.999 = 1, in descrete topology?

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In summary, the conversation discusses the concept of infinitesimals in real numbers and how it relates to the number 0.999... and the number 1. It also explores the difference between hyperfinite and infinite in the context of non-standard models, and how picking a hyperinteger can result in a number that is infinitesimally close to 1. The relationship between infinitely close and equal is also questioned.
  • #1
TylerH
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When one is considering a the real numbers with the extension of an infinitesimal, implying that it is possible for a number to be the highest number lower than a number, would .999... then be the highest number less than 1? (Similar to *R, but I'm generalizing my hypothesis to include any extension to the reals that fits the condition of having an infinitesimal.)
 
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  • #2
TylerH said:
When one is considering a the real numbers with the extension of an infinitesimal, implying that it is possible for a number to be the highest number lower than a number, would .999... then be the highest number less than 1? (Similar to *R, but I'm generalizing my hypothesis to include any extension to the reals that fits the condition of having an infinitesimal.)

The real number 0.999... is the real number one. If you consider other number systems, the real number 0.999... is still the real number one.


As an aside, it is impossible, in an ordered field, for there to be a largest number smaller than 1. Between any two numbers, there is always another number.


In the hyperreal numbers, the hyperreal number 0.999... (this time the notation is referring to a hyperdecimal number, rather than a decimal number) is the hyperreal number 1. What the hyperreal numbers do have is the ability to consider a number 0.999...9 with a hyperfinite (but transfinite) number of 9's, rather than having a 9 in every hyperdecimal position as in the numeral 0.999... Such a number would be infinitessimally close to 1.
 
  • #3
What is the difference between hyperfinite and infinite?
 
  • #4
TylerH said:
What is the difference between hyperfinite and infinite?
They are more or less independent concepts.

Hyperfinite plays the same role in the non-standard model as finite does in the standard model. e.g. every finite set of real numbers has a largest element, and so does every hyperfinite (internal) set of hyperreal numbers. In the case of counting (hyper)decimal places, to say that 0.99...9 has hyperfinitely many 9's just says there is a hyperinteger number of 9's, and the rest of the digits are zero.

Infinite, here, has to do with comparing standard things to non-standard things -- in this case, a positive infinite hyperinteger is simply hyperinteger that isn't also a standard integer. (and, thus, is larger than all standard integers)



In the standard model, you can pick any positive integer and write down a numeral with that many 9's after the decimal place, and the rest of the digits zero. This number will be less than 1.

Transferring this to the non-standard model means you can pick any positive hyperinteger and write down a hypernumeral with that many 9's after the decimal place, and the rest of the digits zero. This number will also be less than 1.

If your hyperinteger is infinite, then the number so written will be infintiessimally close to 1.
 
  • #5
If your hyperinteger is infinite, then the number so written will be infintiessimally close to 1.
Is this supposed to mean they are equal? If so, why wouldn't infinitely close imply not equal?
 

FAQ: .999 = 1, in descrete topology?

1. What is the meaning of ".999 = 1" in discrete topology?

In discrete topology, ".999 = 1" means that in a discrete space, where every subset is open, the open set containing 1 and the open set containing 0.999 are the same, making them indistinguishable from each other.

2. How is ".999 = 1" proven in discrete topology?

To prove ".999 = 1" in discrete topology, we can use the definition of discrete topology which states that every subset is open. In this case, the open set containing 1 and the open set containing 0.999 are both equal to the set {1}, showing that they are the same and ".999 = 1".

3. Is ".999 = 1" true in all topologies?

No, ".999 = 1" is not true in all topologies. It is only true in discrete topology, where every subset is open. In other topologies, such as the standard topology, ".999" and "1" are considered as distinct points and not equal to each other.

4. What implications does ".999 = 1" have in discrete topology?

The implication of ".999 = 1" in discrete topology is that it shows the equality of two points in a discrete space, even though they may appear to be different values. This highlights the importance of understanding the underlying topology of a space in mathematical analysis.

5. Why is it important to consider ".999 = 1" in discrete topology?

It is important to consider ".999 = 1" in discrete topology because it challenges our intuition and common understanding of numbers. In discrete topology, ".999" and "1" are considered as the same point, which may seem counterintuitive at first. However, understanding this concept is crucial in correctly analyzing and solving problems in discrete topology.

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