$$p = a_0 + a_1i$$

in which $i^2 = -1$ and where the coefficients $a_0, a_1$ are real numbers.

Perhaps you have also heard about the quaternions:

$$p = a_0 + a_1i + a_2j + a_3k$$

with $ijk = -1$ and $i^2 = j^2 = k^2 = -1$.

If you are an algebra enthusiast, you most likely also know about the octonions!

$$p = a_0 + a_1i + a_2j + a_3k + a_4I + a_5J + a_6K + a_7E$$

the multiplication rules of the imaginary units are slightly more complicated here... but they can be neatly visualized through the Fano plane.

All the previous numbers and the algebras they are part of, including the reals, belong to a family of algebras named the Cayley-Dickon algebras. As you may have noticed, the dimension of each algebra doubles that of the previous, i.e., the algebra of the reals is of dimension 1, that of the complex of dimension 2, quaternions, 4, octonions, 8... In fact, each algebra is obtained from the previous through what's called the Cayley-Dickson construction process, and it goes on forever. The next algebra in this series is the algebra of the sedenions, of dimension 16, and I don't really know the names for the consecutive ones!

Anyway, I'm sure by now you must be very eager to operate on the elements of Cayley-Dickon algebras! And, since I know things get tricky when the dimension goes up, I have made a calculator to do the job for you! So, go ahead, input your numbers in the input fields—they must be members of valid Cayley-Dickon algebras—choose your favorite operator... et voilà! there you go the result!