This post on R2 roots covers the definition of roots and how roots are derived from the R2 multiplication table. Examples of square, cube and higher roots are given.
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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 n√ symbols, where n ≥ 2, represents the root of an Opposite Value. 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 ?√x?. For example: 2√1^ = 1^ and 2√iv = 2√1iv = iv . In other words, ignore the rotation expressed by the Opposite Values 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 the Opposite Value. For example: √(4v) = 2i^ or 2iv.
Calculate the full root as the square root of the Counter times the square root of the Unitary. For example: √(9v) = √9*√(1v) = 3*i^ = 3i^ or = 3*iv = 3iv.
Designate the degree of the root in the same way as for classical maths. So, 3√8i^ = 2i^ and is the cube root of the Counter 8 and holds the same Opposite Sign.
Rotation Table
The R2 multiplication table defines the following:
- 1^2 = 1^*1^ = 1^ => 1^n = 1^
- 1v2= 1v*1v = 1^ and 1v3 = 1v*1^ = 1v
- => for even values of n, 1vn = 1^
- => for odd values of n, 1vn = 1v
- i^2 = i^*i^ = 1v, i^3 = i^*1v = iv
- i^4 = i^*iv = 1^; i^5 = i^*1^ = i^
- iv2 = iv*iv = 1v, iv3 = iv*1v = i^
- iv4 = iv*i^ = 1^; iv5 = iv*1^ = iv
The following can be deduced from the above:
- √(1^) = 1^ or 1v
- √(1v) = i^ or iv
- n√(1^) = 1^, 1v for even values of n
- = 1^ for odd values of n and
- = i^ or iv for values of 4n
- n√(1v) = 1v for odd values of n and
- = i^ or iv for values of (2 +4n)
Other Roots Based on Spherical Coordinates
Other roots can be calculated by looking at the spherical coordinates. 1^ is the cube root of 1^. The spherical co-ordinates of 120o^, (½v + √3/2i^), are also the cube root because 3 counterclockwise rotations of 120o from 1^ bring a point back to the location 1^. i.e. (½v + √3/2i^)* (½v + √3/2i^)* (½v + √3/2i^) = 1^.
The spherical co-ordinates of 240o^, (½v + √3/2iv), are the cube root because 3 clockwise rotations of 120o from 1^ bring a point back to 1^. i.e. (½v + √3/2iv)* (½v + √3/2iv)* (½v + √3/2iv) = 1^.
Number of Roots
The number of roots available are dependent on the degree. For example 3√ has 3 roots. Other examples are:
- 3√(1^) = 1^ or (½v + √3/2i^) or (½v + √3/2iv)
- 5√(1^) = 1 ^ or (0.309^ + 0.951i^) or (0.309^ + 0.951iv)
- or (0.809v + 0.588i^) or (0.809v + 0.588iv)
- 4√(1v) = (√2/2^ + √2/2i^) or (√2/2v + √2/2i^) etc..
- 8√(1v) = 0.924^ + 0.383i^ etc.;
Conclusion
Try these examples of roots with our online calculator.
Next: Simple Roots
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