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 the sedenions, there exist x and y such that |xy|≤(√2)|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