Does 0.9999999... = 1?

[Ryan2065]

If 0 = 0.0^, then there would be no need for 0.0^ to exist.

So since 1/2 exists there is no reason for .5 to exist... Therefore .5 MUST be different than 1/2

I think the problem here is that if you think 1 = 0.9^ its because you can't comprehend infinity, so must round up for it to make sense to you.

Again... Either you do not comprehend infinity or every number theorist math professor does not understand infinity.

 

[Fluffy]

Thanks Fluffy. I don't understand what Halycon is trying to say, if I did I would give a rebuttal. Halycon basically told me to ignore the first line in my proof, and then did some arithmetic. It wasn't clear to me what he was saying. If you could rephrase his points for me, I'll see what I can do.

Halcyon made a post which had the following maths (or something akin to this, can't find the specific post):

0.9^ = 0.9^
9.9^ = 9.9^ (Multiply by 10)
9 = 9 (Subtract 0.9^)
1 = 1 (Divide by 9)

Essentially he has taken the 1st proof of the OP (the one you posted a few times) and replaced x with 0.9^. This should be valid since we have defined x = 0.9^. However, instead of getting the same answer, he gets 1 = 1.

Now I suppose it could be argued that this is the same answer since 0.9^ = 1 but it doesn't seem to prove that 0.9^ = 1. In fact it doesn't seem to prove anything at all.

I don't understand how to explain where/if he has gone wrong and why replacing x with 0.9^ is not valid or if it is valid, why he is coming up with something different.

 

[KingNothing]

There's nothing there that resembles a proof. It's just a bunch of eqalities. A proof is a logical set of equations that follow from previous equations.

Often we don't understand why something is true, but if we can understand the individual steps in the proof of it, then the whole must be true.

In my post I have set x=.^9, and from that I have derived, I repeat derived and NOT SET, x=1. Because x=.^9 and x=1, it follows that .^9=1.

 

[KingNothing]

In other words, Suppose I say x= apple. Halcyon is saying, "no get rid of x and replace it with apple. Then apple = apple and therefore you haven't shown x=apple." That makes no sense.

 

[Fluffy]

There's nothing there that resembles a proof. It's just a bunch of eqalities. A proof is a logical set of equations that follow from previous equations.

Often we don't understand why something is true, but if we can understand the individual steps in the proof of it, then the whole must be true.

In my post I have shown that x=.^9, and from that I have derived, I repeat derived and NOT SET, x=1. Because x=.^9 and x=1, it follows that .^9=1.

Gotcha thanks! . Just couldn't quite get my head around it!

 

[SoyLeche]

Line 1 x = .^9
Line 2 10x = 9.^9
Line 3 10x-x = 9x= 9.^9-.^9 = 9
Line 4 9x=9
Line 5 x=1

I still haven't seen anyone prove this wrong. Which step do you disagree with?

Look at post 234. I explained which step they disagree with there.

 

[KingNothing]

Oh I know where they disagree. You got it right. I wanted to see if they still disagreed when put in the context above though. I don't think any rational person can.

 

[SoyLeche]

Oh I know where they disagree. You got it right. I wanted to see if they still disagreed when put in the context above though. I don't think any rational person can.

And yet, I know that Halcyon is rational...

 

[SoyLeche]

Halcyon made a post which had the following maths (or something akin to this, can't find the specific post):

0.9^ = 0.9^
9.9^ = 9.9^ (Multiply by 10)
9 = 9 (Subtract 0.9^)
1 = 1 (Divide by 9)

Essentially he has taken the 1st proof of the OP (the one you posted a few times) and replaced x with 0.9^. This should be valid since we have defined x = 0.9^. However, instead of getting the same answer, he gets 1 = 1.

Now I suppose it could be argued that this is the same answer since 0.9^ = 1 but it doesn't seem to prove that 0.9^ = 1. In fact it doesn't seem to prove anything at all.

I don't understand how to explain where/if he has gone wrong and why replacing x with 0.9^ is not valid or if it is valid, why he is coming up with something different.

