Special algebra :

 a^2 + b^2 + c^2

There exists an algebra such that there are no zero divisors and with norm 

a^2 + b^2 + c^2.

It is a nondistributive algebra and it extends the complex numbers.

x = a + b i + c j

the norm of x = a^2 + b^2 + c^2

Sum is as ordinary, and product is most easily defined in these coordinates :

x = a + b i + c j

x = r ( sin(u) cos(w) , sin(u) sin(w) , cos(u) ) 

with r = sqrt( a^2 + b^2 + c^2)

u = arccos (c/r) = arctan ( sqrt(a^2 + b^2) / c )

w = arctan(b/a)

and then 

x^n = r^n ( sin(n u) cos(n w) , sin(n u) sin(n w) , cos(n u) )

and ofcourse products are defined by 

X = A + B i + C j

X = R ( sin(U) cos(W) , sin(U) sin(W) , cos(U) ) 

with R = sqrt( A^2 + B^2 + C^2)

u = arccos (C/R) = arctan ( sqrt(A^2 + B^2) / C )

w = arctan(B/A)

x X = r R ( sin(u + U) cos(w + W) , sin(u + U) sin(w + W) , cos(u + U) )

notice how x X cannot equal 0 unless x or X equals 0.

There are no zero-divisors in this algebra !

Notice how a + b i is the traditional complex number.

!! This is the only unital algebra without zero divisors that has commutative and associative multiplication and a solution to x^ = -1 !!  

Thank you for your intrest.

tommy1729