Comments by "MC116" (@angelmendez-rivera351) on "Welch Labs"
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Actually, yet again we stand corrected. Not even division by zero is a problem difficult enough to remain unsolved. As of 1998, mathematicians who study group and field theory have finally managed to build a form of algebra which modifies the distributive rules of commutative and associative rings such that division by zero is defined. This is what they have called wheel algebra theory, and it is merely an extension of the current number system that exists, not a brand new one from scratch. There are at least two possible ways in which this functions properly: we either assign a value to all the indeterminate forms, and then let ξ = 1/0, where M < ξ < L, M representing a number of infinite magnitude within the positives and L representing a number of infinite magnitude within the negatives. We get rid of multiplicative associativity in the general case and allow an additive nullifier ξ to exist uniting the positives with the integers (turning the real number line into a circle and the complex plane into a torus).
The other form we can do this is to define a divisor operator / as unary rather than binary, we modify the basic algebraic distribution laws so that they are consistent both in the general case and in the special cases (which would simplify to the ones we currently have in non-zero division fields) and it would leave us with a number system known as the extended/completed Riemann sphere. /0 = Infinity ("Infinity" as in the symbol used to represent Infinity, not as in the actual concept of Infinity. The two are very distinct. Our concept of infinity still isn't a number, but /0 produces a number of infinite magnitude which is unsigned, meaning it is neither positive nor negative. Thus the real number line is still folded and united in a circle. The complex plane is united in a sphere since /0 is both real and imaginary. Then, in the very center of the sphere, lying outside the plane itself, exists the element W = 0/0. So division by zero is a matter of modifying distributive laws involving the division operator, leaving division as unary, and introducing two new numbers to the system, that way it preserves every other law of algebra.
Both systems are consistent in themselves and the axioms are regular and stable.
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Raphael Schmidpeter Sorry, but no, that is wrong. Have you ever read the basic definition of exponentiation? It is an iteration of the multiplication hyper-operator, and by definition, iterations are only defined for natural numbers. The exponentiation function was defined not in terms of the complex numbers, but the natural. Roots and logarithms are mere inverses of it. Exponentiation does tie back to arithmetic, and exponentiation is not closer under the reals for a completely distinct reason than you mention. Otherwise, you are claiming that the absolute value function is not defined in the reals simply because the functional definition that accounts for complex number requires the structure of the complex plane to begin with. But that is quite ridiculous since the absolute value function which gives an output for complex inputs is nothing but an extension of the pre-existing defined function, in the same way that complex exponentiation and complex logarithmic operations are nothing more than extensions of the pre-existing definitions for those functions on the reals.
And no, it is not unfair to say that the complex numbers are not closed under log simply because log is undefined, because that would imply that one could not claim the reals are not closed under even fractional exponents (I.e., square roots) to begin with. That is how the concept of closure works in mathematics. Log 0 could easily be defined as a new number and one could create axioms such that this number preserves the common properties of algebra and arithmetic we all know. But that does not mean the complex numbers are closed under logarithms: whatever log 0 is, it isn't a complex finite number, so the complex finite number system is not closed under log. That's the end of it.
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1. The exponential function can be defined for the reals without necessity of analysis. The limit definition of e^x can be expanded by the binomial theorem into its infinite series without any form of analytic continuation, and the approach suffices to define the function for the real numbers without the existence of complex numbers in them. The problem only truly arises when defining e^ix, which requires trigonometric functions already defined for the complex plane.
2. The trivial set {0=1} allows 1/0 to exist while following the commutativity, associativity, and distributivity of algebra. I agree. Where we disagree is that there exists a non-trivial set where this is also possible, and this set is invoked by wheel algebra theory. The involution operator, "/", is a modified form of division such that division is a unary operation, and in the general case, x^(-1) is not equivalent to /x. However, commutativity, associativity and distributivity are all preserved, albeit with a tiny bit of modification, and whenever 0x=0 holds for any x, the identities all simplify down to identities that we use in non-wheel algebras, thus proving the wheel as an extension of a field. The extended Riemann sphere, which is the best projected model for this structure, is the set C in union with {/0, 0/0}. This is a mathematically sound structure, and you should read on it. My point is, there exists a non-trivial set for which it works. But my conjecture is there is no non-trivial set which is closed under all operators.
3. You ask what is that I define as an operation, and furthermore, what do I imply by all operations. That is a very good question. For our purposes, we can speak of all the integral hyper-operators and their respective inverses.
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