The Sedenion Product Calculator

Software © (2009) John Wayland Bales under the GNU General Public License

x0  y0  c z0 
x1  y1  r z1 
x2  y2  z2 
x3  y3  z3 
x4  y4  z4 
x5  y5  z5 
x6  y6  z6 
x7  y7  z7 
x8  y8  z8 
x9  y9  z9 
x10 y10 z10
x11 y11 z11
x12 y12 z12
x13 y13 z13
x14 y14 z14
x15 y15 z15
|x| |y| |z|
|xy|÷(|x|·|y|) =

Sedenions are a sixteen dimensional algebra--the fifth algebra in the sequence of Cayley-Dickson algebras. This calculator computes the product of two vectors in the algebra. Vector components may be entered by hand or may be filled in by pseudo-random numbers on the interval from c-r to c+r. One may opt for discrete integer values for the components or continuous real values.

Sedenions are not normed algebras. This means that |x·y| = |x|·|y| is not an identity since it is untrue for some x and y. In fact, the sedenions contain zero divisors. There are non-zero values of x and y for which xy = 0. A theorem by Joseph Wedderburn published in the Bulletin of the American Mathematical Society in 1925 guarantees that for sedenions |x·y| will never exceed four times the value of |x|·|y|.

The default values of x and y above (obtained by refreshing the page if necessary) satisfy the equation |xy|=(√2)·|x|·|y|. I've been unable to find examples of sedenions x and y for which |xy|>(√2)·|x|·|y|.

I originally conjectured that |xy|≤(√2)·|x|·|y| for all Cayley-Dickson vectors of any dimension. However, this is not true. I've found a counter-example in the 32 dimensional Cayley-Dickson algebra (the 'quinipotentials' for want of a better name). However, I still suspect that the inequality holds for the sedenions.

Newsflash: Just found the following article entitled "Eigentheory of Cayley-Dickson Algebras" by Biss, Christensen, Dugger and Isaksen. Lee lemma 4.4 p. 9 and Thm. 8.3 p 16 which together imply that in the Cayley-Dickson space of dimension 2n, |xy|≤(√2)(n-3)|x|·|y|. This fully explains the results seen.

The calculator can be placed into octonion, quaternion, complex or real mode using the drop-down menu at the top of the calculator. All of these modes use the same multiplication table.

Have a go at it!

See also the Sedenion RPN Calculator and the Octonion RPN Calculator.

For more information and to see my development of the universal Cayley-Dickson algebra as an algebra of real number sequences go here.

©(2009) John Wayland Bales Department of Mathematics, Tuskegee University