0.9 repeater = 1

[Kogloron]

Formula to prove .9 repeating is NOT 1

As has been mentioned, everyone agrees that 1/3 + 2/3 = 1, but if you add the decimal equivalents of 1/3 and 2/3, you get
0.33333... + 0.66666... = 0.99999...
so 0.99999... must be equal to 1.

Here's another proof.

Let x= 0.99999...
Then 10x = 9.99999...
and 10x - x = 9.99999... - 0.99999...
or 9x = 9
So x = 1. But, as defined above, x is also equal to 0.99999...
So 1 = 0.99999...

I've read through this, and i've gotta say math is a delicate thing, 1 screw up at the beginning and you get a tumble effect that changes ALL your results.
What i like about math is that 2 + 2 is always 4... ALWAYS.

First off i will state that 1/3 does not equal .3333.... THAT is an estimate. This fraction does not go into our numerical system.

Second 10 x 9999... does NOT equal 9.9999...

The flaw is these formulas comes from the way we were taught, we're shown a concept that seems to make sense, until we read into it a little bit.

10 x 1.0 = 10.0 the decimal point has moved over to the right one space and putting a decimal then zero is pointless, it's their for this demonstration.
10 x 9.90 = 99.0

10 x 0.9999... = 9.999...0

Try this. [] = cursor

Put your cursor at the end of 0.999[] and start typing 9's, you're on your way to making .999...

NOW multiply 0.999... by 10. Move the decimal over to the right 1 space, now also move your cursor to the left one space like this 9.999[]0
NOW start typing 9's in very basic terms you should get this
Step 1: 9.9990
Step 2: 9.99990
Step 3: 9.999990
etc infinite

But in all seriousness i read this post yesterday and woke up from my evening nap with a eureka moment where i thought outside the box. Has this EVER been brought up, am i eligible for an award? Am i wrong? Show me how if i am!

 

[Ocelot]

10 x 0.9999... = 9.999...0

Assuming your elipsis means recuring then that zero I highlighted does not exist.

You've placed is at the end of an unending sequence. Do you notice the contradiction there?

 

[Ocelot]

Firther more you seem to be adding a zero on the end. WHy. You didn't "add" a zero in the 10 example, you merely explicitly stated one that was there anyway before the decimal shift.

 

[W.D.Clinger]

First off i will state that 1/3 does not equal .3333.... THAT is an estimate. This fraction does not go into our numerical system.

Second 10 x 9999... does NOT equal 9.9999...
The flaw is these formulas comes from the way we were taught, we're shown a concept that seems to make sense, until we read into it a little bit.

10 x 1.0 = 10.0 the decimal point has moved over to the right one space and putting a decimal then zero is pointless, it's their for this demonstration.
10 x 9.90 = 99.0

10 x 0.9999... = 9.999...0

But in all seriousness i read this post yesterday and woke up from my evening nap with a eureka moment where i thought outside the box. Has this EVER been brought up, am i eligible for an award? Am i wrong? Show me how if i am!

Yes, you're wrong. The highlighted texts show where you went wrong. In particular, there is no 0 to the right of 0.999...

Yesterday, Gertrude posted a simple and elegant proof that 0.999...=1 :
http://www.internationalskeptics.com/forums/showpost.php?p=6037942&postcount=197

Vorpal explained the geometric intuition behind the proof of 0.999...=1 using geometric series:
http://www.internationalskeptics.com/forums/showpost.php?p=6037565&postcount=191

I posted a rigorously correct calculation showing why 10x0.999...=9.999... :
http://www.internationalskeptics.com/forums/showpost.php?p=6034784&postcount=163
That calculation fills in the details for Brown's proof (from 2003) that you quoted and rejected.

 

[Kogloron]

Assuming your elipsis means recuring then that zero I highlighted does not exist.

You've placed is at the end of an unending sequence. Do you notice the contradiction there?

Oh yes i see the contradiction, but it's not an impossible contradiction. This is probably going to be a piss poor example, and i know i'll get torn apart, what i'm waiting for is someone who sees what i see and is able to explain it in better detail.

