I pondered about this some more, and Banach-Tarski might seem counter-intuitive, but isn't that true for more things that have to do with infinity? For example, the integers and the natural numbers having the same cardinality is quite strange, since there seem to be more (almost twice as many) integers.
Let's discuss Mathematics
- Thread starter ParadigmShifter
- Start date
ParadigmShifter
Random Nonsense Generator
more than twice as many (I don't consider 0 to be a natural number).
What's weirder is the number of rational numbers (fractions) being the same as the number of natural numbers (Cantor's diagonalisation argument).
What's weirder is the number of rational numbers (fractions) being the same as the number of natural numbers (Cantor's diagonalisation argument).
Infinity isn't exactly a number. (Although we physicists like to treat it that way)
ParadigmShifter
Random Nonsense Generator
I know 
One thing infinity isn't is a real (or complex) number.
One thing infinity isn't is a real (or complex) number.
After many months of mental effort, I think I finally have a fleeting understanding of what imaginary numbers are... (I only had formal math classes through trigonometry, and I wasn't paying all that much attention, if you can believe it).
What are the transcendentals, and are they related to the imaginaries?
What are the transcendentals, and are they related to the imaginaries?
ParadigmShifter
Random Nonsense Generator
Transcendental numbers are not roots of polynomial equations (i.e. they are not algebraic).
e and pi are the best known examples. Those are roots of infinite power series however.
e and pi are the best known examples. Those are roots of infinite power series however.
Does that mean that all transcendentals are roots of infinite power series?
...and can I just interject that it blows my mind that pi and e are so readily related to eachother
reminds me of a favorite comic that shows a decimal expansion of pi that reads: 3.141592653589793helpimtrappedinauniversefactory7108914
...and can I just interject that it blows my mind that pi and e are so readily related to eachother
reminds me of a favorite comic that shows a decimal expansion of pi that reads: 3.141592653589793helpimtrappedinauniversefactory7108914
ParadigmShifter
Random Nonsense Generator
I think so, all transcendentals are roots of power series.
EDIT: probably not, Laurent series (power series with negative terms) are more general.
EDIT: probably not, Laurent series (power series with negative terms) are more general.
Olleus
Deity
Whats a laurent series?
ParadigmShifter
Random Nonsense Generator
It's like a Taylor series but instead of summing from 0 to infinity it sums from -infinity to +infinity.
http://en.wikipedia.org/wiki/Laurent_series
http://en.wikipedia.org/wiki/Laurent_series
Olleus
Deity
But whats the point?
If you have a negative power, can you not just expand that as an infinite series of purely positive powers?
So does every Laurent series have an equivalent taylor series?
If you have a negative power, can you not just expand that as an infinite series of purely positive powers?
So does every Laurent series have an equivalent taylor series?
ParadigmShifter
Random Nonsense Generator
First sentence of the wiki:
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.
I skived alot of lectures in my 3rd year but I do believe Laurent series are used to classify functions with discontinuities.
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.
I skived alot of lectures in my 3rd year but I do believe Laurent series are used to classify functions with discontinuities.
Olleus
Deity
Ahh ok, that would make sense (I think). Sorry, I looked at the maths and skipped all the wordy bits of the wiki.
Not quite, Laurent series are mostly used in complex analysis for functions with actual singularities, i.a. a point p with abs(f) not being bounded in any disc around p. That's somewhat more specific.
Although, I suppose if you restrict the function to the real axis it might reduce to a discontunity in some cases.
Mhmm, much too complicated. After all, all transcendentals are roots of polynomials of degree one with non-rational coefficients. The coefficients of a power series are not restricted to rational numbers.
It's also not hard to construct a power series which is zero for a real number and rational coefficients.
ParadigmShifter
Random Nonsense Generator
Yeah, I meant singularities not discontinuities.
Transcendentals are not roots of polynomials with rational coefficients, I believe.
Transcendentals are not roots of polynomials with rational coefficients, I believe.
Yes, that's the definition of transcendental. But a power-series is not a polynomial. Or did you refer to another part of my post?
ParadigmShifter
Random Nonsense Generator
Yeah, just clarifying this:
After all, all transcendentals are roots of polynomials of degree one with non-rational coefficients
After all, all transcendentals are roots of polynomials of degree one with non-rational coefficients
True enough, my point was that a polynomial of degree one is always a (short) power-series regardless of whether the coefficients are rational or not.
But as I said, you can find a power-series with just rational coefficients which is zero for a given real number as well.
Similar threads
