Complex and Rational Numbers

Julia ships with predefined types representing both complex and rational numbers, and supports all standard mathematical operations on them. 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 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 juxtaposed with identifiers as 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
2.729624464784009 - 6.9606644595719im

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
0.20689655172413793 + 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

As is common, the absolute value 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 full gamut of other 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 - 3.0518977991517997im

julia> exp(1 + 2im)
-1.1312043837568138 + 2.471726672004819im

julia> sinh(1 + 2im)
-0.48905625904129374 + 1.4031192506220407im

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:

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:284

julia> sqrt(-1 + 0im)
0.0 + 1.0im

The literal numeric coefficient notation 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

Hoever, this is not recommended; Use the 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.

Inf and NaN propagate through complex numbers in the real and imaginary parts of a complex number as described in the Special floating-point values section:

julia> 1 + Inf*im
complex(1.0,Inf)

julia> 1 + NaN*im
complex(1.0,NaN)

Rational Numbers

Julia has a rational number type to represent exact ratios of integers. Rationals are constructed using the // 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 num and 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
Inf

julia> -3//0
-Inf

julia> typeof(ans)
Rational{Int64} (constructor with 1 method)

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

julia> 0//0
ERROR: invalid rational: 0//0
 in Rational at rational.jl:7
 in // at rational.jl:17

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//7im

julia> 2//7 * (1.5 + 2im)
0.42857142857142855 + 0.5714285714285714im

julia> 3//2 / (1 + 2im)
3//10 - 3//5im

julia> 1//2 + 2im
1//2 + 2//1im

julia> 1 + 2//3im
1//1 - 2//3im

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