# Multiplication by the complex conjugate: mult_c_conjugate

### 5.10.7 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 | ⎞ ⎠