what reuben said
Quote:
Originally Posted by popsicle_3000
I haven't seen him actually attempt to prove his point/theory. stargroup that is
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What point/theory would you like me to prove? If you're talking about analytic continuation I'm not an expert. Google it yourself to find all the information you could want about it.
If you're talking about the notion that different objects in different domains can have different properties and therefore many different statements can be justified/proven under certain contexts/conditions/systems, here's an example:
Polynomials of degree n with real (or even complex) coefficients have exactly n compelx roots, but not necessarily n real roots. Certain applications of polynomials, such as most geometry problems, disregard the complex solutions because many kinds of complex lengths/measurements cannot be physically realized (you can't draw a line i units long). However, the result is fundamentally important to understanding polynomials and is used extensively in other fields of math. Here's a directly related result:
Exponentiation is an example of a function that has different behaviors in different domains. The exponent of e^n tells us to multiply e by itself n times. This makes sense for integers, and if we consider the inverse of this function, also makes sense for rational numbers. However, we can easily plot this relation and make a smooth line to extend it to real numbers. Once we find a real definition for the exponential function, we can observe its properties and extend it to complex numbers (by Taylor expansion for instance) and get results such as e^x = cos x + i sin x.
However, this is where it gets a bit interesting. If x is a rational number, e^x has d solutions, where d is the denominator of x in simplest form. This result is directly related to the extension of polynomials to complex numbers we mentioned above. If x is irrational, then it "probably" (I'll explain in a second) only has one solution in the complex plane. Once we extended our definition, the behavior of rational numbers in this function has changed! Even more bizarre is when you take the relation e^(a*ln(x)) = x^a. Here, we have a case where even though the exponent could be irrational, and yet we can change the base to make the exponent rational again!
And this is only scratching the surface of complex analysis, and complex numbers are just one type of mathematical object. Other mathematical objects include matrices, vectors, tensors, sets, groups, operators, rings, partitions, spaces, manifolds, functions, geometric figures, etc. Even today, we're still investigating all of the properties of these objects and new ones that we've conceptualized.
I know I'm not directly replying to anyone's specific ideas, so I wrote this more for myself as practice. I figured I might as well post it if anyone is interested in reading it.
EDIT: I just realized that I pretty much just did a layman's reiteration of one of reuben's earlier posts. I was wondering why something was bugging me when I posted this LOL