would really suggest reading up on series to get more information. This is a whole topic in and of itself in Calculus III where you do analysis of series and decide whether or not the series converges or is unbounded (or never reaches a value). For instance the following series is not unbounded but it just oscillates back and forth between 1 and 0:

1 + -1 + 1 + -1 + 1 + -1 ...

This series never converges, no matter how many terms you write out, the value will always be 1, then 0, then if you add another term 1, then add another term 0, etc.

An obvious rule for series to converge is that the sequence should go to zero in the limit as it goes to infinity. So for instance:

10 + 100 + 1000 + 10000 + ...

The sequence never goes to 0, it just keeps increasing, so this series clearly is unbounded and diverges.

On the other hand:

9/10 + 9/100 + 9/1000 + ...

This sequence clearly DOES go to 0 as you go to infinity, eventually you'll have nine divided by a gigantic number, which will be very close to 0. So this series MIGHT converge (and you can show that it does).

Here's an example of a divergent series where the sequence converges to 0:

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ...

Well, as you go to infinity, you'll eventually reach one over a very large number and thus the sequence DOES go to 0. HOWEVER, this series (the sum of the sequence) diverges!
Posted by dylancatlow 5 years ago
.999 to the infinity = 1 for the same reason an infinite number of 1's placed next to each other equals infinity.

I concede.
Posted by KroneckerDelta 5 years ago
"there's always infinitely more positions that won't be reached"

That's why an infinite amount of time is being used. What happens when infinity goes against infinity?

How can you say the coordinate infinity is bigger than the time infinity?

It's not infinitely bigger. The problem is it's infinity to begin with. The domain is unbounded so there is nothing to "fill". No matter how long you count for, you will always be able to give me one more integer.

Notice this is a little it different from the case of the reals filling in a FINITE domain. This is more similar to the question about 0.999... = 1.

In this case, you never actually reach one, but you get infinitely close, so in the limit that you actually DO take it to infinity it DOES reach 1. The "..." means you are going to infinity.

What you are talking about with trying to list all of the integers (for example) by listing them for an infinite amount of time is very similar to what I talked about in R2 with the number:

111... = sum(10^i), from i = 0 to i = infinity

No matter how many terms you write down, you are NEVER going to approach ANY number. It's just going to keep growing without bound. You're not approaching ANYTHING! UNLIKE the case of 0.999... in which case, each term you write down gets closer and closer to the value of the limit, which is 1.

My argument, was that the above case 111..., where you have an unbounded number is not really what we normally consider a repeating decimal (again, for one it doesn't HAVE a value!).
Posted by dylancatlow 5 years ago
"The thing about irrational numbers is that they go on forever but NEVER repeat."

But wouldn't an infinite series have a part in it that is infinitely big of whatever you want an infinite number of times?
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