.. _man-complex-and-rational-numbers:
******************************
Complex and Rational Numbers
******************************
Julia ships with predefined types representing both complex and rational
numbers, and supports all :ref:`standard mathematical operations
<man-mathematical-operations>` on them. :ref:`man-conversion-and-promotion`
are defined so that operations on any combination of
predefined numeric types, whether primitive or composite, behave as
expected.
.. _man-complex-numbers:
Complex Numbers
---------------
The global constant ``im`` is bound to the complex number *i*,
representing the principal square root of -1. It was deemed harmful to
co-opt the name ``i`` for a global constant, since it is such a popular
index variable name. Since Julia allows numeric literals to be
:ref:`juxtaposed with identifiers as coefficients
<man-numeric-literal-coefficients>`,
this binding suffices to provide convenient syntax for complex numbers,
similar to the traditional mathematical notation:
.. doctest::
julia> 1 + 2im
1 + 2im
You can perform all the standard arithmetic operations with complex
numbers:
.. doctest::
julia> (1 + 2im)*(2 - 3im)
8 + 1im
julia> (1 + 2im)/(1 - 2im)
-0.6 + 0.8im
julia> (1 + 2im) + (1 - 2im)
2 + 0im
julia> (-3 + 2im) - (5 - 1im)
-8 + 3im
julia> (-1 + 2im)^2
-3 - 4im
julia> (-1 + 2im)^2.5
2.7296244647840084 - 6.960664459571898im
julia> (-1 + 2im)^(1 + 1im)
-0.27910381075826657 + 0.08708053414102428im
julia> 3(2 - 5im)
6 - 15im
julia> 3(2 - 5im)^2
-63 - 60im
julia> 3(2 - 5im)^-1.0
0.20689655172413796 + 0.5172413793103449im
The promotion mechanism ensures that combinations of operands of
different types just work:
.. doctest::
julia> 2(1 - 1im)
2 - 2im
julia> (2 + 3im) - 1
1 + 3im
julia> (1 + 2im) + 0.5
1.5 + 2.0im
julia> (2 + 3im) - 0.5im
2.0 + 2.5im
julia> 0.75(1 + 2im)
0.75 + 1.5im
julia> (2 + 3im) / 2
1.0 + 1.5im
julia> (1 - 3im) / (2 + 2im)
-0.5 - 1.0im
julia> 2im^2
-2 + 0im
julia> 1 + 3/4im
1.0 - 0.75im
Note that ``3/4im == 3/(4*im) == -(3/4*im)``, since a literal
coefficient binds more tightly than division.
Standard functions to manipulate complex values are provided:
.. doctest::
julia> real(1 + 2im)
1
julia> imag(1 + 2im)
2
julia> conj(1 + 2im)
1 - 2im
julia> abs(1 + 2im)
2.23606797749979
julia> abs2(1 + 2im)
5
julia> angle(1 + 2im)
1.1071487177940904
As usual, the absolute value (``abs``) of a complex number is its
distance from zero. The ``abs2`` function gives the square of the
absolute value, and is of particular use for complex numbers where it
avoids taking a square root. The ``angle`` function returns the phase
angle in radians (also known as the *argument* or *arg* function). The
full gamut of other :ref:`man-elementary-functions` is also defined
for complex numbers:
.. doctest::
julia> sqrt(1im)
0.7071067811865476 + 0.7071067811865475im
julia> sqrt(1 + 2im)
1.272019649514069 + 0.7861513777574233im
julia> cos(1 + 2im)
2.0327230070196656 - 3.0518977991518im
julia> exp(1 + 2im)
-1.1312043837568135 + 2.4717266720048188im
julia> sinh(1 + 2im)
-0.4890562590412937 + 1.4031192506220405im
Note that mathematical functions typically return real values when applied
to real numbers and complex values when applied to complex numbers.
For example, ``sqrt`` behaves differently when applied to ``-1``
versus ``-1 + 0im`` even though ``-1 == -1 + 0im``:
.. doctest::
julia> sqrt(-1)
ERROR: DomainError
sqrt will only return a complex result if called with a complex argument.
try sqrt(complex(x))
in sqrt at math.jl:131
julia> sqrt(-1 + 0im)
0.0 + 1.0im
The :ref:`literal numeric coefficient notation <man-numeric-literal-coefficients>`
does not work when constructing complex number from variables. Instead, the
multiplication must be explicitly written out:
.. doctest::
julia> a = 1; b = 2; a + b*im
1 + 2im
However, this is *not* recommended; Use the ``complex`` function instead to
construct a complex value directly from its real and imaginary parts.:
.. doctest::
julia> complex(a,b)
1 + 2im
This construction avoids the multiplication and addition operations.
``Inf`` and ``NaN`` propagate through complex numbers in the real
and imaginary parts of a complex number as described in the
:ref:`man-special-floats` section:
.. doctest::
julia> 1 + Inf*im
1.0 + Inf*im
julia> 1 + NaN*im
1.0 + NaN*im
.. _man-rational-numbers:
Rational Numbers
----------------
Julia has a rational number type to represent exact ratios of integers.
Rationals are constructed using the ``//`` operator:
.. doctest::
julia> 2//3
2//3
If the numerator and denominator of a rational have common factors, they
are reduced to lowest terms such that the denominator is non-negative:
.. doctest::
julia> 6//9
2//3
julia> -4//8
-1//2
julia> 5//-15
-1//3
julia> -4//-12
1//3
This normalized form for a ratio of integers is unique, so equality of
rational values can be tested by checking for equality of the numerator
and denominator. The standardized numerator and denominator of a
rational value can be extracted using the ``num`` and ``den`` functions:
.. doctest::
julia> num(2//3)
2
julia> den(2//3)
3
Direct comparison of the numerator and denominator is generally not
necessary, since the standard arithmetic and comparison operations are
defined for rational values:
.. doctest::
julia> 2//3 == 6//9
true
julia> 2//3 == 9//27
false
julia> 3//7 < 1//2
true
julia> 3//4 > 2//3
true
julia> 2//4 + 1//6
2//3
julia> 5//12 - 1//4
1//6
julia> 5//8 * 3//12
5//32
julia> 6//5 / 10//7
21//25
Rationals can be easily converted to floating-point numbers:
.. doctest::
julia> float(3//4)
0.75
Conversion from rational to floating-point respects the following
identity for any integral values of ``a`` and ``b``, with the exception
of the case ``a == 0`` and ``b == 0``:
.. doctest::
julia> isequal(float(a//b), a/b)
true
Constructing infinite rational values is acceptable:
.. doctest::
julia> 5//0
1//0
julia> -3//0
-1//0
julia> typeof(ans)
Rational{Int64} (constructor with 1 method)
Trying to construct a ``NaN`` rational value, however, is not:
.. doctest::
julia> 0//0
ERROR: invalid rational: 0//0
in Rational at rational.jl:6
in // at rational.jl:15
As usual, the promotion system makes interactions with other numeric
types effortless:
.. doctest::
julia> 3//5 + 1
8//5
julia> 3//5 - 0.5
0.09999999999999998
julia> 2//7 * (1 + 2im)
2//7 + 4//7*im
julia> 2//7 * (1.5 + 2im)
0.42857142857142855 + 0.5714285714285714im
julia> 3//2 / (1 + 2im)
3//10 - 3//5*im
julia> 1//2 + 2im
1//2 + 2//1*im
julia> 1 + 2//3im
1//1 - 2//3*im
julia> 0.5 == 1//2
true
julia> 0.33 == 1//3
false
julia> 0.33 < 1//3
true
julia> 1//3 - 0.33
0.0033333333333332993
