====================================================================
Complex
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The ``Complex`` constructor implements complex objects over a commutative
ring R.  Typically, the ring R is Integer, Fraction Integer, Float or
DoubleFloat.  R can also be a symbolic type, like Polynomial Integer.

Complex objects are created by the complex operation. ::

  a := complex(4/3,5/2)
   4   5
   - + - %i
   3   2
                        Type: Complex Fraction Integer

  b := complex(4/3,-5/2)
   4   5
   - - - %i
   3   2
                        Type: Complex Fraction Integer

The standard arithmetic operations are available. ::

  a + b
   8
   -
   3
                        Type: Complex Fraction Integer

  a - b
   5%i
                        Type: Complex Fraction Integer

  a * b
   289
   ---
    36
                        Type: Complex Fraction Integer

If  R is a field, you can also divide the complex objects. ::

  a / b
     161   240
   - --- + --- %i
     289   289
                        Type: Complex Fraction Integer


We can view the last object as a fraction of complex integers. ::

  % :: Fraction Complex Integer
   - 15 + 8%i
   ----------
     15 + 8%i
                       Type: Fraction Complex Integer

The predefined macro %i is defined to be complex(0,1). ::

  3.4 + 6.7 * %i
   3.4 + 6.7 %i
                      Type: Complex Float

You can also compute the conjugate and norm of a complex number. ::

  conjugate a
   4   5
   - - - %i
   3   2
                      Type: Complex Fraction Integer

  norm a
   289
   ---
    36
                      Type: Fraction Integer

The real and imag operations are provided to extract the real and
imaginary parts, respectively. ::

  real a
   4
   -
   3
                      Type: Fraction Integer

  imag a
   5
   -
   2
                      Type: Fraction Integer

The domain Complex Integer is also called the Gaussian integers.  If R
is the integers (or, more generally, a EuclideanDomain), you can compute 
greatest common divisors. ::

  gcd(13 - 13*%i,31 + 27*%i)
   5 + %i
                      Type: Complex Integer

You can also compute least common multiples. ::

  lcm(13 - 13*%i,31 + 27*%i)
   143 - 39%i
                      Type: Complex Integer

You can factor Gaussian integers. ::

  factor(13 - 13*%i)
   - (1 + %i)(2 + 3%i)(3 + 2%i)
                      Type: Factored Complex Integer

  factor complex(2,0)
                2
   - %i (1 + %i)
                      Type: Factored Complex Integer

See Also

* ``)show Complex``