The Sedenion Product Calculator

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.

In the 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|. Thus, for sedenions x and y it is the case that |xy|≤(√2)|x|·|y|. Furthermore if c lies in the interval [0,√2] then there exist sedenions x and y such that |xy|=c|x|·|y|.

The default values of x and y above (obtained by refreshing the page if necessary) satisfy the equation |xy|=(√2)·|x|·|y|. Note that they are both rational points.

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