Orthogonal Rule used to Derive Multiplication Table
The Axioms of Rotation provide the Orthogonal rule that states that multiplication by a unitary is the equivalent of rotation around the unitary’s single point axis by 90o. This allows the derivation of the R2 unitary multiplication table that follows.
Multiplication by i^
Using the post-it note prop, it is easy to see that a counterclockwise rotation of 90^o around the imaginary z-axis moves a point located at (1^, 0) to (0, i^). The Orthogonal rule defines this as multiplication by i^, i.e. i^*1^ = i^. Furthermore, another rotation of 90^o around the imaginary z-axis moves a point located at (0, i^) to (1v, 0). Another rotation of 90^o moves a point located at (1v, 0) to located at (0, iv). A final rotation of 90^o moves a point located at (iv, 0) to the location (1^, 0).
This application of rotation by i^ gives the third row of the R2 multiplication table rules:
i^*1^ = i^, i^*1v = iv, i^*i^= 1v, i^*iv = 1^
Multiplication by iv
Using the post-it note prop, it is easy to see that a clockwise rotation of 90vo around the imaginary z-axis moves a point located at (1^, 0) to (0, iv). The Orthogonal rule defines this as multiplication by iv, i.e. iv*1^ = iv. Further clockwise rotation of 90vo by iv move the point located at (0, iv) first to (1v, 0), secondly to (0, i^) and finally to (1^, 0).
This application of rotation by iv gives the fourth row of the R2 multiplication table rules:
iv*1^ = iv, iv*1v = i^, iv*i^= 1^, iv*iv = 1v
Multiplication by 1^
The R2 rules for multiplication by 1^ can be derived for the table from the rules for i^ and iv as follows:
Given: 1^ = iv*i^ or i^*iv
- => 1^*1^= iv*i^*1^ = iv*i^ = 1^ or
- => 1^*1^ = i^*iv*1^ = i^*iv = 1^
- so 1^*1^ = 1^
- => 1^*i^ = iv*i^*i^ = iv*1v = i^ or
- => 1^*i^ = i^*iv*i^ = i^*1^ = i^
- so 1^*i^ = i^
- => 1^*1v = iv*i^*1v = iv*iv = 1v or
- => 1^*1v = i^*iv*1v = i^*i^ = 1v
- so 1^*1v = 1v
- => 1^*iv = iv*i^*iv = iv*1^ = iv or
- => 1^*iv = i^*iv*iv = i^*1v = iv
- so 1^*iv = iv
This gives the first row of the R2 multiplication table rules:
1^*1^ = 1^, 1^*1v =1v, 1^*i^= i^, 1^*iv = iv
Multiplication by 1v
Finally, the R2 rules for multiplication by 1v can also be derived for the table from the rules for i^ and iv as follows:
Given: 1v = iv*iv or i^*i^
- => 1v*1^ = iv*iv*1^ = iv*iv = 1v or
- => 1v*1^ = i^*i^*1^ = i^*i^ = 1v
- so 1v*1^ = 1v
- => 1v*i^ = iv*iv*i^ = iv*1^ = iv or
- => 1v*i^ = i^*i^*i^ = i^*1v = iv
- so 1v*i^= iv
- => 1v*1v = iv*iv*1v = iv*i^ = 1^ or
- =>1v*1v = i^*i^*1v = i^*iv = 1^
- so 1v*1v = 1^
- => 1v*iv = iv*iv*iv = iv*1v = i^or
- =>1v*iv = i^*i^*iv = i^*1^ = i^
- so 1v*iv = i^
R2 Multiplication Table
Putting all this together gives the R2 unitary multiplication table.
| * | 1^ | 1v | i^ | iv |
1^ | 1^ | 1v | i^ | iv |
1v | 1v | 1^ | iv | i^ |
i^ | i^ | iv | 1v | 1^ |
iv | iv | i^ | 1^ | 1v |
Conclusion
Try these examples of R2 multiplication with our online calculator.
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