I've been thinking about this one a bit.

Overall, this series of equations is pretty uninteresting. It's somewhat equivalent to taking:

2*x + 7 = 13 (subtract 7 from both sides)
2*x = 6 (divide by 2)
x = 3

And then saying - "you didn't do anything. Watch - put 3 back in for x in all the equations"

13 = 13 (subtract 7)
6 = 6 (divide by 2)
3 = 3

"See, it was all just math 'trickery'" - when really all this has done is show that at each step the equation holds, the right side is equal to the left side. But we already knew that, it's the definition of an equation.

The interesting thing about the series of equations in quesiton is that you always end up with the same numbers on each side of the = sign at the end as you did in the beginning as long as in step 2 you subtract the origional number. If you subtract something else, you don't get back to the same place:

2 = 2 (multiply by 100)
200 = 200 (subtract 2)
198 = 198 (divide by 99)
2 = 2

2 = 2 (multiply by 100)
200 = 200 (subtract 3)
197 = 197 (divide by 99)
1.989898... = 1.989898...

1000 = 1000 (multiply by 100)
100000 = 100000 (subtract 1000)
99000 = 99000 (divide by 99)
1000 = 1000

1000 = 1000 (multiply by 100)
100000 = 100000 (subtract 1200)
98800 = 98800 (divide by 99)
997.979797... = 997.979797...

0.1111... = 0.1111... (multiply by 100)
11.1111... = 11.1111... (subtract 0.1111...)
11 = 11(divide by 99)
0.1111... = 0.1111...

etc. etc. etc.

Now, let's play with 0.9999...

0.9999... = 0.9999... (multiply by 100)
99.9999... = 99.9999... (subtract 0.9999...)
99 = 99 (divide by 99)
1 = 1

That's odd - in every case, when we subtract the number we started with in step 2, we end up with the same numbers on the right and left of the bottom equation as we did in the top. Why the difference? Maybe 0.9999... = 1

Let's try subtracting 1 in step 2:

0.9999... = 0.9999... (multiply by 100)
99.9999... = 99.9999... (subtract 1)
98.9999... = 98.9999... (divide by 99)
0.9999... = 0.9999...

That time it worked. Interesting, isn't it? Not a proof, but interesting nonetheless.

 

[SoyLeche]

I've been thinking about this one a bit.

Overall, this series of equations is pretty uninteresting. It's somewhat equivalent to taking:

2*x + 7 = 13 (subtract 7 from both sides)
2*x = 6 (divide by 2)
x = 3

And then saying - "you didn't do anything. Watch - put 3 back in for x in all the equations"

13 = 13 (subtract 7)
6 = 6 (divide by 2)
3 = 3

"See, it was all just math 'trickery'" - when really all this has done is show that at each step the equation holds, the right side is equal to the left side. But we already knew that, it's the definition of an equation.

The interesting thing about the series of equations in quesiton is that you always end up with the same numbers on each side of the = sign at the end as you did in the beginning as long as in step 2 you subtract the origional number. If you subtract something else, you don't get back to the same place:

2 = 2 (multiply by 100)
200 = 200 (subtract 2)
198 = 198 (divide by 99)
2 = 2

2 = 2 (multiply by 100)
200 = 200 (subtract 3)
197 = 197 (divide by 99)
1.989898... = 1.989898...

1000 = 1000 (multiply by 100)
100000 = 100000 (subtract 1000)
99000 = 99000 (divide by 99)
1000 = 1000

1000 = 1000 (multiply by 100)
100000 = 100000 (subtract 1200)
98800 = 98800 (divide by 99)
997.979797... = 997.979797...

0.1111... = 0.1111... (multiply by 100)
11.1111... = 11.1111... (subtract 0.1111...)
11 = 11(divide by 99)
0.1111... = 0.1111...

etc. etc. etc.

Now, let's play with 0.9999...

