This post covers R3 roots including their derivation.
Definition
Wikipedia defines the square root as ‘a square root of a number x is a number y such that y2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or ) is x.’ Wave Numbers uses this definition and adapts it for roots of higher degrees.
The root of an Opposite Value is represented with the n√ symbols where n ≥ 2. The default representation of √ without a value for n represents 2√.
There are two types of root. The first is the root of the Counter in an Opposite Value. The Counter root of an Opposite Value x is the Counter that when raised to the power of n gives the Counter of x and maintains the original Opposite Type and Opposite Sign. The fact that the Opposite Value is not within brackets identifies that it is a Counter root as in ?√n?. For example: 2√j^ = j^ and 2√iv = iv . In other words, the rotation expressed by the Opposite Values is ignored when calculating the Counter root.
The Full root of an Opposite Value x is the Opposite Value that when raised to the power of n gives x. Enclose the Opposite Value by brackets in order to specify the full root of an Opposite Value. For example: √(4jv) = 2jv.
The result of multiplying an Opposite Value by itself in R3 is an Opposite Value with the same Opposite Type and Sign. For example:
- 1^*1^ = 1^, 1v*1v = 1v
- i^*i^*i^ = i^, iv*iv = iv
- j^*j^ = j^, jv*jv*jv = jv
As a result, the root of an Opposite Value always has the same Opposite Type and Sign as the Opposite Value. This means that the Counter and Full roots are always equal, unlike in R1 and R2. This gives the rule:
- n√(1??) = 1??
Designate the degree of roots in the same way as for classical maths. So, 3√8j^ = 2j^.
Number of Roots
There is only 1 root for any expression in R3, independent of the power of the root.
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