# The Octonion RPN Calculator

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

(a,b)×(c,d)=(ac-db*,a*d+cb)
e0
e1
e2
e3
e4
e5
e6
e7
♥♥♥

The Octonion multiplication table used for this calculator is explained here.

#### Calculator Description

This is a postfix calculator with a depth 2 stack. To compute R + S*T, for example: R [enter] S [enter] T [times] [plus]

There are ten storage/retrieval registers, X, Y, Z, A, B, C, D, E, F, G accessed by the dropdown icon.

There are various standard register operations, with [push] replaced with [enter].

Numbers may be entered into the main register by hand, or by using the random number generator.

#### Definition of basis elements

$$e_0=(1,0)=1$$

$$e_1=(0,1)=i$$

$$e_2=(e_1,0)=j$$

$$e_3=(0,e_1)=k$$

$$e_4=(e_2,0)$$

$$e_5=(0,e_2)$$

$$e_6=(e_3,0)$$

$$e_7=(0,e_3)$$

#### Multiplication Table

 e0 e1 e2 e3 e4 e5 e6 e7 e1 –e0 e3 –e2 e5 –e4 e7 –e6 e2 –e3 –e0 e1 e6 –e7 –e4 e5 e3 e2 –e1 –e0 –e7 –e6 e5 e4 e4 –e5 –e6 e7 –e0 e1 e2 –e3 e5 e4 e7 e6 –e1 –e0 –e3 –e2 e6 –e7 e4 –e5 –e2 e3 –e0 e1 e7 e6 –e5 –e4 e3 e2 –e1 –e0

#### A worked example of use

To illustrate one of the Moufang identities, for example Z(X(ZY)) = ((ZX)Z)Y, perform the following operations:

1. Select random vectors X, Y and Z
1. RANDOM X STO
2. RANDOM Y STO
3. RANDOM Z STO
2. Perform the operations on the left side of the identity
1. Z RCL ENTER (Since Z should already be in the register, you don't actually have to recall it.)
2. Y RCL × ENTER
3. X RCL SWAP × ENTER
4. Z RCL SWAP ×
3. Store the result
1. A STO
4. Perform the operations on the right side of the identity
1. Z RCL ENTER
2. X RCL × ENTER
3. Z RCL × ENTER
4. Y RCL × ENTER
5. Subtract the result obtained on the left side of the identity. Result should be 0 (allowing for small errors).
1. A RCL –

John Wayland Bales, Department of Mathematics (Retired), Tuskegee University, Tuskegee, AL 36088 USA