0.9999... = 0.9999... (multiply by 100)
99.9999... = 99.9999... (subtract 0.9999...)
99 = 99 (divide by 99)
1 = 1

That's odd - in every case, when we subtract the number we started with in step 2, we end up with the same numbers on the right and left of the bottom equation as we did in the top. Why the difference? Maybe 0.9999... = 1

Let's try subtracting 1 in step 2:

0.9999... = 0.9999... (multiply by 100)
99.9999... = 99.9999... (subtract 1)
98.9999... = 98.9999... (divide by 99)
0.9999... = 0.9999...

That time it worked. Interesting, isn't it? Not a proof, but interesting nonetheless.

One more thought:

It has been said that using this method we "hide the infinite nature" of 0.9999... and thus work some kind of a trick. Why is it only for 0.9999... that this "trick" works? As has already been shown:

x = 0.8888... (multiply by 100)
100x = 88.8888.... (subtract x)
99x = 88 (divide by 99)
x = 0.8888....

It worked fine. "Hiding the infinite nature" of 0.8888.... did nothing to invalidate the results. Why does it in 0.9999...?

 

[Mercy Not Sacrifice]

What's your point? It's infinitesimally smaller than 1, therefore it is not 1.

Can you please post the mathematical proof of this claim?

What are you talking about? The number bigger than 0.9999^ is 1 and the number smaller than 1 is 0.9999^.

Also, i think those mathematical "proofs" are essentially flawed. They're just fiddling the numbers.
Just because 1/3 = 0.3^ doesn't mean that 3/3 = 0.9^, its just flawed logic - if you divide a group of 3 objects into three groups you get a whole number, if you divide a single object into three you get a fraction if the whole.
When you recombine 3x0.3^ you get 1 because of the infinite 3's, but that doesn't correlate to dividing a group into separate wholes. Those "proofs" are mathematical anomalies that do not represent real world dynamics.

Do you realize that you have just called the very science of mathematics into question?

 

[Mercy Not Sacrifice]

But that goes back to the fact that there is an infinite number of points between two points.

So if you divide 1 by infinity you still can't reach 0.

Actually, 1/inf = 0.

 

[Mercy Not Sacrifice]

The problem is the way math is taught... you will never get me to accept it, because it defies all of the mathematical logic I have been taught... I understand that I am wrong according to your proofs... My brain just can't take it, I was always taught that there is no complete decimal definition of 1/3 because .333333 is incomplete without an addition somewhere, but that the addition cannot exist because then the fraction would then be greater than 1/3...

It seems to me If .99... = 1 then at some point one of the .00...9's(I hope you understand what I mean by that) must become a ten...

What you are saying comes to me like 'at infinity 9=10'... just doesn't process...

I sense you're starting to get it, so I'll say this for the benefit of anyone who's still befuzzled.

If you've got a calculator handy (Windows' version will work), enter 1/9 and see what happens. Notice; it's a stream of 1's after the decimal. Now try it again for 2/9, then 3/9. Notice that whatever whole number between 1 and 8 that you divide by 9, you get a stream of that number after the decimal.

In other words, so far we have:

1/9 = 0.1111...
2/9 = 0.2222...
3/9 = 0.3333...

So we could go on:

7/9 = 0.7777...
8/9 = 0.8888...
9/9 = 0.9999...

Here's where this all comes together. If you enter 9/9 into your calculator, you get a result of 1.
Congradulations, you've just demonstrated that 0.9999... = 1.

 

[KingNothing]

I would stay away from trying to use a calculator to prove something rather than concepts. Most calculators and computer programs will only calculate up to the 14th decimal place or less. They will also round the last digit, so if you enter 7/9 you will actually get .7777777778, or something similar depending on your calculator.

 

[Ozzie]

What you are describing are known as asymptotes. They are usually first encountered when shown the graph of y=1/x. http://en.wikipedia.org/wiki/Asymptote See the second figure from the top.

