Reply to Re: I gave up to the Computers National Olimpiad
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Your 'new' 15873 already exists. It corresponds with the 15873 of the natural numbers (and supersets thereof). My '1' is just a notation for a certain element of a certain set, on which there might not even be such a thing as multiplication.
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I think the general confusion in the math-part of this thread, is that most of you have only done elementary algebra, while this is just a tiny little part of universal algebra (or sometimes just called algebra). Elementary algebra is just a special case of universal algebra, and many of the known, 'obvious', laws in elementary algebra aren't as obvious as they seem in universal algebra. In the broadest sense of the word, "algebra" is the study that deals with sets 'A', combined with (for every set 'A') a collection of operations on A.
Elementary algebra stops at the real axis, not covering complex sets, matrix sets, or even the general terms for these: fields, rings, groups, monoids, etc. Therefor, most people who are known to only a tiny bit of the universal algebra would say other things cannot be, while what they consider a definition to be generally true, is just a definition on the sets it's defined on, because they only know a part of all sets that can possibly be defined, the definitions don't have to be true for those other sets.
For example, (the very same example that lead to this discussion) there are no real numbers x so that x^2 = -1, is true. Stating that there are no numbers x in general so that x^2 = -1, isn't true. Just find a collection of elements, with an operation *, define the notation x^2 to be x*x (so we won't need to have 2 in our set), and define * so that we have an element 'x' and an element '-1', so that x*x = -1. Then there suddenly ís a number so that x^2 = -1.
Another thing that can lead to confusion is the mathematical term for different sets. "natural" numbers, "rational" numbers "real" numbers. When DraconicDink said that negative numbers aren't irrational, he meant a different "irrational" than what algebra-people (algebraicians?) would consider "irrational" (and even more different from what a game theorist would consider "irrational"). DraconicDink's references to "real" numbers as "numbers which exist" is also different from what mathematicians consider "real" numbers. I don't think a number on itself exists. It's always part of a certain algebra, that is, part of a set, on which an operation is defined. Same thing with "natural" numbers. While it might sound like "natural" means "obvious" (which would be a 'natural' line of thought, if you get what I mean), the "natural" numbers aren't quite as obvious as they might seem.
Quick overview (of how mathematicians see things):
Natural numbers
Integers
Rational numbers
Real numbers
---
Complex numbers
--- Stop reading here, unless you really want to dig into advanced algebra stuff
Quaternions
Octonions
Sedenions
Of which the last three are more extreme algebras.
Quaternions don't have a commutative multiplication. This means that for *-for-quaternions the following isn't always true: a*b = b*a
Octonions don't have a commutative multiplication either, but no associative multiplication. This means that a*(b*c) can be different than (a*b)*c. Though it's still true that a*(a*b) = (a*a)*b
The sedenions's multiplication isn't even alternative, which means that a*(a*b) = (a*a)*b.
They are all still power associative, which means that a*(a*a) = (a*a)*a. I haven't found an algebra yet for which this is not true, but I won't exclude anything.
If you're interested in these algebraically beautiful anomalies, you might want to check out free objects. Those are algebras, with one binary operation (operation with two input-variables and one output variable) in which as many axioms and relations between elements are thrown away. I'm possibly the only one here, but I really like to brag about my universitary study, and how I get these things
---
I think the general confusion in the math-part of this thread, is that most of you have only done elementary algebra, while this is just a tiny little part of universal algebra (or sometimes just called algebra). Elementary algebra is just a special case of universal algebra, and many of the known, 'obvious', laws in elementary algebra aren't as obvious as they seem in universal algebra. In the broadest sense of the word, "algebra" is the study that deals with sets 'A', combined with (for every set 'A') a collection of operations on A.
Elementary algebra stops at the real axis, not covering complex sets, matrix sets, or even the general terms for these: fields, rings, groups, monoids, etc. Therefor, most people who are known to only a tiny bit of the universal algebra would say other things cannot be, while what they consider a definition to be generally true, is just a definition on the sets it's defined on, because they only know a part of all sets that can possibly be defined, the definitions don't have to be true for those other sets.
For example, (the very same example that lead to this discussion) there are no real numbers x so that x^2 = -1, is true. Stating that there are no numbers x in general so that x^2 = -1, isn't true. Just find a collection of elements, with an operation *, define the notation x^2 to be x*x (so we won't need to have 2 in our set), and define * so that we have an element 'x' and an element '-1', so that x*x = -1. Then there suddenly ís a number so that x^2 = -1.
Another thing that can lead to confusion is the mathematical term for different sets. "natural" numbers, "rational" numbers "real" numbers. When DraconicDink said that negative numbers aren't irrational, he meant a different "irrational" than what algebra-people (algebraicians?) would consider "irrational" (and even more different from what a game theorist would consider "irrational"). DraconicDink's references to "real" numbers as "numbers which exist" is also different from what mathematicians consider "real" numbers. I don't think a number on itself exists. It's always part of a certain algebra, that is, part of a set, on which an operation is defined. Same thing with "natural" numbers. While it might sound like "natural" means "obvious" (which would be a 'natural' line of thought, if you get what I mean), the "natural" numbers aren't quite as obvious as they might seem.
Quick overview (of how mathematicians see things):
Natural numbers
Integers
Rational numbers
Real numbers
---
Complex numbers
--- Stop reading here, unless you really want to dig into advanced algebra stuff
Quaternions
Octonions
Sedenions
Of which the last three are more extreme algebras.
Quaternions don't have a commutative multiplication. This means that for *-for-quaternions the following isn't always true: a*b = b*a
Octonions don't have a commutative multiplication either, but no associative multiplication. This means that a*(b*c) can be different than (a*b)*c. Though it's still true that a*(a*b) = (a*a)*b
The sedenions's multiplication isn't even alternative, which means that a*(a*b) = (a*a)*b.
They are all still power associative, which means that a*(a*a) = (a*a)*a. I haven't found an algebra yet for which this is not true, but I won't exclude anything.
If you're interested in these algebraically beautiful anomalies, you might want to check out free objects. Those are algebras, with one binary operation (operation with two input-variables and one output variable) in which as many axioms and relations between elements are thrown away. I'm possibly the only one here, but I really like to brag about my universitary study, and how I get these things
