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Forums - General - 0.9999.... = 1.0

 

Are you convinced?

Yes 34 58.62%
 
No 20 34.48%
 
not sure 1 1.72%
 
Total:55
dsgrue3 on 01 May 2013
ebw said:

You are deeply misguided about what a proof is.  Your calculation begins with the premise that x=2 and deduces that x = ±2, which I completely agree with.  You seem to misunderstand what the ± notation means.  The last sentence reads "x must be either 2 or -2", which of course it is, since we happen to know it is 2.  There is nothing in the sentence "x = ±2" that says that x could be -2, only that x cannot take any other value.  Likewise, Pezus's example seeks to show that if x has any value whatsoever, it must be 1.  Reading this as "-2 = 2" is utter nonsense.

Actually +/- means either one is a solution for x, which it obviously isn't.

How am I misguided about what a proof is? I'm the one suggesting this type of variable declaration manipulation is utter nonsense. There is no equation to solve, only an assertion that x is 2, or in the OP case that x = 0.999...

Soleron said:

His explanation doesn't contain any invalid steps. Yours does. Neither are formal proofs but no one here would understand the formal proof.

Mine does not contain invalid steps because there is no rule that describes variable declaration manipulation.

You are missing the fundamental point here, which is that it isn't an equation. X is already known, any other value you associate to x is invalid by this very reason, which is why logic goes out the window.

I'm sure many people can understand a formal proof here.



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AbbathTheGrim on 01 May 2013

Seems like when you argue about infinite numbers the discussions extends to the infinite. Who would've seen that coming?



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Nem on 01 May 2013
Jay520 said:

This is a fairly old topic so I'm not sure if all of you have heard about this. I just found out about it and found it interesting.

The claim is that 0.9999.....(infinite 9s) is equal to 1.

Here are some proofs

-

x = 0.9999… given
10x = 9.9999…. multiply by 10
9x = 9 subtract x
x = 1 divide by 9
0.999... = 1 substitution

- 

What's the difference between 1 and 0.9999....? Their difference is 0.0000....The zeroes will stretch out infinitely and we will never reach 1. Therefore, 0.00000...can be simplified to 0.

If the difference between any two numbers is 0, then they are of the same value.

- 

On the number line, there is no real number between 0.99999..... and 1. If there is no real number between any two numbers, then those two numbers are of the same value. There is no number higher than 0.9999... and lower than 1, therefore they are of the same value.

- 

1= 7/7 
1= 5/7 + 2/7 
1= 0.714285714285.... + 0.285714285714......
1= 0.999999999999

- 

Sum of an infinite convergent geometric series


and many more

Maybe someone already said this but can you really subtract an X like that when you get to 9X? I was convinced the equation rules didnt permit that, even though i see what you're saying.



Jaydi on 01 May 2013
dsgrue3 said:
ebw said:

You are deeply misguided about what a proof is.  Your calculation begins with the premise that x=2 and deduces that x = ±2, which I completely agree with.  You seem to misunderstand what the ± notation means.  The last sentence reads "x must be either 2 or -2", which of course it is, since we happen to know it is 2.  There is nothing in the sentence "x = ±2" that says that x could be -2, only that x cannot take any other value.  Likewise, Pezus's example seeks to show that if x has any value whatsoever, it must be 1.  Reading this as "-2 = 2" is utter nonsense.

Actually +/- means either one is a solution for x, which it obviously isn't.

How am I misguided about what a proof is? I'm the one suggesting this type of variable declaration manipulation is utter nonsense. There is no equation to solve, only an assertion that x is 2, or in the OP case that x = 0.999...

Soleron said:

His explanation doesn't contain any invalid steps. Yours does. Neither are formal proofs but no one here would understand the formal proof.

 

Mine does not contain invalid steps because there is no rule that describes variable declaration manipulation.

You are missing the fundamental point here, which is that it isn't an equation. X is already known, any other value you associate to x is invalid by this very reason, which is why logic goes out the window.

I'm sure many people can understand a formal proof here.

 

(x=2) implies that (x²=4). (x²=4) implies that (x=2 or x=-2). Now we "sum up" the two implications: (x=2) implies that (x=2 or x=-2) which is totally true. Logic is respected and so do the mathematics.

 

Edit: Of course you cannot "reverse" these two implications because (x=2) is not equivalent to (x²=4).



dsgrue3 on 01 May 2013
Jaydi said:

x=2 implies that x²=4. x²=4 implies that (x=2 or x=-2). Now we "sum up" the two implications: x=2 implies that (x=2 or x=-2) which is totally true. Logic is respected and so do the mathematics.

x = 2 cannot ever be -2, it will always be 2 and no other value. It has already been declared as such.

