complex-and-rational-numbers.rst

.. currentmodule:: Base

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.

Complex Numbers

The global constant :const:`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:

julia> 1 + 2im
1 + 2im

You can perform all the standard arithmetic operations with complex numbers:

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:

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:

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 (:func:`abs`) of a complex number is its distance from zero. :func:`abs2` gives the square of the absolute value, and is of particular use for complex numbers where it avoids taking a square root. :func:`angle` 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:

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, :func:`sqrt` behaves differently when applied to -1 versus -1 + 0im even though -1 == -1 + 0im:

julia> sqrt(-1)
ERROR: DomainError:
sqrt will only return a complex result if called with a complex argument. Try sqrt(complex(x)).
 in sqrt(::Int64) at ./math.jl:278
 ...

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:

julia> a = 1; b = 2; a + b*im
1 + 2im

However, this is not recommended; Use the :func:`complex` function instead to construct a complex value directly from its real and imaginary parts.:

julia> complex(a,b)
1 + 2im

This construction avoids the multiplication and addition operations.

:const:`Inf` and :const:`NaN` propagate through complex numbers in the real and imaginary parts of a complex number as described in the :ref:`man-special-floats` section:

julia> 1 + Inf*im
1.0 + Inf*im

julia> 1 + NaN*im
1.0 + NaN*im

Rational Numbers

Julia has a rational number type to represent exact ratios of integers. Rationals are constructed using the :obj:`//` operator:

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:

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 :func:`num` and :func:`den` functions:

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:

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:

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:

julia> isequal(float(a//b), a/b)
true

Constructing infinite rational values is acceptable:

julia> 5//0
1//0

julia> -3//0
-1//0

julia> typeof(ans)
Rational{Int64}

Trying to construct a :const:`NaN` rational value, however, is not:

julia> 0//0
ERROR: ArgumentError: invalid rational: zero(Int64)//zero(Int64)
 in Rational{Int64}(::Int64, ::Int64) at ./rational.jl:8
 in //(::Int64, ::Int64) at ./rational.jl:22
 ...

As usual, the promotion system makes interactions with other numeric types effortless:

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