July 22, 2024
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R3 – Division – Definition

What is R3 Division?

R3 division is the inverse of R3 multiplication. The result of division  of an operand by an operator is a math expression. Multiplying the operator with the result gives the operand. For example:

  • 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.

It is a scalar type operation that takes into account the Opposite Signs. As the inverse of multiplication, it requires the reversal of both the circular motion of multiplication through the plane and the linear motion along a number line.

The inverse of the circular motion is worked out using the division division table for R3 below.

The inverse of the scalar part of the operation works the same as division in classical maths. For example:

  • for 12^/4iv = 3jv, the scalar part is 12/4 = 3

So, putting the two steps of division together, 12^/4iv = 3jv is calculated as:

  1. 12/4 = 3
  2. 1^/iv = jv
  3. Therefore, combining the two, 12^/4iv = 3jv


In division, the divisor is the Operator and the Numerator is the Operand. The result takes its Opposite sign from the division table. For example: 12^/4iv = 3jv. Here 4iv is the operator, 12^ is the operand and 1^/iv in the division table gives a result of jv.

The Operator can be an Opposite Value or it can be a Counter. The Operand can only be an Opposite Value. For example: 6i^/2 = 3i^  is valid but 6/2i^ is not.

As in classical numbers, you cannot divide by 0.

  • 1^/0   – operation is undefined
  • jv/0     – operation is 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.

R3 Division Table

The division table is derived very simply from the multiplication table. For example:

  • 1^*iv = jv
    • Remember that the format of multiplication is Operator * Operand
    • So, dividing both sides by the Operator 1^ gives:
      • iv = jv/1^

  • jv*i^ = 1^ => 1^/jv = i^
    • So, dividing both sides by the Operator jv gives:
    • i^ = 1^/jv



A Counter cannot be divided into another Counter as a stand alone calculation because the result of any calculation has to be an Opposite Value. Terms can be divided by Counters.

Division by Counters operates on the individual Opposite Value elements. For example:

  • (6^ + 9iv + 12j^)/3 = (2^ + 3iv + 4j^)
  • As 3*(2^ + 3iv + 4j^) = (6^ + 9iv + 12j^)


Try these examples of R3 division with our online calculator.

Next: Self Division and Unitaries

Previous: Cross Product

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