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A growing basis? 
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#1
May214, 04:09 AM

P: 70

Hi i was just wondering if there is any concept/theory/idea (or anything really) that relates to a growing basis (primarily in R) .
What i mean by a growing basis is the following; say you start off with one element in your basis and you encounter a number/vector that cannot be built by the element, this number that cannot be built then gets incorporated into the basis and becomes an element of it and this process goes on and on picking up more building elements as it encounters numbers that cant be built from the existing elements. For example, in R, say the number you begin with is 2, then the next (integer) number you encounter is 3, which cant be built by 2, so 3 gets including in the basis. If this process were to go on and on, you would have a basis consisting of all primes (2,3,5,7,11,13,....) since all integers can be built from the primes. I hope that makes sense. Thank you! 


#2
May214, 04:20 AM

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PF Gold
P: 1,908

Sounds like physics  the experimental approach to mathematics!
You will find something analogous in the creation of imaginary and complex numbers; here is a brief summary: http://www.math.uri.edu/~merino/spri...umbers2006.pdf 


#3
May214, 10:48 AM

P: 70

Thank you for the links Ultra. Do you imply there is no such approach that exists to date? That's interesting you say it sounds like physics, I've heard that before!



#4
May214, 10:54 AM

Mentor
P: 18,225

A growing basis?
Check out the Sieve of Eratosthenes: http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes



#5
May214, 01:53 PM

P: 70

That is very comparable to the ideas I'm thinking of, cheers for that. Another question, the set i gave as an example before, the set of primes, that isn't a basis is it? since a basis under the current definitions and axioms in linear algebra say that a basis is linearly independent, however, 2,3 and 5 are elements of the set of primes but 2+3=5 so 5 is a linear combination of other elements in the basis, and hence the set isn't linear independent.
This brings me to ask another question. How is R a subspace and how can it have 1 dimension (or am I completely wrong and R is not 1 dimensional?)? since by definition a subspace needs to be closed under scalar multiplication. In R, all numbers are a linear combination of 1, so you could say a basis for R is {(1)}, but i don't see how this basis is closed under scalar multiplication since it has, to me, no meaning to multiply (1) by a scalar quantity, c, as all scalar quantities are not defined within the basis {(1)}. Except for maybe ( 1 +/ 1 +/ 1 +/ 1 +/...=c c [itex]\in[/itex] R) but then if instead you used the basis {(2)} of R, there are scalar multiples, c, that cant be built by purely adding or subtracting 2 (ie 2 +/ 2 +/ 2 +/...≠ c c [itex]\in[/itex] R). 


#6
May214, 02:00 PM

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P: 18,225

All we demand for a basis is that each element in ##\mathbb{R}## can be written as a unique linear combination of the basis ##\{1\}##. And indeed, any element ##x\in \mathbb{R}## can be written as ##x = x\cdot 1## and this is unique. Of course, ##\{2\}## is a basis too and so is ##\{3\}##. In fact, if ##x## is a nonzero real, then ##\{x\}## is basis for ##\mathbb{R}##. 


#7
May314, 02:25 AM

P: 70

Also, When designing a new mathematical model, is it acceptable and sometimes necessary to change the axioms that we assume in current mathematics, provided that we deduce logical and sound conclusions from these new axioms. And as long as the axioms are "sensible". I think this may relate to Godel's Incompleteness theory, but I'm not entirely sure how to apply it. Cheers, Michael 


#8
May314, 06:47 AM

Math
Emeritus
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PF Gold
P: 39,490




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