Let's discuss Mathematics
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ParadigmShifter
Random Nonsense Generator
In modulo 4 arithmetic 2+2 = 0.
In the ring with only 1 element (the trivial ring), x+x = x and x*x = x so 1 = 0
In the ring with only 1 element (the trivial ring), x+x = x and x*x = x so 1 = 0
ParadigmShifter
Random Nonsense Generator
Not in my book. Some people say it is though.
ParadigmShifter
Random Nonsense Generator
Probably because we start at 1 when counting.
Unless you are a programmer of course
0 isn't nice to divide by either.
Unless you are a programmer of course
0 isn't nice to divide by either.
ParadigmShifter
Random Nonsense Generator
Sure they are, 5/3 = 1 remainder 2, lots of people forget that although that's how we generally learn about division. It's more general than fractions, e.g. polynomials can be divided the same way.
I don't think a 0-dimensional vector space is very useful
In Topology, any countable set has dimension zero however.
What is the degree of the zero polynomial?
I don't think a 0-dimensional vector space is very useful
In Topology, any countable set has dimension zero however.
What is the degree of the zero polynomial?
Spoiler :
ParadigmShifter
Random Nonsense Generator
True. It's very important for the dimension theorem in vector spaces.
Not if the topological space is weird enough.
ParadigmShifter
Random Nonsense Generator
Have you got an example? I'm pretty sure that the rational numbers and the Cantor set are considered dimension zero.
Are you talking about Hausdorff dimension by any chance?
Are you talking about Hausdorff dimension by any chance?
Hmm, it's just a thought that I just had. You are right that both rational numbers and the Cantor Set do have dimension zero, but that's because they are in R which has a pretty nice topology.
For example, consider the space S:={1,2,3} and define a subset of S as open if it contains the 1 or is empty. Yields a valid topology on S if I'm not mistaken. Now consider the covering of S consisting of {1,2} and {1,3}. It's not refinable and the 1 is in both sets, so S has a topological dimension of 1.
Note that the space isn't very nice, it's not even Hausdorff. I'm pretty sure that your statement holds on nicer spaces (uncertain which exact attributes they needs to have) and that the topological dimension doesn't even make much sense on the sort of space I have given. Still, it should be formally valid. It's some times since I heard topology, so there might be mistakes somewhere.
ParadigmShifter
Random Nonsense Generator
It's years since I did Topology and it was really hard to understand at the time
Here's a link I found anyway...
http://hypertextbook.com/chaos/32.shtml
Here's a link I found anyway...
http://hypertextbook.com/chaos/32.shtml
The answers on previous page didn't perhaps tell very well what's the point of trancendentals, focusing more on the power series business...
Consider numbers in general. Children first learn that numbers are 1,2,3,... Then they find out about 0 and negative numbers, -1,-2,-3,... and then the rational numbers, and finally about real numbers like squareroot 2, pi and e and so on, even though they're not very aware what real number really means.
Now as square root 2 can easily be proven to be irrational and is the first number known to be irrational, question arises: are all irrationals squareroots? Or more generally are all irrationals n-roots? The answer to both these questions is no, and can most easily be seen when considering numbers like sqrt 2+ sqrt 3 (exercise: prove that sqrt n is rational if and only if n is a square of a natural number).
All of these counter examples are nonweird, because they are roots of polynomials, sqrt 2 +sqrt 3 for example is root of (x^2 -5)^2 -24. While they are perfectly good answers, they aren't entirely satisfying, because we wanted to ask if there's really weird real numbers. So it's natural to reformulate the question: are there any real numbers which aren't roots of polynomials (with rational coefficients)? Numbers which are such roots are called algebraic, and the rest are called trancendental. For all I know, no number was known to be trancendental before late 19th century when pi and e were proven to be ones.
Now back to the numbers in general, the concept of number has been upgraded by adding something to the previous definition: First you had just the natural numbers, then you add 0 and the negative numbers and get integers, and then you add the fractions and get the rationals. Now tempting expansion would be to add the roots of polynomials, but the existence of trancendentals shows that it's insufficient.
Trancendetality of pi is also important, because as a corollary a circle can't be squared (with compass and ruler). The problem is pretty much as old as the western civilization.
I wouldn't be too sure about it, although it isn't used explicitly. Think this standard undergraduate exercise as an example: You have to prove that x in R^n is at the boundary of a nonempty set A if and only if there's a sequence of A's points which converges to x.
The proof of course is that you take x_k from the intersection of A and B(x;1/k), where k=1,2,3,..., and the sequence (x_k) converges to x. But: Do you use the axiom of choice when picking that x_k? In all honesty, I don't know if it's used. However this example, I think, is enough to show that the matter isn't as easy and clear cut as it is often seen to be.
Hmm, is that actually true?
Let F(z) be the function that gives the complex conjugate of z, zo f(x+iy)=x-iy
F(z)=U(z)+iV(z) with U(z)=Re(z)=x and V(z)=-Im(z)=-y
dU/dx=1
dV/dy=-1
?
Or am I not properly understanding you?
Let F(z) be the function that gives the complex conjugate of z, zo f(x+iy)=x-iy
F(z)=U(z)+iV(z) with U(z)=Re(z)=x and V(z)=-Im(z)=-y
dU/dx=1
dV/dy=-1
?
Or am I not properly understanding you?
ParadigmShifter
Random Nonsense Generator
Conjugation isn't a holomorphic function.
Found this, I'll read it after the Everton match
http://en.wikipedia.org/wiki/Holomorphic_function#Examples
Found this, I'll read it after the Everton match
http://en.wikipedia.org/wiki/Holomorphic_function#Examples
ParadigmShifter
Random Nonsense Generator
Right, Chelsea 0 - 0 Everton.
So a natural number of goals was not scored
Back to complex derivatives:
* Unlike real derivatives if a function can be complex differentiated once it can be differentiated infinitely many times, and is called holomorphic
* The Cauchy-Riemann equations give the necessary and sufficient criteria for a function to be complex differentiable
* Complex conjugation mustn't have a derivative over C but I haven't found a proof of this. I might do some back of the envelope calculations.
However, complex differentiation is more complicated than real differentiation since the limit must be the same in all directions (unlike the real case where it is only approaches from the left and the right).
So a natural number of goals was not scored
Back to complex derivatives:
* Unlike real derivatives if a function can be complex differentiated once it can be differentiated infinitely many times, and is called holomorphic
* The Cauchy-Riemann equations give the necessary and sufficient criteria for a function to be complex differentiable
* Complex conjugation mustn't have a derivative over C but I haven't found a proof of this. I might do some back of the envelope calculations.
However, complex differentiation is more complicated than real differentiation since the limit must be the same in all directions (unlike the real case where it is only approaches from the left and the right).
ParadigmShifter
Random Nonsense Generator
Yeah that's true, those are the Cauchy-Riemann equations you posted. If it's in a physics textbook it probably uses "analytic" instead of holomorphic.
I'm saying real functions that can't be differentiated infinitely many times aren't complex differentiable when embedded in C.
I'm looking for an example
I'm saying real functions that can't be differentiated infinitely many times aren't complex differentiable when embedded in C.
I'm looking for an example
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