Talk:LCM numeral system

Please have your comment signed and timestamped by using four tildes ~~~~. — Daniel Forgues 20:29, 1 May 2014 (UTC)

Most people are familiar with fixed radix numbering systems like base ten and base two. A fixed radix numbering system uses the coefficients of a power series to represent a number. 4,321 (base 10) represents $4*10^{3}+3*10^{2}+2*10^{2}+1*10^{0}$.

There are also product based numbering systems. These are a type of mixed radix numbering systems (http://en.wikipedia.org/wiki/Mixed_radix). A product base uses two series: a base series, B, and the partial products of the base series, P. All fixed radix numbering systems are also product based numbering system. The two series for base 10 are:

S = $1,10,10,10,...$

P = $1,10,100,1000,...$

Product based numbering systems are positional numbering systems. The allowable coefficients for the ith lowest order position are 0 through S_(i+1)-1.

The factorial numbering system is an example of a product based numbering system.

S = $1,2,3,4,...$

P = $1!,2!,3!,4!,...$

The allowable coefficients for the ith position in this base are 0 through i. The lowest order position can be 0 or 1. The second lowest order position can be 0 through 3-1 = 0, 1, or 2.

4321 (base !) = $4*4!+3*3!+2*2!+1*1=119(base10)$

Any series of positive integers that starts with 1 can be the base series for a product based numbering systems. We can use the prime series to create the primorial numbering system (http://en.wikipedia.org/wiki/Mixed_radix):

S = $1,2,3,5,...$

P = $1,2,6,30,...$

4321 (base prime) = $4*30+3*6+2*2+1*1=143(base10)$

In a fixed radix numbering system there are always rational numbers that have no finite representations. In base 10, 1/3 = 0.333... (base 10) and there is no finite representation. Some product based numbering systems have finite representations for every rational number. The factorial base is such a base. The primorial base does not have this property. A product based numbering system has a finite representation for every rational number if and only if every prime power is a factor of some product in the series P.

We can construct a base with this property using the root of prime powers series:

S = $1,2,3,2,5,7,...$

P = $1,2,6,12,60,420,...$

4121 (base LCM) = $4*12+1*6+2*2+1*1=59(base10)$

These series give us the LCM numbering system. Every rational number has a finite representation in the LCM numbering system.

With all these different numbering systems it seems reasonable to ask what is the the most "efficient" system. If we just consider the number of coefficients needed to represent some number then the largest radix bases are the most efficient. This would mean base $1,000,000$ is more efficient than base $2$. A better way to measure efficiency would be to consider the number of coefficients as the width of a representation and the largest radix used as the height of the representation. We can then determine the "area" of a representation. A five digit base 10 number would have an area of $5*10=50$. A 10 bit binary number would have an area of $10*2=20$. The binary numbering system is very efficient using this measure. The LCM numbering system is very efficient base for a numbering system with finite representations for every rational number.

Russell Easterly 20:03, 2 May 2014 (UTC)