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Re: (Newbie question): New types of numbers

• To: mathgroup at smc.vnet.net
• Subject: [mg66556] Re: [mg66545] (Newbie question): New types of numbers
• From: "Carl K. Woll" <carlw at wolfram.com>
• Date: Sat, 20 May 2006 04:46:51 -0400 (EDT)
• References: <200605190740.DAA12871@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

craigugoretz at gmail.com wrote:
> Hello,
> 
>      I am new to Mathematica, but I have a potentially complex problem
> to solve.  My goal is get Mathematica to understand a new type of
> number called a "neutrosophic number".  Neutrosophic numbers have a
> similar form to complex numbers, except that I^2 = I instead of I^2=-1.
>  Is it possible to add a new type for the neutrosophic numbers in
> addition to the four types that Mathematica currently supports
> (Integer, Real, Rational, Complex per Appendix A.1.5 in the help
> browser for Mathematica 5.1)?  If so, what bases would I need to cover
> to make the implementation possible for user defined operators (to be
> developed) and the system operators '+' and '*' (to begin with).  If it
> is not possible to create new types of numbers, is there a way to "fake
> it"?  One possible example may involve intercepting input from the main
> evaluation loop and coercing it somehow.
> 
>                                                              Sincerely,
>                                                              Craig
> Ugoretz

One possibility is to create your own data structure and give rules for 
multiplication, addition and powers of the data structure:

neuro /: neuro[a_, b_] neuro[c_, d_] := neuro[a c, a d + b c + b d]
neuro /: neuro[a_, b_]^n_Integer := neuro[a^n, ((a + b))^n - a^n]
neuro /: neuro[a_, b_] + neuro[c_, d_] := neuro[a + c, b + d]
neuro /: x_ neuro[a_, b_] := neuro[a x, b x]
neuro /: x_ + neuro[a_, b_] := neuro[a + x, b]

For example:

In[17]:= neuro[0, 1] neuro[0, 1]

Out[17]= neuro(0, 1)

In[18]:= a + b neuro[0, 1]

Out[18]= neuro(a, b)

In[19]:= neuro[a, b] neuro[c, d]

Out[19]= neuro(a c, b c + a d + b d)

In[20]:= neuro[a, b]^3

Out[20]= neuro(a^3, ((a + b))^3 - a^3)

One can also create formatting rules so that neuro[a,b] displays as

a+b *n*

where *n* is your I element.

Carl Woll
Wolfram Research

• References:
  • (Newbie question): New types of numbers
    • From: "craigugoretz@gmail.com" <craigugoretz@gmail.com>