What is R3 Division?
In the context of R3, division is defined as the inverse operation of multiplication. The result of dividing an operand by an operator is an expression such that when the operator is multiplied by this result, it yields the original operand.
R3 division is a scalar operation that incorporates rotation through Opposite Types and Signs. As the inverse of multiplication, it necessitates reversing both the circular motion of multiplication through the plane and the linear motion along a number line.
The circular motion’s inverse is determined using the R3 division table provided below.
The scalar part of the operation inverses similarly to division in classical mathematics.
The division table is derived very simply from the multiplication table. For example:
![\[1\hat{ }*i^v = j^v \implies i^v = j^v/1\hat{ }\]](https://wavenumbers.com/wp-content/ql-cache/quicklatex.com-2a4441d8fbd93c7edeba6c8381e31f1f_l3.png "Rendered by QuickLaTeX.com")
![\[j^v*i\hat{ } = 1\hat{ }\implies i\hat{ } =1\hat{ }/j^v\]](https://wavenumbers.com/wp-content/ql-cache/quicklatex.com-a0278050c1d30c434acd6e6734ffd245_l3.png "Rendered by QuickLaTeX.com")
| / | 1^ | 1v | i^ | iv | j^ | jv |
| 1^ | 1^ | 1^ | j^ | jv | iv | i^ |
| 1v | 1v | 1v | jv | j^ | i^ | iv |
| i^ | jv | j^ | i^ | i^ | 1^ | 1v |
| iv | j^ | jv | iv | iv | 1v | 1^ |
| j^ | i^ | iv | 1v | 1^ | j^ | j^ |
| jv | iv | i^ | 1^ | 1v | jv | jv |
Syntax
In R3 division, the divisor acts as the Operator, and the numerator serves as the Operand. The symbol / represents the operation. The result derives its Opposite Sign from the division table.
In the equation, 12^/4iv = 3jv, 4iv is the Operator and 12^is the Operand and 1^/iv in the division table gives a result of jv. The scalar part of the division is 12/4 = 3. Thus, putting the two steps of division together, 12^/4iv = 3jv is calculated as:
- 12/4 = 3
- 1^/iv = jv
The Operator can be an Opposite Value or a Counter, while the Operand can only be an Opposite Value. For example: 6i^/2 = 3i^ is valid but 6/2i^ is not.
As in classical mathematics, you cannot divide by zero. So, equations such as 1^/0 or j^/0 are undefined.
Division is neither commutative or associative.
- 6j^/3iv = 2^ and 3iv/6j^ = 0.5v
- Division is not commutative
- (24j^/8iv)/2v = 3^/2v = 3/2^
- 24^/(8iv/2v) = 24^/4jv = 6i^
- Division is not associative.
Counters
A Counter cannot be divided directly by another Counter in a standalone calculation because the result of such a division must be an Opposite Value. However, it is possible to divide terms by Counters.
When dividing by Counters, the operation is performed on the individual Opposite Value elements. For example:
- (6^ + 9iv + 12j^)/3 = (2^ + 3iv + 4j^)
- As 3*(2^ + 3iv + 4j^) = (6^ + 9iv + 12j^)
Examples
- 12^/4iv = 3jv because 4iv*3jv = 12^.
- Here 12^ is the operand, 4iv the operator and 3jv the result.
- Operator * Result = Operand
- 4iv*3jv = 12^
- 12i^/4 = 3i^because 4*3i^ = 12i^.
- Operator * Result = Operand
- 4*3i^ = 12i^
- Here 12i^ is the operand, 4 the operator and 3i^ the result.
Conclusion
Try these examples of R3 division with our online calculator.
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