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

[cathalcummins]

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 $$73=7\times \vec{v}_1+3\times\vec{v}_0$$ where $$\vec{v}_n=10^n$$, so its representation is on $$\mathbb{Z}^2$$ lattice.

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

Is there an obvious relationship between the vector in both spaces? I would be very interested in some feedback!

 

[dodo]

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, $w_n = 2 w_{n-1}$.

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.

 

[cathalcummins]

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, $w_n = 2 w_{n-1}$.

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 $v_n v_m= v_{n+m}$ 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.

 

[zgozvrm]

Just for fun...

Convert the number CAT₃₆ to base 35


What do you get?

 

[HallsofIvy]

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!