R3 – Multiplication – Definition

What is R3 Multiplication?

Multiplication in R3 is a scalar operation that incorporates rotation through Opposite Types and Signs. It follows a “times-add” process, with the Opposite Sign of the result determined by the multiplication table. It differs from R1 and R2 multiplication because it now brings in the concept of rotation in real space compared to the imaginary space of R1 and R2.

R3 Unitary Multiplication Table

The Orthogonal Rule states that multiplication by a unitary is equivalent to rotation around the unitary’s axis by 90^o. R3 fully supports the Orthogonal rule. This principle enabled the development of the multiplication table below directly from the rotations of a 3d sphere, as derived in an earlier post.

*	1^	1v	i^	iv	j^	jv
1^	1^	1v	j^	jv	iv	i^
1v	1^	1v	jv	j^	i^	iv
i^	jv	j^	i^	iv	1^	1v
iv	j^	jv	i^	iv	1v	1^
j^	i^	iv	1v	1^	j^	jv
jv	iv	i^	1^	1v	j^	jv

R3 Wave Numbers fully implement Cardano’s (1501-1576) alternative rule of signs, where he proposed that multiplying a minus by a minus gives a minus. Cardano viewed positives and negatives as two distinct areas that should be kept separate. According to his rule, multiplying positives results in a positive, while multiplying negatives results in a negative.

It also allows R3 to support the reversal of multiplication through division.

In essence, the multiplication of two Opposite Values involves both circular motion through space and linear motion.

Syntax

The Operator is defined as the first term in a multiplication, which acts on the Operand, the second term of the multiplication. The symbol * represents the operation. For example:

3i^v*4j^{^} = 12^v

Here, 3i^v is the operator and 4j^{^}is the operand. The operator can be an Opposite Value or a Counter, while the operand can only be an Opposite Value. For example:

4*3i^{^} = 12i^{^}.

Assume multiplication when an Opposite Value precedes or follows a round bracket or when two round brackets are together. For example:

4(5^{^} + 2i^v) = 20^{^} + 8i^v

(5i^{^} + 2^v)4j^v = 20^v + 8i^v

(5^{^} + 2i^v)(4i^v + 7j^{^}) = 14^v + 43i^v + 20j^v 

Counters cannot be multiplied together as the result of any operation has to be an Opposite Value. They can be multiplied with terms.

Next: Simple Multiplications

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