I understand (in R2) the integration of an integrable function over a finite interval conceptually as the process of adding up an infinite number of infinitesimally sized areas to arrive at a finite number which represents the area under the curve. dx represents the infinitesimal width of these areas and is multiplied by the height which is given by the value of the function at the point. So it is a product of the height given by the value of the function and the width which is given by the infintesimal dx. Summing all of these infinitesimal areas over infinity yields a finite area result
That’s only true in the limit. For any particular value of dx, the “width” has a real number value and there are (for the case of a definite integral on a finite ball) a finite number of “areas” to sum.
You have to go to the limiting case to get that finite value. And that limiting case involves taking a side trip outside the real numbers to get a result that is in the real numbers (for well behaved functions).
Just to add to 74Westy’s question.
If the integer one is thought to be infinitely precise…the existence of the concept of .000…0001 in the reals as a the fundamental representation of an infinitely precise interval would seem to be required to satisfy this assessment and balance the reals.
Just making sure I get thispost. Wouldn’t want it be wasted on any old rubbish.
No such thing. Beyond what I noted about integrals, such a number would break closure of the real numbers under addition and multiplication.
There’s no real bouncing. Metaphor is a poor substitute for the actual math, and it’s easy to stretch the metaphor too far.
Basically, the “dx” in an integral is a shorthand way to express the limit. There’s no real number “dx” the specific way you are thinking. But it is a very useful (and not entirely inaccurate) metaphor to think of it that way - as long as you don’t stretch the metaphor past the breaking point.
“side trip”
I can kind of see where you are going with this as in bouncing back and forth between that which is conceptual and that which is real…if you want to expand on it I would be curious to see a compact form of how you see it.
Depending on what definition you’re using, the ‘dx’ in the integral is meaningless notation, a measure, a differential form, etc. It is not a real number. It’s evocative of the geometric definition of a Riemann sum, but that doesn’t imply that it’s literally a multiplication of f by some infinitesimal dx, without any limit process involved.
Please cite an example of this…others in this thread have stated that the concept would lead to contradictions…I have yet to see a concrete example that could not be refuted conceptually. You guys are comfortable saying things like pi converges to pi etc. (meaningless tautologies). So it is clear that there is ambiguity in your thinking.
Let me just add.
I see the statement “pi converges to pi” as the most arrogant and presumptuous statement that can be made.
Convergence is directly tied to pattern recognition.
Pi contains no pattern as far as we know.
What pi converges to is unknown.
And in the real world 3.14 and 3.14159 are two ENTIRELY different things.
Take your “number” 0.0000…001 and let’s call it ε. What is ε/2? Is this bigger than, smaller than, or the same value as ε? Does the result exist?
How about ε*ε? Is the result bigger than, smaller than, or equal to ε? Does the result exist?
I can invoke the same tautologies these guys use
What is pi over 2, why jeepers, its pi over 2
What is epsilon over 2, why jeepers, its epsilon over 2.
There are many operations for which the exact result is unknown…why does that
bar .000…0001?
It’s not impossible to infer a well formed meaning of 0.000…0001 (eg. as the limit of 0.0001 x 10[SUP]n[/SUP] as n tends to infinity. (the pre-multiplication by 0.0001 is just a nicety to “preserve” the leading 0.000 before the ellipsis, we could equally well discard it)).
This number (so defined) is equal to 0. And incidentally is also equal to
0.000…00099999999999999999999999999999999999999999999999999999999999999999999999999999…
among other representations.
What needs to happen is a proof has to be submitted showing how the concept yield to a contradiction within the system…RIGOROUSLY
This is complete and utter nonsense. A single number doesn’t “converge” to anything; convergence has absolutely nothing to do with pattern recognition; the BBP formula gives a simple algorithm for computing the base-16 digits of \pi; and \pi is exceedingly well-understood. At the very least, you’re confusing real numbers with computable numbers, but that’s just one example.
“It’s not impossible to infer a well formed meaning of 0.000…0001 (eg. as the limit of 0.0001 x 10n as n tends to infinity.”
(I think you mean possible here as opposed to “impossible” be careful, freudian slips apply to math too)
Can you do this for pi?
nm
OK, lets say I don’t need to know exactly. Answer these questions then:
ε/2 > ε ?
ε/2 < ε ?
εε > ε ?
εε < ε ?
If ε is an element of the reals you should be able to answer this. For example, I know that
π/2 < π
π*π > π
or do you argue that we don’t know that the above two are true?
Not true. Go away and read some mathswill you, it’s tedious to have to even field such nonsense.
In order to perform infinite precision calculations with pi I would need a decimal representation involving an ellipsis such as .999… has… What is that representation for pi?
that spigot formula has already been discussed, it gives one digit,
since you seem to know it show me the decimal representation…
With the ellipsis