Definitions are not supposed be proved. It makes no sense to "prove" a definition. I think what you're looking for is a proof of the equivalence of two definitions of e.
Exactly! Also how does an exponent x of e transfer to the specific position in the definitions. How these connections where discovered?
What is the proof for the two common definitions of e? The continuous compounding and the sum over inverse factorials?
Also what is the proof of the exponential function e^x pertaining the above?
I tried by myself but failed.
Of course I would like elementary (no complex analysis) proofs.
Thank you very much!
those would be the numbers.
to both of you,
i noticed that that number is always either (b-a) div 2,(b-a) div 2 +1 or (b-a) div 2 -1.
That seems to be true. I didn't find any counter example yet after 10. Those repeats offset each other. If I didn't have exam tomorrow I would work with it. Modular arithmetic seems to do the trick and hopefully we can prove also why the formula works to be sure.
The correct phrasing is more transcendental numbers than natural numbers. Real numbers are not "real" but the name historically came about to make the distinction between them and imaginary numbers.
Conceptually what Cantor spectacularly proved was that infinity extends not just outwards from the number line, towards +∞ and -∞, but also "inwards". The number line is more like an infinite fractal-like 1-dimensional grid.
This is also equivalent to points not having size. That's why no matter the line segment or area, the cardinality of the points is the same, C, the continuum.
However, in practice we have never used but computable rational numbers. Only mathematicians will leave a root of 2 at the end of an equation. An engineer will carry out that root of 2 from equation to equation but at the very end he will round of to any degree of accuracy he needs. Same goes for π ≈ 355/113.
Back to Cantor, his discovery makes explicit something many would have thought but not be able to prove. That there is no next real number. From this also follows, 0.999... = 1.
Possible? Indeed. Practical to find it anytime soon? Near impossible.
The statistics suggest you have to search for googol^googol digits of pi to find such "ordered" digits; be it the millions of repeats of the same digit, or a progression.
But, if we ever harness the power of quantum mainframe supercomputers, I can easily see newspaper news about those special sequences of digits of pi. When, and if, this happens it will make Chudnovski brothers' efforts seem like play in the sand!
Nice food for thought: "We are in Digits of Pi and Live Forever": http://sprott.physics.wisc.edu/pickover/pimatrix.html
As you count they generally alternate odd, even, odd, even etc. But for every 10 numbers (starting with 1) they repeat. E.g. 9 (odd) , 1+0 (odd), and 3+9 (even) , 4+0 (even)
Generally it looks like for every 10^(n) for odd n there is a repeat. E.g. 9+9+9 (odd) 1+0+0+0 (odd). But 9+9 (even) and 1+0+0 (odd).
It remains for it to be proven, then apply iteration and modular arithmetic and get the algorithm.
1. Yes, it won't happen any time soon, but knowing the facts helps things toward that direction. 12 is perfect in that it's not to far from 10 but has more and "better" factors.
2. It's arbitrary from the point of view of numbers. Aliens with 6 fingers quite probably will have a base-6 number system. But the math remain the same.
3. Just the semantics will change not the actual math, but even then they will look prettier. Of course the drawback is you have to revert all past papers written in the decimal system.
4. Compact length. Numbers in the binary are lengthy while for larger bases you progressively gain more compact displays. This argument is thus in favor of the duodecimal compared to decimal.
You have a point in 1. and 3. but they are historical reasons and not mathematical. It's like the transition (but not as important) to having zero, if people weren't willing to change math would suffer.
True reasons about the drawbacks of the duodecimal system are that it doesn't divide perfectly 5, though dividing 3 is a greater advantage, and you have to memorize more multiplications. However, the paper I cited describes why in the end it is better that the decimal. It is not by chance that Leibniz and Pascal supported it.
you're welcome.i find those interesting,and wonder why mathematicians don't care about them at all.
Apart from topology (why?), maybe it's that these numbers are dependent on the decimal base system which is arbitrary.
Maybe only the binary system might offer some insight whenever the number base is directly involved due to it being the minimal base available.
I'm an advocate of the duodecimal (12) system. We lose perfect division with 5 but gain that with 3, 4 and 6. Especially 3, it is a more important prime factor than 5 since it comes earlier. Google The Case Against Decimalisation pdf.
I read the article. So the definition changes for that domain with gamma functions and what not.
Thanks for clearing up the definition does not hold for Re(s) < 1. In other forums people were just stating it is an unexpected result and we have to live with it and I was thinking WHAT??!, the equality is altogether wrong, meaning the sum over ones.
My question is how analytic continuation does not address a new function all together? Is it because the transformation is unique; that is the key?
Alright, ζ(0) = 1 + 1 + 1 + ... = -1/2. Miracles do happen!
I want to comprehend what this means but it seems to me this just supersedes quantitative math.
Is there supposed to be a qualitative limited meaning to it?
Because if you forget the outcome is from zeta and just write 1 + 1 + 1 + ... = -1/2 this clearly is nonsense alone like that, as Hardy and Littlewood thought initially of it mailed by Ramanujan.
Math gurus enlighten me.
Hello. I'm glad to now be a member of this cool forum. I love math and physics.
My contribution to this topic.
Make it as simple as possible but not simpler - Einstein
-Extra elementary definition of primes:
Prime numbers are those positive integers that cannot be written as a repeated summation of any smaller such number excluding 1 because it already creates all positive integers.
E.g. 10 can be 2+2+2+2+2 or 5+5. On the other hand prime number 7 cannot be written like that.
Ancient greeks visualized numbers as pebbles (before modern tradition with the number line). Prime number of pebbles cannot be organized to form a rectangle (plus its special case the square). Greeks for this reason called them linear numbers.
-Prime numbers are "attracted" next to numbers having as factors both 2 and 3, thus multiples of 6.
This is in other words the fact that prime numbers greater than 3, thus almost all of them, are of the form 6n + 1 or 6n -1, for n=positive integer.
In a table of positive integers arranged in 6 columns, prime numbers greater than 3 all fall in the 1st and 5th columns.
A definite non primality test for a number X is to divide X+1 and X-1 with 6. If in both cases there is a remainder, X is not prime.
-Explicit formula for e in terms of prime counting function π(χ).
This is a beautiful derivation that for some reason is not popular and not found in Wiki, Wolfram and other sites.
Starting with the Prime number Theorem and solving for e the following formula is derived:
Where the power of n can be interpreted as the prime number density. Indeed. n grows to infinity, but in the meantime the prime number density gets very very close to 0 limiting the expression to the constant of growth e.
The beauty of this formula is offset by the painfully slow convergence rate. Even for n=1E21 (a billion trillion) there is still a mistake of the order of 2%.
It's more like a beautiful connection than a workhorce of an equation.