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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
UltimateUnknown on 28 April 2013
MDMAlliance said:
Jay520 said:
MDMAlliance said:
Jay520 said:
MDMAlliance said:
.999... =/= 1
It only approximately equals to it. It would be viewed the same way as a number approaches a limit, but never actually touching it. Calculus stuffs.


What number is between 1 and 0.999999...(?)


How is that relevant?  

 

Because if there is no number higher than 0.9999r and less than 1, then the two numbers are of the same value.


There is no "number higher than 0.9999r"  
When you go by that scale, you never reach 1 because you set your limit to 1.  There are an endless amount of points between 0 and 1.

This right here is pretty much the answer. It's setting your limit.

I think I understood the concept of this when looking at the exponential graph (as above), which keeps getting closer and closer to y=0 but never actually touches it. It only touches y=0 at infinity. But if you set the limit for x to infinity, then you can say exp(x)=0. In reality, the graph will never touch the x axis (because we can't really get to infinity) but by setting infinity as your limit you can say it can take a value of 0.

The same thing for the argument of 0.99999.... being equal to 1. If you set the limit for 0.99999... to be 1, then your defined function will be 1 at infinity, just as alliance said.



 

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Jay520 on 28 April 2013
MDMAlliance said:


You completely missed the point.  I did not say non-whole numbers don't exist.  

Also, the decimal system is a form of expressing a quantity but it is flawed.  However, we can still express those numbers even though they do not fit into one system.  1/9 is expressed as a fraction.  Pi is expressed as the ratio of the diameter to the circumference of a circle.  

.999... (repeated) is expressed through series that go on "infinitely" 

There's a major difference between those.

So you have a problem with the decimal system in general then? What's your opinion of 1/9. Writting in decimal form, it is written as 0.111111 (repeating). Do you think this is flawed representation of an actual number? If that's the case, why can't you accept that 0.9999999 (repeating) is a flawed representation of an actual number? That number would be 9 * 0.11111111 (1/9) which is 9/9 which is 1. What's wrong with that logic? 

In fact, the decimal system is only flawed if you assume 0.999999 doesn't equal one. If you accept that it does, then you are free to write fractions like 1/9, 3/9, etc. in decimal form and they would be perfectly valid.

And speaking of series. 0.99999 (repeating) can be expressed as an infinite series. It can be expressed as 9(1/10) + 9(1/10)^2 + 9(1/10)^3 + 9(1/10)^4.....which can be expressed as a sum of an infinite geometric sequence:

 9(1/10)
1-(1/10)

which equals 1



Player2 on 28 April 2013
MDMAlliance said:
Jay520 said:
MDMAlliance said:


I do not think you understand the concept.  The 9's (as we represent it as) will continue forever.  There's no end to it.  Infinity is not a number.  You may not be able to wrap your mind around that, but it's okay.  Not many people can.  

There is no "number higher than 0.9999r"  
When you go by that scale, you never reach 1 because you set your limit to 1.  There are an endless amount of points between 0 and 1.

 The 9's (as we represent it as) will continue forever.  There's no end to it.  Infinity is not a number. 

You must also concede that pi isn't a number either, since it repeats forever AND it doesn't have a pattern.

You must also concede that 1/9 isn't a number since (in decimal form), it repeats forever. (Even though 1/9 definitely IS a number).

When you go by that scale, you never reach 1 because you set your limit to 1.  There are an endless amount of points between 0 and 1.

Huh? I know there's an infinite amount of points between 0 and 1. There is an infinite amount of numbers between any two real different numbers. That's not the case for 0.9999... and 1.


You completely missed the point.  I did not say non-whole numbers don't exist.  

Also, the decimal system is a form of expressing a quantity but it is flawed.  However, we can still express those numbers even though they do not fit into one system.  1/9 is expressed as a fraction.  Pi is expressed as the ratio of the diameter to the circumference of a circle.  

.999... (repeated) is expressed through series that go on "infinitely" 

There's a major difference between those.

And the number e in non-decimal form looks like this:

or this:

or alternatively:

Yeah, all those look great.



Jay520 on 28 April 2013
Jay520 said:
MDMAlliance said:


You completely missed the point.  I did not say non-whole numbers don't exist.  

Also, the decimal system is a form of expressing a quantity but it is flawed.  However, we can still express those numbers even though they do not fit into one system.  1/9 is expressed as a fraction.  Pi is expressed as the ratio of the diameter to the circumference of a circle.  

.999... (repeated) is expressed through series that go on "infinitely" 

There's a major difference between those.

So you have a problem with the decimal system in general then? What's your opinion of 1/9. Writting in decimal form, it is written as 0.111111 (repeating). Do you think this is flawed representation of an actual number? If that's the case, why can't you accept that 0.9999999 (repeating) is a flawed representation of an actual number? That number would be 9 * 0.11111111 (1/9) which is 9/9 which is 1. What's wrong with that logic? 

In fact, the decimal system is only flawed if you assume 0.999999 doesn't equal one. If you accept that it does, then you are free to write fractions like 1/9, 3/9, etc. in decimal form and they would be perfectly valid.

And speaking of series. 0.99999 (repeating) can be expressed as an infinite series. It can be expressed as 9(1/10) + 9(1/10)^2 + 9(1/10)^3 + 9(1/10)^4.....which can be expressed as a sum of an infinite geometric sequence:

 9(1/10)
1-(1/10)

which equals 1


Better picture:



MDMAlliance on 28 April 2013
Jay520 said:
MDMAlliance said:


You completely missed the point.  I did not say non-whole numbers don't exist.  

