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olcott <polcott333@gmail.com> writes:Whether it is a real number or not is moot to me.On 4/1/2024 6:11 PM, Keith Thompson wrote:"IDK, probably not."olcott <polcott333@gmail.com> writes:>
[...]Since PI is represented by a single geometric point on the number line[...]
then 0.999... would be correctly represented by the geometric point
immediately to the left of 1.0 on the number line or the RHS of this
interval [0,0, 1.0). If there is no Real number at that point then
there is no Real number that exactly represents 0.999...
In the following I'm talking about real numbers, and only real
numbers -- not hyperreals, or surreals, or any other extension to the
real numbers.
You assert that there is a geometric point immediately to the left
of
1.0 on the number line. (I disagree, but let's go with it for now.)
Am I correct in assuming that this means that that point corresponds
to
a real number that is distinct from, and less than, 1.0?
>
IDK, probably not. I am saying that 0.999... exactly equals this number.
Did you even consider taking some time to *think* about this?
--Do you believe that to be responsive to my question? It isn't. At all.More generally, does each real number correspond to a point on the>
number line, and does each point on the number line correspond to a real
number? (The real numbers can be formally defined without reference to
geometry, but let's go with your geometric model for now.)
>
The line segment [0.0, 1.0] is exactly one geometric point longer than
[0.0, 1.10), having all points in common besides the last point.
You have answered none of my questions. I'm mildly disappointed, butIf so, let's call that real number (immediately to the left of 1.0) x.
Consider ((x + 1.0)/2.0). Let's call that number y. (The intent is
to
construct a real number that is exactly halfway between x and 1.0.)
Is y a real number? (If not, the real numbers are, unexpectedly,
not
closed under common arithmetic operations.)
Is y less than, equal to, or greater than x?
Is y less than, equal to, or greater than 1.0?
Again, I am talking *only* about real numbers.
Given your past history, I do not expect straight answers to these
questions, but I'm prepared to be pleasantly surprised.
not really surprised, that you didn't even try to answer any of them.
My conclusion remains the same: you don't know what you're talking
about, and your statements about real numbers and limits are wrong,
incoherent, or both.
I encourage others to consider this when considering trying to explain
things to olcott.
"I can explain it to you, but I can't comprehend it for you."
-- Edward I. Koch
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