6.10.7  Multiplication by the complex conjugate: mult_c_conjugate

The denominator of a complex expression can be made a real number by multiplying the numerator and denominator of the expression by the complex conjugate of the denominator. The mult_c_conjugate can perform this multiplication.

• mult_c_conjugate takes one argument:
  expr, a complex expression.
• mult_c_conjugate(expr) returns the following:
  • If expr is a fraction with a complex (non-real) denominator, then this expression is returned with the numerator and denominator multiplied by the complex conjugate of the denominator.
  • If expr is a fraction with a real denominator (if expr is not a fraction, it is regarded as a fraction with a denominator of 1), then this expression is returned with the numerator and denominator multiplied by the complex conjugate of the numerator.

Examples.

• Input:
  mult_c_conjugate((2+i)/(2+3*i))
  Output:

  ⎛ ⎝ 2+i ⎞ ⎠ ⎛ ⎝ 2−3 i ⎞ ⎠ ⎛ ⎝ 2+3 i ⎞ ⎠ ⎛ ⎝ 2−3 i ⎞ ⎠

• Input:
  mult_c_conjugate((2+i)/2)
  Output:

  ⎛ ⎝ 2+i ⎞ ⎠ ⎛ ⎝ 2−i ⎞ ⎠ 2  ⎛ ⎝ 2−i ⎞ ⎠