R2 – Multiplication – Definition

What is R2 Multiplication?

Multiplication in R2 is similar to multiplication of complex numbers in classical maths. It is a scalar type operation that takes into account rotation through the Opposite Signs. It can be considered a ‘times-add’ operation with the Opposite Sign of the result dependent on the R2 multiplication table.  

This means that multiplication comprises of movement both in a circular motion and linear motion along a number line.

R2 Unitary Multiplication Table

The Orthogonal rule  states that multiplication by a unitary is the equivalent of rotation around the unitary’s axis by 90^o. Unitary rotation is not standard in R2 as there is no free axis of rotation. However, multiplication by i^{^} and i^v follow the orthogonal rule.

Given that i^{^} and i^v follow the orthogonal rule, then it can be deduced that no rotation occurs when multiplying by the unitary 1^{^} and 180^vo rotation occurs when multiplying by unitary 1^v.

Below is the R2 multiplication table as derived from rotation in an earlier post.

Syntax

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

• 3i^v*4^{^} = 12i^v. Here 3i^v is the operator, * is the operation and and 4^{^}is the operand.

The Operator can be an Opposite Value or it can be a counter. However, the Operand can only be an Opposite Value. For example:

• 4*3i^{^} = 12i^{^}
• 4i^{^}*3  is invalid

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)4^v = 8^{^} + 20i^v
• (5^{^} + 2i^v)(4^v + 7i^{^}) = 6^v + 43i^{^}

Try these examples of multiplication with our online calculator.

Counters

Counters cannot multiply together because the result of any operation has to be an Opposite Value. However, counters can multiply with terms.

Next: Simple Multiplications

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