But i guess the zero is the caboose of a train. If you have infinite carts on the train then there is no caboose. That is unless all the carts were added in between the last cart (zero) and all the other carts (nines)

The fact is ALL numbers multiplied by 10 have a zero at the end. and all numbers multiplied by 100 have 2 zeros at the end. The reason for this is that you will always keep the same amount place holders after a decimal point to be correct.
This means that 1.234 (3 digits after the decimal point) multiplied by 10 would be 12.340

Looking for a good public speaker who gets me to help me out a little. Or looking for someone to show me undeniably why i'm wrong. (I'm a skeptic too, and will accept proof that contradicts my equation)
1.23400

 

[Giggywig]

The fact is ALL numbers multiplied by 10 have a zero at the end. and all numbers multiplied by 100 have 2 zeros at the end. The reason for this is that you will always keep the same amount place holders after a decimal point to be correct.
This means that 1.234 (3 digits after the decimal point) multiplied by 10 would be 12.340

Your facts are not facts. If you multiply Pi by 10 does it suddenly terminate and have a 0 at the end? No.

 

[Rasmus]

The fact is ALL numbers multiplied by 10 have a zero at the end.

No.

and all numbers multiplied by 100 have 2 zeros at the end. The reason for this is that you will always keep the same amount place holders after a decimal point to be correct.

no.

This means that 1.234 (3 digits after the decimal point) multiplied by 10 would be 12.340

But everyone in the whole world would give you the result as 12.34 and not as 12.340

Just like you started out from 1.234 and not - as you could have done - 1.23400000000000000

You are starting out with a decimal that ends in an infinite string of zeroes - even though we're not usually writing that down because it would be pointless and inconvenient. But the zeroes are there.

If you were to write down the multiplication for 10 x 0,999.... you could *tell* that there would never be a zero. where should it come from? (You know there's only going to be more nines, even if you could never write them all down, just like you know that 1/3 will have an endless strings of 3's after the decimal point, even if you can never write them all down.)

Looking for a good public speaker who gets me to help me out a little. Or looking for someone to show me undeniably why i'm wrong. (I'm a skeptic too, and will accept proof that contradicts my equation)
1.23400

sit down with a piece of paper and do 1/3.
What is your result?

I get 0.33333 with an infinite string of threes.. I can't write that down but I know that is what the result is. Do you agree?

 

[sol invictus]

NOW multiply 0.999... by 10. Move the decimal over to the right 1 space, now also move your cursor to the left one space like this 9.999[]0
NOW start typing 9's in very basic terms you should get this
Step 1: 9.9990
Step 2: 9.99990
Step 3: 9.999990
etc infinite

Does .9999*10 = 9.9990? Sure, but it also equals 9.999, and 9.99900, and even 9.9990000... . And .9999 = .99990, etc., so it's got nothing to do with multiplying by 10.

Another major problem with your logic is that while it's true that 9.999... is the limit of the series defined by
Step 1: 9.9990
Step 2: 9.99990
Step 3: 9.999990,
it's also true that it's the limit of the series defined by
Step 1: 9.9991
Step 2: 9.99991
Step 3: 9.999991,
or
Step 1: 9.99942
Step 2: 9.999942
Step 3: 9.9999942,
or even
Step 1: 9.999314159...
Step 2: 9.9999314159...
Step 3: 9.99999314159... .

The limit of all of those sequences is the same number, namely 9.999.... = 10.

 

[Rasmus]

But i guess the zero is the caboose of a train. If you have infinite carts on the train then there is no caboose.

Right. There is no last digit in a number with infinitely many digits.

That is unless all the carts were added in between the last cart (zero) and all the other carts (nines)

But that is not how numbers are made. A number doesn't fill a template that has to end with a zero. 5.437 doesn't start out as 0 or 0.00000 or 0.0000.... or 3.5 or any such thing. It is not build up from somewhere.

5.437 doesn't have to be 5.4 before it can be 5.437, much less does it have to be 5.47

You are confusing the act of writing down a number on a piece of paper with the number itself. But one doesn't have to do anything with the other. It would be like saying that a cabbage (green thing, grows in a field) must at some stage f it's existence be a yellow car in New York or at least a famous Swedish rock band.

