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Mickey1
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No of dimensions for simple equation
« on: Feb 22nd, 2010, 5:48am » |
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Consider the number of dimensions, 2, of the complex numbers which is the field where all simple equation can be solved (such A*A=B) for all B, natural, rational or real.
Can we conjecture that this number, 2, is a result of the 2 available composition rules and generalize this to say that similar equations with N composition rules, suitably defined, would also be solvable in an N-dimensional field?
Obviously this would not be the case if the composition rules were not independent in some way. We would not expect clockwise and anti-clockwise movements of an object to be independent in this respect.
This conjecture seems to be true for addition: dim=1!
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Obob
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Re: No of dimensions for simple equation
« Reply #1 on: Feb 23rd, 2010, 9:43am » |
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It's a bit hard to really make sense of what you mean (not that it's hard to understand you; just many terms are undefined), but the answer is certainly no, for the following reason:
Suppose you have some way of introducing a new composition rule on complex numbers. One possibility is you can solve all the equations with the 3 composition rules over the complex numbers already. Otherwise, we get some new "field" where we can solve such equations, call it K. By forgetting about the third composition rule, we get an honest field. This field is an extension field of the complex numbers C. But every extension field of C is infinite dimensional, since C is Algebraically Closed. So to solve polynomials in 3 composition rules, you either need to be able to do so over C to begin with (in which case you might say the new operation is "not independent"), or you need infinitely many dimensions.
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| « Last Edit: Feb 23rd, 2010, 9:44am by Obob » |
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rmsgrey
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Re: No of dimensions for simple equation
« Reply #2 on: Feb 24th, 2010, 5:20am » |
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Is multiplication really independent of addition? If so, why isn't exponentiation independent of multiplication?
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Obob
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Re: No of dimensions for simple equation
« Reply #3 on: Feb 24th, 2010, 6:49am » |
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Addition and multiplication are interwined by the distributive law a(b+c) = ab+ac, so they aren't "completely" independent. You wouldn't be able to define multiplication in terms of only finitely many additions, though, so multiplication is a "new" operation.
Similarly, exponentiation is interwined with addition and multiplication by the law exp(a+b) = exp(a)exp(b). But again you couldn't define exponentiation using only finitely many additions and multiplications. Exponentiation is "independent" of addition and multiplication.
And in the sense of this question, you can consider the equation exp(z) = 0. This has no solution in C. You can ask though if there is some number system extending C for which
1) the usual field axioms are satisfied,
2) there is an operation exp extending the usual one, and
3) there is a number z with exp(z) = 0
You can define such an object formally by letting K = C((z)) be the field of Formal Laurent Series with finite order tails in the variable z, and then simply defining exp(f(z)) = exp(a_0), where f(z) = sum a_i z^i is a Laurent series with a finite tail.
This new exponential function still satisfies the fundamental law
exp(f+g) = exp(f)exp(g),
so it still intertwines exponentiation with addition and multiplication in the same way.
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| « Last Edit: Feb 24th, 2010, 9:55am by Obob » |
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Mickey1
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Re: No of dimensions for simple equation
« Reply #4 on: Mar 3rd, 2010, 3:06am » |
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The article about algebraically closed fields to which a link was given sounds convincing and its use of polynomial roots looks almost as if the subject was designed explicitly to answer my question. I thank the senior riddler for that.
However, I still wonder about the origin of the number 2, the number of dimensions of complex numbers. We now have dimension 1 for one composition rule, 2 for 2 rules and then infinity. It gives me an unsatisfactory impression of lack of harmony.
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Obob
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Re: No of dimensions for simple equation
« Reply #5 on: Mar 3rd, 2010, 4:19am » |
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In some sense your original question is just posed incorrectly. Instead of starting with the real numbers (which are essentially an "analytic" object, not an "algebraic" one), notice that you can already solve addition equations in the rational numbers. In the sense of algebra, the rational numbers are already a 1-dimensional object. If you want to solve polynomial equations, you need to add in numbers like square roots and i; when you do this, you get the field known as the algebraic closure of the rationals, or the "algebraic numbers." This field is infinite dimensional over the rational numbers (although it is smaller than the field of complex numbers; for instance pi is not an algebraic number), so the jump from 1 dimension to infinitely many dimensions already occurs when you want to solve polynomials instead of just additive equations. If you wanted to solve equations with another composition law, you would have to add in infinitely more dimensions over the algebraic numbers.
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| « Last Edit: Mar 3rd, 2010, 4:22am by Obob » |
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ThudnBlunder
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Re: No of dimensions for simple equation
« Reply #6 on: Mar 3rd, 2010, 5:43am » |
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on Feb 22nd, 2010, 5:48am, Mickey1 wrote:
Can we conjecture that this number, 2, is a result of the 2 available composition rules and generalize this to say that similar equations with N composition rules, suitably defined, would also be solvable in an N-dimensional field?
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on Mar 3rd, 2010, 3:06am, Mickey1 wrote:
However, I still wonder about the origin of the number 2, the number of dimensions of complex numbers. We now have dimension 1 for one composition rule, 2 for 2 rules and then infinity. It gives me an unsatisfactory impression of lack of harmony. |
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Quaternions have four composition rules in four dimensions.
Octonions have eight composition rules in eight dimensions.
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| « Last Edit: Mar 3rd, 2010, 5:47am by ThudnBlunder » |
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Obob
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Re: No of dimensions for simple equation
« Reply #7 on: Mar 3rd, 2010, 6:07am » |
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Quaternions and Octonians still only have two composition rules: addition and multiplication.
Another fault they have is that they don't still satisfy the normal field axioms: multiplication isn't commutative in the quaternions, and it isn't even associative in the octonians.
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