Proof: 0.9999 does not equal 1

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In summary: To see a few of the many threads, click here.In summary, after a conversation about the equality of 0.999... and 1, it was concluded that 0.999... does indeed equal 1 due to the definition of decimal place notation and the limit of an infinite geometric series. This topic has been previously discussed on PhysicsForums and can be found by searching or looking at related discussions.
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Or rather counter proof.
They said x=0.999...
10x=9.999...
9x=9.999...-x
9x=9
x=1
but this is obviously wrong, you can't substract infinity from infinity unless you consider infinity a number and if so then you would get 8.99...1 and not 9. either way 0.999...= 1 is wrong. and is not different than saying (0.999...) +x=1.99...8 you can't add an infine amount of nines to an infinite amount of nines or subtract. if you could then you would consider infinity as a number and in that case the proof is also wrong
 
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This subject has been discussed many times on PhysicsForums. Please do a search of the forum or look at the "Similar Discussions" links below.

Yes, 0.999... = 1.

HallsofIvy said:
0.999... = 1 because, by the definition of "decimal place notation", 0.999... is the limit of the infinite series .9+ .09+ .009+ ... That's a geometric series and it's easy to show that the limit is 1.

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Math401 said:
but this is obviously wrong, you can't substract infinity from infinity
We are not subtracting infinity from infinity here. The subtraction is 9.999... - 0.999..., with each number having an infinite number of 9 digits to the right of the decimal point. The result is 9.000..., with an infinite number of 0 digits to the right of the decimal point.
As DrClaude said, this has been discussed many times here at PF.
 

What is the proof that 0.9999 does not equal 1?

The proof is a mathematical concept that shows that two values are not equal. In this case, the proof involves using the concept of limit to show that 0.9999 is infinitely close to 1 but never actually reaches it, therefore they are not equal.

How can 0.9999 be equal to 1 if they look the same?

It is important to understand that in mathematics, numbers can have multiple representations. The decimal representation of 0.9999 is just one way of expressing the number, but it does not accurately represent its value. Just like 1/3 can also be represented as 0.3333, but it is not equal to 0.3333.

Why is it important to prove that 0.9999 does not equal 1?

Proving that 0.9999 does not equal 1 is important because it helps us understand the concept of limit and the difference between actual value and representation. It also helps us avoid mathematical errors and misunderstandings in more complex equations.

Can't we just round 0.9999 to 1 and consider them equal?

No, rounding is a mathematical operation that is used to simplify numbers, but it does not change their actual value. In this case, rounding 0.9999 to 1 would mean ignoring the infinite number of decimal places that make it different from 1.

Is there any real-life application of this proof?

Yes, the concept of limit and the difference between actual value and representation are important in various fields such as engineering, physics, and computer science. For example, in computer programming, understanding this concept helps avoid rounding errors and ensures accurate calculations.

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