 

[Kogloron]

No.



no.



But everyone in the whole world would give you the result as 12.34 and not as 12.340

Just like you started out from 1.234 and not - as you could have done - 1.23400000000000000

You are starting out with a decimal that ends in an infinite string of zeroes - even though we're not usually writing that down because it would be pointless and inconvenient. But the zeroes are there.

If you were to write down the multiplication for 10 x 0,999.... you could *tell* that there would never be a zero. where should it come from? (You know there's only going to be more nines, even if you could never write them all down, just like you know that 1/3 will have an endless strings of 3's after the decimal point, even if you can never write them all down.)



sit down with a piece of paper and do 1/3.
What is your result?

I get 0.33333 with an infinite string of threes.. I can't write that down but I know that is what the result is. Do you agree?

I agree and i concede. This was more of a strong gut notion of logic then anything. I just needed to really hear a few opinions after hearing my point. I thought my argument had merit, and i still have a feeling I MAY be correct, but until i have another moment of genius i'm going to let this go. I'm outmatched here, i don't even understand the formula that W.D Clinger provided. So since i'm out of my league i'm goign to monitor this for any interesting discoveries. Thanks for hearing me out

-Kog

 

[Fnord]

As in "why is Gödel's incompleteness theorem at all important?" Well, before Gödel, there was a believe that Mathematics could prove everything mathematical. Hilbert, around the turn of the last century, speculated that even Physics could be reduced to a few axioms and then everything true in Physics could be derived and proven mathematically. Gödel ended all that. Anything at the level of arithmetic and above cannot be both consistent (i.e. free of contradiction) and complete (able to prove all true statements about itself).

.
No ... more like "Why does Gödel's Incompleteness Theorem merit so many pages in a single thread?" I mean, regardless if its alleged validity as a theorem, how does it apply to the real world? Will it get me through truck-driving school? Does it help anyone determine how many times to flip a burger? Can I use it as both a floor wax and a dessert topping? Will it help me find the TV remote?

Seriously, though ... what are the practical applications of Gödel's Incompleteness Theorem?

 

[roger]

First off i will state that 1/3 does not equal .3333.... THAT is an estimate. This fraction does not go into our numerical system.

Others have addressed your trailing zero problem, so I won't. But this quoted statement is the root of the problem.

You are wrong. By definition .3333.... is 1/3. Well, it falls out from the definition of decimal representation. Remember, math is not a description of reality (though it certainly does in some cases), it is a formal system of rules. The rules say .33333.... = 1/3.

From there, the .999999..... = 1 falls out quite easily, as does the rest of the mathematics in this thread.

Since you claim to have trouble with the mathematics expressed in this thread, let me offer you a hand-waving analogy . Suppose instead of decimal representation we use roger representation. In roger representation I write 1/3 as aBq. (assume I now type a bunch of rules that does in fact let 1/3 be typed as aBq). It would make no sense for someone to argue that this can't be right, as a is the first letter in the alphabet, thus =1, or that the ASCII representation of B is 0x42, or whatever, because those have nothing to do with my rules. So I can summarily dismiss any objection of this kind.

It may not be clear to you, but this is what you are doing when you dismiss .33333..... = 1/3. We've defined a representation, decimal representation, where .3333.... does equal 1/3. Not arbitrarily close or whatever, but equal.

 

[Ocelot]

.
Seriously, though ... what are the practical applications of Gödel's Incompleteness Theorem?

You're in the middle of writing an important report. The application you're running freezes.

It's an all too familiar scenario. Sometimes you wait a while and it comes back to life. Sometimes a few well chosen key presses coax it back into a responsive state.

However sometimes you wait a while and it hasn't come back yet. Should you continue to wait?

Godel's incompleteness theorum allows you to find out that the answer to that question is often:...

...there's no way of telling so don't beat yourself up over it.

 

[W.D.Clinger]

Seriously, though ... what are the practical applications of Gödel's Incompleteness Theorem?

Gödel's incompleteness theorems are more important for philosophy than for practical applications.

