July 22, 2024
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R2 – Division – Advanced

This post covers more advanced math division in R2 including multiplicative inverses and the use of simultaneous equations.

R2 Division Table

/1^1vi^iv
1^1^1vivi^
1v1v1^i^iv
i^i^iv1^1v
ivivi^1v1^

Inverses

The theorems of multiplicative inverses describe the features of the multiplicative inverse. An Opposite Value in R2 is the multiplicative inverse of another Opposite Value when the product of the two Opposite Values equals 1^.

a*a-1 = 1^ where a-1 = 1?^/a. For example: 1^/i^ is the multiplicative inverse of i^ as i^*1^/i^ = 1^.

  • 1^*1^-1 = 1^(1^/1^)= 1^*1^ = 1^
  • 1v*1v-1 = 1v(1^/1) = 1v*1v = 1^
  • i^*i^-1 = i^(1^/i^)= i^*iv = 1^
  • iv*iv-1 = iv(1^/i) = iv*i^ = 1^
  • 3i^*3i^-1 = 3i^(1^/3i^) = 3i^/3i^ = 1^
  • 3iv*3iv-1 = 3iv(1^/3iv) = 3iv/3iv = 1^

Multiplication by the multiplicative inverse of an Opposite Value is the equivalent of dividing by the Opposite value. For example:

  • The multiplicative inverse of 3iv is 1^/3iv = 0.3333i^
  • 9i^/3iv =3v = 9i^*0.3333i^ = 3v

There are multiplicative inverses for expressions:

  • (4^ + 3iv)-1 = 1^/(4^ + 3iv) = (0.16^ + 0.12i^)
    • (42.5v + 87.32iv)/(4^ + 3iv) = (3.68^ + 19.07iv)
    • (0.16^ + 0.12i^)*(42.5v + 87.32iv) = (3.68^ + 19.07iv)

R2 Advanced Division using Simultaneous Equations

A more advanced method of R2 division uses simultaneous equations. Take the expression (10v + 14iv)/(18^ + 3iv). The answer to the division is (0.4144v + 0.8468iv). This is calculated by solving the equations below that are derived with the help of the R2 multiplication table:

  • A:           18x + 3y + 10  = 0   and
  • B:           18y + 3x +14 = 0

Deriving Equation A

It is known that (10v + 14iv)/(18^ + 3iv)  = x? + yi?. This implies that:

  • (10v + 14iv) =   (18^ + 3iv)( x? + yi?
    • = 18^x?+ 18^yi? + 3ivx? + 3ivyi?.

Because x represents the ^ or v Opposite Value and y represents the i^ or iv Opposite Value, we can deduce from the R2 multiplication table that 18^x?+ 3ivyi? + 10^ = 0 and 18^yi? + 3ivx? + 14i^ = 0.

Equation A is based on the ^ or v part of the result which in this case is 0.4144v. The R2 multiplication table table indicates that 18^ needs to be multiplied with an ^ or v Opposite Value in order to produce a ^ or v result. For example:

  • 18^*2^  = 36^ and 18^*2v = 36v
    • Note that here the signs of the results are the same as the operand
      • The result for the operand 2^  is 36^and for 2v  is 36v.

The multiplication table indicates that 3iv needs to be multiplied with an i^ or iv in order to produce a ^ or v result. For example:

  • 3iv*2i^  = 6^ and 3iv*2iv = 6v
    • Note that here the signs of the results are the same as the operand
      • The result for the operand 2i^  is 6^and for 2iv  is 6v

This gives the simultaneous equation A:

  • 18^x + 3ivy + 10^  = 0

Deriving Equation B

Equation B is based on the i^ or iv part of the result which in this case is 0.8468iv. The R2 multiplication table indicates that 18^ needs to be multiplied with an i^ or iv Opposite Value in order to produce an i^ or iv result. For example:

  • 18^*2i^  = 36i^ and 18^*2iv = 36iv
    • Note that here the signs of the results are the same as the operand
      • The result for the operand 2i^  is 36i^and for 2iv  is 36iv

The multiplication table indicates that 3iv needs to be multiplied with a ^ or v in order to produce an i^ or iv result. For example:

  • 3iv*2^ = 6iv and 3iv*2v = 6i^
    • Note that here the signs of the results are different to the operands
      • Each operand and result (2^ and 6iv) and (2v and 6i^) has a different Opposite Sign.
    • The flip in 3x in the simultaneous equation represents that the sign of the result is different to the operand.

This gives the simultaneous equation B:

  • 18^y + 3ivx +14^ = 0

Solving the Equations

The two simultaneous equations 18^x + 3ivy + 10^  = 0 and 18^y + 3ivx +14^ = 0 can now be solved using the classical methodology. The Opposite Types and Signs in the equation can be ignored, so the equations to be solved are:

  • 18x + 3y + 10 = 0
  • 18y + 3x +14 = 0

 

This results in x = 0.4144v and y = 0.8468iv which satisfy the equations as follows:

  • 18^*0.4144v+ 3iv*0.8468iv + 10^ = 0
    • = 7.46v + 2.54v +10^ = 0
  • 18^*0.8468iv+ 3iv*0.4144v + 14i^ = 0
    • = 15.24iv + 1.24i^ +14i^ = 0

R2 Advanced Division using Multiplicative Inverse

The Multiplicative Inverse of (18^ + 3iv) gives the same result:

  • (18^ + 3iv)-1 = 1^/(18^ + 3iv)
    • = (0.05405^ + 0.00901i^)
    • (0.05405^ + 0.00901i^)*(10v + 14iv) = (0.4144v + 0.8468iv)

Conclusion

Try these examples of advanced R2 division with our online calculator.

Next: Roots

Previous: Flipping

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