The Cayley-Dickson Calculator

GNU Public License 3 Software © (2009) John Wayland Bales under the GNU General Public License

The universal real Cayley-Dickson algebra A is an infinite dimensional vector inner product space with Euclidean norm consisting of all sequences of real numbers of the form

x0, x1,  . . .xn, 0, 0, 0,  . . .

The canonical basis vectors for this space are denoted i0, i1, i2, i3,  . . . where

i0 = 1, 0, 0, 0,  . . .

i1 = 0, 1, 0, 0,  . . .

i2 = 0, 0, 1, 0,  . . .

i3 = 0, 0, 0, 1,  . . .


Real numbers are identified with sequences of the form

x0, 0, 0, 0,  . . .

Complex numbers are identified with sequences of the form

x0, x1, 0, 0, 0,  . . .

Quaternions are identified with sequences of the form

x0, x1, x2, x3, 0, 0, 0,  . . .

Octonions are identified with sequences of the form

x0, x1, x2, x3, x4, x5, x6, x7, 0, 0, 0,  . . .

The Sedenions are identified with sequences of the form

x0, x1,  . . .x15, 0, 0, 0,  . . .

There is an infinite sequence of Cayley-Dickson algebras, each with twice the dimension of the previous algebra and each containing all previous algebras as proper sub-algebras.

If the ordered pair (x, y) of two sequences

x = x0, x1, x2,  . . .


y = y0, y1, y2,  . . .

is identified with the "shuffled" sequence

(x, y) = x0, y0, x1, y1, x2, y2,  . . .

then each algebra in the sequence beginning with the complex numbers consists of all ordered pairs of elements of the previous algebra.

And for each element a,b ε A the ordered pair (a,b) ε A and if x ε A, then there are unique elements a,b ε A such that x = (a,b).

The basis vectors obey the identities

i2p = (ip, 0)

i2p+1 = (0, ip)

The conjugate of an element

x = x0, x1,  . . .xn, 0, 0, 0,  . . .

is the sequence

x* = x0, –x1,  . . .–xn, 0, 0, 0,  . . .

Thus for sequences x and y it follows that (x,y)* = (x*,–y).

For sequences a, b, c and d for which multiplication is defined, the Cayley-Dickson product of the shuffled sequences (a,b) and (c,d) is defined by

(a,b)·(c,d) = (ac-db*,a*d+cb).

Using this product definition (there are alternate definitions), the product of the basis vectors is determined recursively as follows:

If p and q are non-negative integers then

i2p·i2q = (ip·iq, 0)

i2p·i2q+1 = (0, ip*·iq)

i2p+1·i2q = (0, iq·ip)

i2p+1·i2q+1 =  –(iq·ip*, 0)

The ultimate result of this recursive definition is a product of Cayley-Dickson basis vectors ip and iq given by the formula

ip· iq = γ(p,q) ip^q

where γ(p,q) is either +1 or -1 and where p^q is the bit-wise 'exclusive or' of the binary representations of p and q.

[For example if p = 5=101B and q = 6=110B then p^q=101B^110B=011B=3. So i5·i6 =  γ(5,6)i3.]

The sign function γ is called a "twist" and can be recursively defined as follows:

The proof of the Cayley-Dickson twist is given in "Cayley-Dickson and Clifford Algebras as Twisted Group Algebras (2003)" by John W. Bales.

You may use the following javascript applet to find the product of any two of the first 109-1 basis vectors. Larger values of p and q can trigger the sign bit when rendering the value of p^q.

Enter a value of p:
Enter a value of q:
The value of p^q is:
The value of γ(p,q) is:

For the universal Cayley-Dickson algebra, if 0 ≠ p≠ q≠ 0 and if ip· iq = ir then

These may be called the "quaternion" properties.

If 0 ≠ p≠ q≠ 0 and if ip· iq = ir then we denote this by the ordered number triple (p,q,r).

For the universal Cayley-Dickson algebra (p,q,r) implies (q,r,p) and (r,p,q) by the quaternion properties, and by the properties of γ implies (2p,2q,2r), (2q,2p+1,2r+1), (2q+1,2p,2r+1) and (2q+1,2p+1,2r). Beginning with (1,2n,2n+1) for all n, these recursively generate the multiplication table for the universal Cayley-Dickson algebra when 0 ≠ p≠ q≠ 0. To complete the table one needs only to note that i0 = 1 and ip· ip = –1 for p ≠ 0.


The last four identities are the inductive identities and are implied by the nature of γ. The inductive identities are summarized in the following table:

(p,q,r) 2p 2p+1
2q –2r 2r+1
2q+1 2r+1 2r

[ Note to Octonion specialists: This indexing of the octonion basis vectors, automatically satisfies index doubling, but not index cycling. For more on this topic go here.]

Given the Euclidean inner product <x,y>, the universal Cayley-Dickson algebra A satisfies the adjoint properties

<x·y, z> = <x, z·y*>

<x·y, z> = <y, x*·z>

Thus the kth component of the product x·y, is <x·y, ik> = <x, ik·y*>.

Also check out the Sedenion Reverse Polish Notation (RPN) Calculator,

the Octonion RPN calculator,

and the Sedenion Norm Comparison Calculator.

And the Color-coded Clifford Algebra Multiplication Table

Reference "Cayley-Dickson and Clifford Algebras as Twisted Group Algebras" (by J. W. Bales (2003); Dept. of Mathematics, Tuskegee University

Reference "Properly twisted groups and their algebras" by J. W. Bales (2006); Dept. of Mathematics, Tuskegee University

Reference "A catalog of Cayley-Dickson-like products" by J. W. Bales (2011); Dept. of Mathematics, Tuskegee University

Reference "A tree for computing the Cayley-Dickson twist" by J. W. Bales (2007) published in the Missouri Journal of Mathematical Sciences Vol. 21 No. 2 (2009)

© John Wayland Bales (2009)

Reference "The eight Cayley-Dickson doubling products" by J. W. Bales (2015); Advanced in Applied Clifford Algebras (Springer)

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