R1 – Roots – Definition

This post on R1 roots covers the definition of roots and how roots are derived from the multiplication table. It includes examples of square, cube and higher roots.

Definition

Wikipedia defines a root as: ‘an nth root of a number x is a number r (the root) which, when raised to the power of the positive integer n, yields 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 Sign. Identify the Counter root by the fact that the Opposite Value is not within brackets as written in ^?√n^?. For example: ²√1^{^} = 1^{^} or ²√1^v = 1^v . In other words, ignore the rotation expressed by the Opposite Sign 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.  To specify the Full root of the Opposite Value, the Opposite Value needs to be enclosed by brackets. For example: √(4^{^}) = 2^{^} or 2^v.

Calculate the full root as the root of the Counter times the root of the Unitary. For example: √(9^{^}) = √9*√(1^{^})  = 3*1^{^} = 3^{^} or 3*1^v =3^v.

Designate the degree of roots in the same way as for classical maths. So ³√8^{^} = 2^{^} and is the cube root of the Counter 8 and holds the same Opposite Sign.

                     

Multiplication Table

The multiplication table for R1 defines:

• 1^{^2} = 1^{^}*1^{^} = 1^{^}, 1^{^3} = 1^{^}*1^{^}*1^{^} = 1^{^}, and so on
  • => 1^{^n} = 1^{^}
• 1^v2= 1^v*1^v = 1^{^},    1^v3 =  1^v*1^{^} = 1^v, 1^v4 =  1^v*1^v = 1^{^} and so on
  • => for even values of n, 1^vn = 1^{^}
  • => for odd values of n, 1^vn = 1^v.

The √(1^v) does not exist in R1 because no Opposite Value multiplied by itself results in 1^v.

The following can be deduced from the above:

• √(1^{^}) = 1^{^} or 1^v
• ⁿ√(1^{^}) = 1^{^} or 1^v for even values of n
• ⁿ√(1^v) = 1^v  for odd values of n
• √(9^{^}) = 3^{^}or 3^v
• √9^{^}= 3^{^} as this is the square root of a Counter
• √(9^v) does not exist in R1
• √9^v = 3^v  as this is the square root of a Counter.  

Conclusion

Try these examples of R1 roots with our online calculator.

Next: Roots of Expressions

Previous: Division