Also, the decimal system is a form of expressing a quantity but it is flawed.  However, we can still express those numbers even though they do not fit into one system.  1/9 is expressed as a fraction.  Pi is expressed as the ratio of the diameter to the circumference of a circle.  

.999... (repeated) is expressed through series that go on "infinitely" 

There's a major difference between those.

So you have a problem with the decimal system in general then? What's your opinion of 1/9. Writting in decimal form, it is written as 0.111111 (repeating). Do you think this is flawed representation of an actual number? If that's the case, why can't you accept that 0.9999999 (repeating) is a flawed representation of an actual number? That number would be 9 * 0.11111111 (1/9) which is 9/9 which is 1. What's wrong with that logic? 

In fact, the decimal system is only flawed if you assume 0.999999 doesn't equal one. If you accept that it does, then you are free to write fractions like 1/9, 3/9, etc. in decimal form and they would be perfectly valid.

And speaking of series. 0.99999 (repeating) can be expressed as an infinite series. It can be expressed as 9(1/10) + 9(1/10)^2 + 9(1/10)^3 + 9(1/10)^4.....which can be expressed as a sum of an infinite geometric sequence:

 9(1/10)
1-(1/10)

which equals 1


.9 repeating is not a real number considering the fact that it is, unlike numbers like 1/9 and pi, essentially a form of infinity.  

Also, 9(1/10) = .1 * 9 = .9/ .9  (no repeating)

However, if you were to have .9 repeating over .9 repeating, it only equals one because they cancel each other out.  They are infinity starting at the same point at the same rate.



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Jay520 on 28 April 2013
KylieDog said:
Jay520 said:
AbbathTheGrim said:
But the difference between 0.9999... and 1 is .1 not 0. 


Nope, it's zero. 

1 - 0.9 = 0.1

1 - 0.99 = 0.01

1 - 0.9999 = 0.0001

But when you have infinite nines, you also have infinite zeros and never reach the 1. 

  1.0000000000000000........
- 0.9999999999999999.........
=0.0000000000000000........

At no point will you ever reach ....0001.


At no point will that 0.9999 ever become 1 either.

If you accept that there are infinite 9s then it is. 

Just like the fraction 5/7 doesn't equal 0.714285.... or 0.714285714285, but it does equal the the repeating decimal 0.714285 (with an infinite number of 714285s).



MDMAlliance on 28 April 2013
Jay520 said:
Jay520 said:
MDMAlliance said:


You completely missed the point.  I did not say non-whole numbers don't exist.  

Also, the decimal system is a form of expressing a quantity but it is flawed.  However, we can still express those numbers even though they do not fit into one system.  1/9 is expressed as a fraction.  Pi is expressed as the ratio of the diameter to the circumference of a circle.  

.999... (repeated) is expressed through series that go on "infinitely" 

There's a major difference between those.

So you have a problem with the decimal system in general then? What's your opinion of 1/9. Writting in decimal form, it is written as 0.111111 (repeating). Do you think this is flawed representation of an actual number? If that's the case, why can't you accept that 0.9999999 (repeating) is a flawed representation of an actual number? That number would be 9 * 0.11111111 (1/9) which is 9/9 which is 1. What's wrong with that logic? 

In fact, the decimal system is only flawed if you assume 0.999999 doesn't equal one. If you accept that it does, then you are free to write fractions like 1/9, 3/9, etc. in decimal form and they would be perfectly valid.

And speaking of series. 0.99999 (repeating) can be expressed as an infinite series. It can be expressed as 9(1/10) + 9(1/10)^2 + 9(1/10)^3 + 9(1/10)^4.....which can be expressed as a sum of an infinite geometric sequence:

 9(1/10)
1-(1/10)

which equals 1


Better picture:


Oh, you were trying to say this?  That one seems to just be .9 + .09 + .009 repeating forever.  

However 9(1/10)/1-(1/10) = 1 

is not the same thing as .9 repeating.



Soleron on 28 April 2013
Jay520 said:
...


Better picture:

This explanation, and the realisation that a single number can have more than one representation in decimal numbers (example: 0.35, 0.350, 0.3500, 0.35000... are all the same number) should be enough for anyone numerate. There is no dispute, it's the same quantity as 1 represents.



Chark on 28 April 2013

1 = 0.9... + ( 1 - 0.9...)

1 - 0.9... = x

1 = 0.9... + x

I don't see how this is reconcilable. The absence of an answer is not an answer.

In OP, wouldn't 9x = 8.9... not 9?



Before the PS3 everyone was nice to me :(

MDMAlliance on 28 April 2013
Soleron said:
Jay520 said:
...


Better picture:

This explanation, and the realisation that a single number can have more than one representation in decimal numbers (example: 0.35, 0.350, 0.3500, 0.35000... are all the same number) should be enough for anyone numerate. There is no dispute, it's the same quantity as 1 represents.


I take it math isn't your strong subject?  That or you haven't taken higher levels of math.  The whole distinction for 0.35 and 0.350 and going on is mostly a distinction made for science.  They use it to keep their "significant figures" as they need to keep their measurements as accurate as they were able to measure, whatever they were measuring, with.

You can try to make "proofs" that .9 repeating = 1, but this only can work if you ignore the fact that our decimal system doesn't work perfectly.