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

[I_am_learning]

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.

 

[hilbert2]

An open interval does not have a maximum number. We can say, though, that the supremum of the set \left\{x\in\Re|x&lt;1\right\} is 1.

 

[Number Nine]

It doesn't have one. For any a < 1, there is some b such that a < b < 1.

 

[I_am_learning]

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.

 

[micromass]

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 $b<1$.

 

[I_am_learning]

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?

 

[micromass]

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.