Does 0.999... Exist? Evidence & Argument

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In summary, the conversation discusses the existence of the number 0.999... and whether it can be considered to have an independent existence on the real-number line. The argument presented is that since 0.999... would have to be located just before 1 on the line, but no such point exists, it must be relegated to the realm of metaphysics. However, it is argued that since 0.999... is equal to 1, and 1 exists, then 0.999... must also exist.
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Opus131
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We know that analytically 0.999... = 1.0. But does 0.999... have any sort of "independent" existence (on the real-number line or otherwise)? Here's my argument against that:

Consider the semi-open line segment [0,1). This set of points has no greatest member. Let's say it did have a greatest member. Then surely that would equal 0.999... Put another way, if 0.999... doesn't equal 1, then we have to conclude that 0.999... only equals 0.999..., which is to say that it's in a class by itself. But where on the real-number line is it located? Intuitively, it would have to be the point "just before" unity. But no such point exists because no two points are "next to" each other on the line. Therefore the idea that 0.999... exists is vague and has to be relegated to the realm of metaphysics.

My argument is intuitive, not analytical. But is the conclusion correct?
 
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Welcome to the PF.

Please see the FAQ thread about this in the General Math forum:

https://www.physicsforums.com/showthread.php?t=507002

.
 
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  • #3
I'm not sure what you mean by "analytically 0.999... = 1.0". It is a fact that 0.999... is equal to 1 and since 1 exists, yes, 0.999... exists.
 

Related to Does 0.999... Exist? Evidence & Argument

1. What is the definition of "0.999..." in mathematics?

In mathematics, "0.999..." is known as a repeating decimal, which means that the digit 9 is repeated infinitely. It can also be written as 0.9̅ or 0.(9).

2. Is there any concrete evidence that supports the existence of "0.999..."?

Yes, there are several mathematical proofs that demonstrate the existence of "0.999..." as a real number. One of the most well-known proofs is the geometric series, which shows that the sum of an infinite sequence of numbers can converge to a finite value, in this case, 0.999...

3. How can it be possible for "0.999..." to equal 1?

Although it may seem counterintuitive, the concept of limits in calculus can help explain this. As the number of 9s in "0.999..." increases infinitely, the value gets infinitely closer to 1, but it will never actually reach 1, as it is an infinite decimal. Therefore, we can say that 0.999... is the limit of the sequence as the number of 9s approaches infinity, and thus, it is equal to 1.

4. Are there any real-world examples that demonstrate the existence of "0.999..."?

Yes, there are many real-world situations where "0.999..." can be observed. For example, if you divide 1 by 3 using a calculator, the result will be 0.333333... with the digit 3 repeating infinitely. Similarly, if you divide 2 by 3, the result will be 0.666666... and so on. These numbers, when added together, will equal 0.999...

5. Can the existence of "0.999..." be proven using different mathematical methods?

Yes, there are multiple ways to prove the existence of "0.999...". Other than the geometric series mentioned earlier, one can also use algebraic proofs, such as multiplying both sides of the equation 0.999... = x by 10, which gives 9.999... = 10x, and then solving for x to get x = 1. This shows that 0.999... is equal to 1.

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