9.11 Complex

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. For more information about the numerical and graphical aspects of complex numbers, see ugProblemNumeric .

Complex objects are created by the complexcomplexComplex operation.

a := complex(4/3,5/2)

\[\]

43+52i

_{Type: Complex Fraction Integer}

b := complex(4/3,-5/2)

\[\]

43-52i

_{Type: Complex Fraction Integer}

The standard arithmetic operations are available.

a + b

\[\]

83

_{Type: Complex Fraction Integer}

a - b

\[\]

5i

_{Type: Complex Fraction Integer}

a * b

\[\]

28936

_{Type: Complex Fraction Integer}

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

a / b

\[\]

-161289+240289i

_{Type: Complex Fraction Integer}

Use a conversion (ugTypesConvertPage in Section ugTypesConvertNumber ) to view the last object as a fraction of complex integers.

% :: Fraction Complex Integer

\[\]

-15+8i15+8i

_{Type: Fraction Complex Integer}

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

3.4 + 6.7 * %i

\[\]

3.4+6.7i

_{Type: Complex Float}

You can also compute the conjugateconjugateComplex and normnormComplex of a complex number.

conjugate a

\[\]

43-52i

_{Type: Complex Fraction Integer}

norm a

\[\]

28936

_{Type: Fraction Integer}

The realrealComplex and imagimagComplex operations are provided to extract the real and imaginary parts, respectively.

real a

\[\]

43

_{Type: Fraction Integer}

imag a

\[\]

52

_{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)

\[\]

143-39i

_{Type: Complex Integer}

You can factorfactorComplex Gaussian integers.

factor(13 - 13*%i)

\[\]

-(1+i)(2+3i)(3+2i)

_{Type: Factored Complex Integer}

factor complex(2,0)

\[\]

-i(1+i)2

_{Type: Factored Complex Integer}