R3 – Division – Self Division and Unitaries

This post covers self division in R3. It looks at the cases where the result of division is a unitary and where it is not.

Self Division

In quaternions, R1 and R2, the division of an expression by itself results in the answer 1^{^}. For example in R2: 

• (6^{^}+ 11i^v)/ (6^{^}+ 11i^v) = 1^{^}

However, in Wave Numbers R3, division of an expression by itself does not always yield a unitary result. The result of dividing an operand by an operator is another expression. Multiplying the operator by this result returns the original operand. For example:

This result can be verified by multiplying it by the original operator:

Unitaries

Division in Wave Numbers is the inverse of multiplication, meaning it reverses both scalar multiplication and a rotation. Multiplication in Wave Numbers moves a point from one location to another, while division moves a point back, effectively reversing scalar multiplication and circular motion.

In R3, self-division results in a unitary value when the division is by the same, single Opposite Value. For example:

• 1^{^}*1^{^} = 1^{^} =>  1^{^}/1^{^} = 1^{^}
• 7^{^}*1^{^} = 7^{^} =>  7^{^}/7^{^} = 1^{^}
• 1^v*1^{^} = 1^{^} =>  1^{^}/1^v = 1^{^} 
• 9^v*1^{^} = 9^{^} =>  9^{^}/9^v = 1^{^}
• 9^{^}*1^v = 9^v =>  9^v/9^{^} = 1^v
• 1j^v*1j^{^} = 1j^{^} =>  1j^{^}/j1^v = 1j^{^} 
• 9j^{^}*1j^v = 9j^v =>  9j^v/9j^{^} = 1j^v

Moreover, division can never result in the Counter 1 as it cannot be the location of a point in R3.

Some answers to other types of division can consist of unitaries. For example:

• (4^{^} + 5i^{^} + 6j^{^})*(^{^} ) = (4^{^} + 6i^{^}+ 5j^v)
  • =>  (4^{^} + 6i^{^}+ 5j^v) /(4^{^} + 5i^{^} + 6j^{^}) = ^{^}
• (4^{^} + 5i^{^} + 6j^{^})(^v + i^) = (10^v + i^v+ 9j^{^})
  • =>  (10^v + i^v+ 9j^{^})/(4^{^} + 5i^{^} + 6j^{^}) = (^v + i^{^})
• (4^{^} + 5i^{^} + 6j^{^})(^{^} + i^{^} + j^{^}) = (3^{^} + 7i^{^}+ 5j^{^})
  • => (3^{^} + 7i^{^}+ 5j^{^})/(4^{^} + 5i^{^} + 6j^{^}) = (^{^} + i^{^} + j^{^})

Conclusion

Try these examples of R3 division with our online calculator.

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