What Are the Advantages of Base Twelve Over Other Number Bases?

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As far as conversions between bases go: is there any advantage representing a number as a vector in an n-th dimensional vector space?

So, say I wish to represent the number 73 in base 10, it'd be [tex]73=7\times \vec{v}_1+3\times\vec{v}_0[/tex] where [tex]\vec{v}_n=10^n[/tex], so its representation is on [tex]\mathbb{Z}^2[/tex] lattice.

The equivalent representation in base 2 is 1001001, so as a vector, it'd be [tex]73=1\times \vec{w}_6+1\times\vec{w}_3+1\times\vec{w}_0[/tex] where [tex]\vec{w}_n=2^n[/tex], so its representation is a point on a [tex]\mathbb{Z}^6[/tex] lattice.

Is there an obvious relationship between the vector in both spaces? I would be very interested in some feedback!
 
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I don't think the vector space paradigm applies. A basis of a vector space is made of independent vectors; while, in your binary example, [itex]w_n = 2 w_{n-1}[/itex].

Edit: oh, that was possibly a stupid remark of mine. You mean a vector space over the trivial field {0,1}. Then the question is, over which fields you mean, when using a base greater than 2? Since, for non-prime bases, I think, modular arithmetic does not form a field.
 
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Dodo said:
I don't think the vector space paradigm applies. A basis of a vector space is made of independent vectors; while, in your binary example, [itex]w_n = 2 w_{n-1}[/itex].

Edit: oh, that was possibly a stupid remark of mine. You mean a vector space over the trivial field {0,1}. Then the question is, over which fields you mean, when using a base greater than 2? Since, for non-prime bases, I think, modular arithmetic does not form a field.

That has cleared the picture up considerably - I was aware of the connection between the "unit vectors" and was more concerned with [itex]v_n v_m= v_{n+m}[/itex] idea.

With regards the underlying field in each case; it seems a little much to use a different field for each conversion and expect it to go smoothly.

Thank you for your response.
 
Just for fun...

Convert the number CAT36 to base 35


What do you get?
 


zgozvrm said:
Convert the number CAT36 to base 35


What do you get?
Oh, cute! That's the first worthwile thing I have seen in this thread!