1 is by definition 0.999999999 9?

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In summary, the conversation discusses the equality between .999... and 1. It is explained that .999... is defined as the infinite sum of 9/10 + 9/100 + 9/1000 + ..., which is proven to be equal to 1 in first-year calculus. It is also stated that there is no positive difference between the two numbers, and thus they must represent the same number. Possible objections to this reasoning are addressed, and it is concluded that there are numerous discussions about this topic available online.
  • #1
adarpodracir
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Hi there,

I have a question regarding this statement:

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My question is whether we can say so...

Thank you very much!
 
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  • #2
OP, the answer is that .999... = 1. It's an equality. They're two expressions that represent the same number.

The reason this is so is that .999... is defined as the infinite sum

9/10 + 9/100 + 9/1000 + ...

This is a geometric series whose sum is 1. This is proven in first-year calculus.

Another way to see it is that there's no distance between the number denoted by .999... and the number denoted by 1. That is, suppose you say, well, .999... is 1/zillion away from1. But I'll just point out that if you take enough 9's, you'll eventually get WITHIN 1/zillion of 1.

So if there's no conceivable positive difference between .999... and 1, then they must represent the same number.

Possible conceptual objections to this reasoning are things like:

* "But how can you have two different expressions for the same number?" Easy. 4 and 2 + 2 are two different expressions for the same number. It happens all the time.

* There must be an "infinitesimal" difference between 1 and .999..." In the standard real number system, there are no infinitesimals. A distance is either zero or positive. Since there's no positive distance between .999... and 1, the distance between them is zero and they're the same number.

Hope this helps. There are discussions of this topic all over the net.
 
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  • #3
There is already a thread about this. Please visit the Frequently Asked Questions subforum.
 
  • #4
Do you realize the question you ask in your post and the question you ask in the title are quite different?
 
  • #5
Please read this: https://www.physicsforums.com/showthread.php?t=507001 [Broken]
 
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  • #6
Many thanks to all of you for reply. Everything is clear now.
 

1. What does "1 is by definition 0.999999999 9" mean?

This statement means that the number 1 and the repeating decimal 0.999999999 are equivalent and represent the same value.

2. How can 1 be equal to 0.999999999?

In mathematics, numbers can be written in different ways, but they can still represent the same value. In this case, 1 and 0.999999999 are two different representations of the same number.

3. Is 1 exactly equal to 0.999999999?

Yes, mathematically speaking, 1 and 0.999999999 are exactly equal. However, when we use decimal notation, there may be a very small difference due to the limitations of decimal representation.

4. Why is 1 equal to 0.999999999?

This is a fundamental concept in mathematics that has been proven and accepted by mathematicians. It can be explained through various mathematical proofs, such as the geometric series or the limit of a sequence.

5. Can you provide an example to demonstrate that 1 is equal to 0.999999999?

Sure, let's take the fraction 9/9. When we divide 9 by 9, we get the decimal 0.999999999. However, 9 divided by 9 is also equal to 1. Therefore, 0.999999999 and 1 are equivalent.

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