Complex and Rational Numbers

    The global constant im is bound to the complex number i, representing the principal square root of -1. (Using mathematicians’ or engineers’ j for this global constant was rejected since they are such popular index variable names.) Since Julia allows numeric literals to be , this binding suffices to provide convenient syntax for complex numbers, similar to the traditional mathematical notation:

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

    1. julia> (1 + 2im)*(2 - 3im)
    2. 8 + 1im
    4. julia> (1 + 2im)/(1 - 2im)
    5. -0.6 + 0.8im
    7. julia> (1 + 2im) + (1 - 2im)
    8. 2 + 0im
    10. julia> (-3 + 2im) - (5 - 1im)
    11. -8 + 3im
    13. julia> (-1 + 2im)^2
    14. -3 - 4im
    16. julia> (-1 + 2im)^2.5
    17. 2.729624464784009 - 6.9606644595719im
    19. julia> (-1 + 2im)^(1 + 1im)
    20. -0.27910381075826657 + 0.08708053414102428im
    22. julia> 3(2 - 5im)
    23. 6 - 15im
    25. julia> 3(2 - 5im)^2
    26. -63 - 60im
    28. julia> 3(2 - 5im)^-1.0
    29. 0.20689655172413796 + 0.5172413793103449im

    The promotion mechanism ensures that combinations of operands of different types just work:

    1. julia> 2(1 - 1im)
    2. 2 - 2im
    4. julia> (2 + 3im) - 1
    5. 1 + 3im
    7. julia> (1 + 2im) + 0.5
    8. 1.5 + 2.0im
    10. julia> (2 + 3im) - 0.5im
    11. 2.0 + 2.5im
    13. julia> 0.75(1 + 2im)
    14. 0.75 + 1.5im
    16. julia> (2 + 3im) / 2
    17. 1.0 + 1.5im
    19. julia> (1 - 3im) / (2 + 2im)
    20. -0.5 - 1.0im
    22. julia> 2im^2
    23. -2 + 0im
    25. julia> 1 + 3/4im
    26. 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:

    1. 1 + 2im
    3. julia> real(1 + 2im) # real part of z
    4. 1
    6. julia> imag(1 + 2im) # imaginary part of z
    7. 2
    9. julia> conj(1 + 2im) # complex conjugate of z
    10. 1 - 2im
    12. julia> abs(1 + 2im) # absolute value of z
    13. 2.23606797749979
    15. julia> abs2(1 + 2im) # squared absolute value
    16. 5
    18. julia> angle(1 + 2im) # phase angle in radians
    19. 1.1071487177940904

    As usual, the absolute value (abs) of a complex number is its distance from zero. gives the square of the absolute value, and is of particular use for complex numbers since it avoids taking a square root. angle returns the phase angle in radians (also known as the argument or arg function). The full gamut of other is also defined for complex numbers:

    1. julia> sqrt(1im)
    2. 0.7071067811865476 + 0.7071067811865475im
    4. julia> sqrt(1 + 2im)
    5. 1.272019649514069 + 0.7861513777574233im
    7. julia> cos(1 + 2im)
    8. 2.0327230070196656 - 3.0518977991517997im
    10. julia> exp(1 + 2im)
    11. -1.1312043837568135 + 2.4717266720048188im
    13. julia> sinh(1 + 2im)
    14. -0.4890562590412937 + 1.4031192506220405im
    1. julia> sqrt(-1)
    2. ERROR: DomainError with -1.0:
    3. sqrt will only return a complex result if called with a complex argument. Try sqrt(Complex(x)).
    4. Stacktrace:
    5. [...]
    7. julia> sqrt(-1 + 0im)
    8. 0.0 + 1.0im

    The literal numeric coefficient notation does not work when constructing a complex number from variables. Instead, the multiplication must be explicitly written out:

    However, this is not recommended. Instead, use the more efficient function to construct a complex value directly from its real and imaginary parts:

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

    This construction avoids the multiplication and addition operations.

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

    1. julia> 1 + Inf*im
    2. 1.0 + Inf*im
    4. julia> 1 + NaN*im
    5. 1.0 + NaN*im

    Rational Numbers

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

    1. julia> 2//3
    2. 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:

    1. julia> 6//9
    2. 2//3
    4. julia> -4//8
    5. -1//2
    7. julia> 5//-15
    9. julia> -4//-12
    10. 1//3
    1. julia> numerator(2//3)
    2. 2
    4. julia> denominator(2//3)
    5. 3

    Direct comparison of the numerator and denominator is generally not necessary, since the standard arithmetic and comparison operations are defined for rational values:

    Rationals can easily be converted to floating-point numbers:

    1. julia> float(3//4)
    2. 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:

    1. julia> a = 1; b = 2;
    3. julia> isequal(float(a//b), a/b)
    4. true

    Constructing infinite rational values is acceptable:

    1. julia> 5//0
    2. 1//0
    4. julia> x = -3//0
    5. -1//0
    7. julia> typeof(x)
    8. Rational{Int64}

    Trying to construct a NaN rational value, however, is invalid:

    1. julia> 0//0
    2. ERROR: ArgumentError: invalid rational: zero(Int64)//zero(Int64)
    3. Stacktrace:
    4. [...]

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

    1. julia> 3//5 + 1
    2. 8//5
    4. julia> 3//5 - 0.5
    5. 0.09999999999999998
    7. julia> 2//7 * (1 + 2im)
    8. 2//7 + 4//7*im
    10. julia> 2//7 * (1.5 + 2im)
    11. 0.42857142857142855 + 0.5714285714285714im
    13. julia> 3//2 / (1 + 2im)
    14. 3//10 - 3//5*im
    16. julia> 1//2 + 2im
    17. 1//2 + 2//1*im
    19. julia> 1 + 2//3im
    20. 1//1 - 2//3*im
    22. julia> 0.5 == 1//2
    23. true
    25. julia> 0.33 == 1//3
    26. false
    28. julia> 0.33 < 1//3
    29. true