On the other hand, Gödel's first incompleteness theorem is connected to the undecidability of particular problems, such as the halting problem^WP, that have enormous implications for practical applications. Although Gödel proved his incompleteness theorems in 1931, and Turing didn't prove the undecidability of the halting problem until 1936, a version of Gödel's first incompleteness theorem can be proved as a corollary of the undecidability of the halting problem.

One consequence of the undecidability of the halting problem is that there is no algorithm that can take two arbitrary computer programs as input and decide whether the programs are equivalent. Another consequence is that there is no general algorithm for deciding whether a computer program is correct with respect to its specification; in other words, the naive formulation of the grading problem for freshman computer science is undecidable.

That affects everyone who writes or relies upon computer programs. To take just one example, almost all computer programs are written in human-readable source languages and then translated by a compiler^WP into nearly-unreadable machine languages. There are infinitely many correct translations of a program, and there are usually quite a few plausible translations; the problem of choosing one of the more efficient plausible translations is the central problem of compiler optimization^WP. It turns out that many of the compiler optimizations we would like for our compilers to perform would require the compilers to decide undecidable problems; hence we can't write compilers that perform those optimizations as well as we would like, and have to settle for suboptimal translations into machine code.

That may give you some idea of why undecidability is a required topic in most undergraduate computer science curricula, but is not part of the standard curriculum for mathematics, science, and most engineering majors.

 

[jsfisher]

Lindenbaum's lemma and Gödel's incompleteness theorems involve the same notion of completeness: a theory is complete if and only if it is consistent but adding any new sentence to the theory would make it inconsistent.

You are correct, of course. That's what I get for relying on (a distant) memory for what type of completeness the lemma used.

 

[Vorticity]

...Skeptics who (quite properly!) regard the axiom of choice as unproven...

Why should it be a problem that an axiom is unproven?

 

[W.D.Clinger]

Why should it be a problem that an axiom is unproven?

That's not the problem, and my mention of it was a bit of a joke: Paul Cohen proved that AC can't be proved from ZF.

As an occasional skeptic, I don't think axioms should be multiplied beyond necessity. To my knowledge, there are no practical applications for which the axiom of choice is actually necessary; the axiom is hardly self-evident, and is in fact controversial; it is inconsistent with axioms that are just as elegant and no less plausible, such as the axiom of determinacy^WP; in short, its implicit assumption by many mathematicians during the 20th century appears to have been an unfortunate accident of history. I have no objection to theorems whose proof requires the axiom of choice, but I argue that a correct statement of such a theorem would include the axiom of choice as an explicit hypothesis.

If anyone really wants to discuss that, we should start another thread.

 

[gnome]

When I think of an axiom, I don't think of something that is self-evident exactly, but something that HAS to be true in order for the formal system to be useful for its intended purpose.

 

[Kwalish Kid]

But in all seriousness i read this post yesterday and woke up from my evening nap with a eureka moment where i thought outside the box. Has this EVER been brought up, am i eligible for an award? Am i wrong? Show me how if i am!

It isn't original. In a way, it is brought up all the time in standard mathematical computation classes. You are dead wrong.

The repeating decimal is a means of indicating the construction of a series. There are many ways of calculating multiplication, but they do not usually break down into specific methods like yours, where whenever one multiplies by 10 one simply moves a decimal place and adds a zero. This may be equivalent in some cases to an application of multiplication, but it misses the essence of multiplication.

Your cursor example is exactly the wrong way to think about calculating with numbers. In standard computational theory, a computation cannot revise a digit once placed. In your proceedure, we cannot do any multiplication with 0.999... until we have figured out the final place of your "0". (Of course, there are many calculations that we cannot do with decimal representations of numbers, which is why they are probably a sub-standard representation of real numbers.)

 

[Soapy Sam]

Soapy's Simplification :- " Number is imaginary. Quantity is real. Any number may be turned into a quantity by adding the words "Functional cows", to the number.

In any case n where the resulting quantity "n functional cows" makes no sense - examples being fractions, roots , infinities and similar suspect operations, it should be assumed that the equivalent number makes no sense either - and anyone suggesting otherwise should be slapped with a wet turbot.


This will greatly simplify mathematics .

It may be easily demonstrated that 0.999... functional cows is not a cow.

And if you don't believe me, try selling same to a farmer.