You're abusing the word "or" here which actually not mean what you're saying. It is really an "either" rather than an "or."

"or" as you imply is intended to be an exlusive or. It isn't in this case as both solutions are proper.

EDIT: I don't want to discuss this any further. It isn't helping with the concept of the OP that 0.999... = 1, which I fully agree with. I just disagree with the initial "proof."



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Jaydi on 01 May 2013
dsgrue3 said:
Jaydi said:

x=2 implies that x²=4. x²=4 implies that (x=2 or x=-2). Now we "sum up" the two implications: x=2 implies that (x=2 or x=-2) which is totally true. Logic is respected and so do the mathematics.

x = 2 cannot ever be -2, it will always be 2 and no other value. It has already been declared as such.

You're abusing the word "or" here which actually not mean what you're saying. It is really an "either" rather than an "or."

"or" as you imply is intended to be an exlusive or. It isn't in this case as both solutions are proper.

Of course x will never be equal to 2. But, mathematical logic use the word "or" in the very manner I used it. If you alone give it another meaning, you won't make it to the right conclusion.

First of all, your computations are meaningless in a mathematical point of view. You should've write

x=2 => x²=4 => (x=2 or x=-2).

In this case, there is no mathematical flaw and everything is correct: the "or"  has to be understood in a mathematical way, i.e. the assertion is true if and only if x is equal to one of the answers.

If I say that you're alive or you're dead, is that sentence right or not?

 

It reminds me an old joke about a logician who just had a baby. One of his/her friend calls him and ask "Is it a he or a she?" and the logician just answers "Yes". And from a logical point of view, he's true.



dsgrue3 on 01 May 2013
Jaydi said:

It reminds me an old joke about a logician who just had a baby. One of his/her friend calls him and ask "Is it a he or a she?" and the logician just answers "Yes". And from a logical point of view, he's true.

Lol that is a funny joke.



ebw on 01 May 2013
dsgrue3 said:
ebw said:

You are deeply misguided about what a proof is.  Your calculation begins with the premise that x=2 and deduces that x = ±2, which I completely agree with.  You seem to misunderstand what the ± notation means.  The last sentence reads "x must be either 2 or -2", which of course it is, since we happen to know it is 2.  There is nothing in the sentence "x = ±2" that says that x could be -2, only that x cannot take any other value.  Likewise, Pezus's example seeks to show that if x has any value whatsoever, it must be 1.  Reading this as "-2 = 2" is utter nonsense.

Actually +/- means either one is a solution for x, which it obviously isn't.

No, that is exactly what ± does not mean, as I have already stated.  Your interpretation leads to fuzzy logic, while the standard interpretation is really quite consistent.  Your bald assertion amounts to "I choose to interpret this statement in this non-standard way so as to criticise the logic which is only flawed by my invalid interpretation".



dsgrue3 on 01 May 2013
ebw said:
dsgrue3 said:

Actually +/- means either one is a solution for x, which it obviously isn't.

No, that is exactly what ± does not mean, as I have already stated.  Your interpretation leads to fuzzy logic, while the standard interpretation is really quite consistent.  Your bald assertion amounts to "I choose to interpret this statement in this non-standard way so as to criticise the logic which is only flawed by my invalid interpretation".

x^2 = 4. You're saying that -2 is not a solution? Just trying to understand where you're coming from.



Zkuq on 01 May 2013
dsgrue3 said:

I don't want to discuss this any further. It isn't helping with the concept of the OP that 0.999... = 1, which I fully agree with. I just disagree with the initial "proof."

The initial proof has no errors in it, it's standard equation manipulation and if we interprete your 'counter-example' your way, I'm fairly scared about the consequences to solving equations in general. Your 'counter-example' (or rather, your interpretation of it) pretty much says that whenever we modify equations, there's a chance we screw up just by modifying the equation even if all steps are correct. To me, it seems Jaydi is right here.

And besides, this whole thing is on pretty shady ground in the sense that you should be able to point out the error in the original proof considering it's so simple. For example, in 1=2 "proofs" you can find a step where the equation was multiplicated or divided by zero, but in this case, I doubt you can find a single step where an error is made. This explanation doesn't exactly prove anything but it should make you think if you've done something wrong when you can't actually find the error in such a simple process. And then there's the fact that two seemingly different things are often equal, even if the seem unrelated to each other.