This is a similar idea because it is generally stated by mathematicians that at infinity, the line will touch the axis. It is similar because it deals with a concept of infinity that developed due to the internal inconsistencies of the previous concept and this version of infinity is the same as is being used here. However, beyond that, the similarities end.


Hey michel, I am uncertain if you are approximising when you say "0.9999999" but I just want to make sure that you understand that I am not talking about 0.9999999 but "0." followed by an infinite number of 9s (for which the mathematical notation is 0.9^)


Keeping the above in mind, it is impossible to have 0.9^8 because this would involve placing an 8 at the end of an infinite sequence of 9s. "at the end of infinity" is similar to ideas such as "outside of space" and "before time". It is certainly not a concept that is mathematically supported. Infinity has no end at which the 8 may be placed.

Pi is one of the mathematical nuggets that I have not yet studied in depth. I understand it to be an irrational number (so it cannot be accurately written as a fraction). However, it is also a transcendental number and so it is part of an entire realm of mathematics I have yet to study. I am uncertain how you would go about calculating a transcendental number (the definition of one is sufficiently complex so I will not post it here) but as far as I am aware, it is not possible to write them without approximising.

However, the decimal to a sufficient degree will always be more accurate than the fraction. The fraction is useful for "simple" calculations in which relatively imprecise answers are sufficient (eg understanding the use and impact of Pi does not require an accurate output)


Thank you but I was specifically after a reference that stated that 0.9^ is "a number approaching 1 that can never quite get there". Apologies for my skepticism but I do not believe that you will be able to find a respectable mathematician who will endorse that statement.

You should also know that an "infinite regress", as you have defined it, involves convergent series and that it is well documented that when you sum and infinitely long series made up of convergent terms, you will get a finite number. http://en.wikipedia.org/wiki/Geometric_series#Infinite_geometric_series


And a piece of cake is a fraction of the entire cake. It is 1/3 of the entire cake and 1/3 is an infinitely recurring decimal. That you have 3 pieces of cake makes sense since 3 x 1/3 = 3/3 = 1 and you have 1 cake.

I agree but "I can't fly" does not equate to "flight is impossible" which appears to be analagous to what is being argued in that particular case.

If under "this" version of infinity, the assymptote touches the axis at infinity, then you are defining .9' = 1 through your definition of infinity, and there seems little point in engaging in algebraic gymnastics to proove otherwise. In this conception, .9' =1 is treated as a priori knowledge and cannot be disproven.

You asked in a previous post what is a defined quantity.( I had pointed out that neither .3' nor 1/3 are defined quantities). I would respond by pointing out that defined quantities in abstract allow drawing of equality between them. Your OP proofs are based on this idea.

Logically the statement .9' =1 only holds if the asymptote reaches the axis at infinity.

Therefore, .9' = 1 is an assumption of mathematicians concept of infinity, not that .9' = 1.

Otherwise we can say .9' = 1 by convention.

 

[KingNothing]

I'm done debating this. I just thought this was funny:

Therefore, .9' = 1 is an assumption of mathematicians concept of infinity, not that .9' = 1

Perhaps you meant differently, but that sounds like you're saying .9^=1 only mathematically. What else are we talking about here besides math?

The next person to use the phrase 'at infinity' needs to be whacked. It's a logical oxymoron and represents a lack of understanding.

 

[Ozzie]

I'm done debating this. I just thought this was funny:


Perhaps you meant differently, but that sounds like you're saying .9^=1 only mathematically. What else are we talking about here besides math?

The next person to use the phrase 'at infinity' needs to be whacked. It's a logical oxymoron and represents a lack of understanding.

Yes and exactly.

 

[Ryan2065]

The next person to use the phrase 'at infinity' needs to be whacked. It's a logical oxymoron and represents a lack of understanding.

All the math professors are saying it!

 

[Ozzie]

All the math professors are saying it!

So considering infinity in a time-space context is an oxymoron, is it?

 

[Ryan2065]

So considering infinity in a time-space context is an oxymoron, is it?

Consider it for an infinite amount of time! ^-^