If x<1 what is the maximum value of x?

In summary, there is no maximum value for open intervals, and therefore the comparison of a maximum value and 1 cannot be done. The question regarding the least value of x when |x - 3| < 5 does not make sense. There is also no complex solution in this context, but it is possible to extend the real numbers to include a solution. However, this has not been studied extensively.
  • #1
I_am_learning
682
16
It appears the maximum value of x = 0.999... . But 0.999... = 1. But max value of x cannot be 1 based on the criteria. What is the maximum value of x then? It's puzzling me.
 
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  • #2
An open interval does not have a maximum number. We can say, though, that the supremum of the set [itex]\left\{x\in\Re|x<1\right\}[/itex] is 1.
 
  • #3
It doesn't have one. For any a < 1, there is some b such that a < b < 1.
 
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  • #4
Is it that it does not have maximum or is it that we cannot numerically represent it?
If we are asked to compare "the maximum value" and 1, we still say 1 is the larger number, won't we?
Thanks for the feedback.
 
  • #5
I_am_learning said:
Is it that it does not have maximum or is it that we cannot numerically represent it?
If we are asked to compare "the maximum value" and 1, we still say 1 is the larger number, won't we?
Thanks for the feedback.

There is no maximum value. That is, there is no real number ##a## such that ##a<1## and such that ##b\leq a## for all ##b<1##.
 
  • #6
micromass said:
There is no maximum value. That is, there is no real number ##a## such that ##a<1## and such that ##b\leq a## for all ##b<1##.

Thank you micromass.
I now get that there isn't any number which you can claim to be the maximum, because whatever number (<1) you claim, I always got another number (<1) but greater than yours. Since there isn't any number, the comparison cannot be done. But I have this gut feeling that even if we can't have any number as 'the maximum', we can still compare and say, 'the maximum' is less than 1.

If there is no maximum for such open intervals, is someone 'correct' to ask this question? If yes what is your answer?

Provided |x - 3| < 5
Quantity A: Least value of x
Quantity B: -2
Compare Quanity A and B and mention which of them is greater or if they are equal or if the relation cannot be determined.


P.S.: You mentioned 'real'. Is there a complex solution?
 
  • #7
I_am_learning said:
Thank you micromass.
I now get that there isn't any number which you can claim to be the maximum, because whatever number (<1) you claim, I always got another number (<1) but greater than yours. Since there isn't any number, the comparison cannot be done. But I have this gut feeling that even if we can't have any number as 'the maximum', we can still compare and say, 'the maximum' is less than 1.

If there is no maximum for such open intervals, is someone 'correct' to ask this question? If yes what is your answer?

Provided |x - 3| < 5
Quantity A: Least value of x
Quantity B: -2
Compare Quanity A and B and mention which of them is greater or if they are equal or if the relation cannot be determined.

No, the question doesn't make any sense, since there is no maximum value.
In your second example, the "Quantity A" does not exist, so we can't compare it with -2.

P.S.: You mentioned 'real'. Is there a complex solution?

No, there is no complex solution.
However, I would think that it is possible to extend the real numbers. So we add new elements to ##\mathbb{R}##. Then we get some kind of new number system in which a solution does exist. This is in principle possible. But I know of nobody that has studied this.
 

1. What is the maximum value of x if x is less than 1?

The maximum value of x would be 1, as it is the closest number to 1 that is still less than 1.

2. Is there a numerical value for x if x is less than 1?

Yes, there can be a numerical value for x if it is less than 1. It can be any decimal number between 0 and 1, but the maximum value would still be 1.

3. Can x be equal to 1 if x is less than 1?

No, if x is less than 1, it cannot be equal to 1. It can only approach 1 as a limit, but it will never reach 1.

4. How does the maximum value of x change if the inequality changes to x<2?

If the inequality changes to x<2, the maximum value of x would change to 2. This is because 2 is the closest number to 2 that is still less than 2.

5. Can x be a negative value if x is less than 1?

Yes, x can be a negative value if it is less than 1. It can be any negative decimal number between 0 and 1, but the maximum value would still be 1.

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