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Julia 复数和分数

# Complex and Rational Numbers

Julia includes predefined types for both complex and rational numbers, and supports
all the standard Mathematical Operations and Elementary Functions on them. [Conversion and Promotion](@ref 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. (Using mathematicians' i or engineers' j for this global constant were rejected since they are such popular index variable names.) Since Julia allows numeric literals to be [juxtaposed with identifiers as coefficients](@ref 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
2.729624464784009 - 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
0.20689655172413796 + 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> z = 1 + 2im
1 + 2im

julia> real(1 + 2im) # real part of z
1

julia> imag(1 + 2im) # imaginary part of z
2

julia> conj(1 + 2im) # complex conjugate of z
1 - 2im

julia> abs(1 + 2im) # absolute value of z
2.23606797749979

julia> abs2(1 + 2im) # squared absolute value
5

julia> angle(1 + 2im) # phase angle in radians
1.1071487177940904

As usual, the absolute value (abs) of a complex number is its distance from zero.
abs2 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 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, sqrt behaves differently
when applied to -1 versus -1 + 0im even though -1 == -1 + 0im:

julia> sqrt(-1)
ERROR: DomainError with -1.0:
sqrt will only return a complex result if called with a complex argument. Try sqrt(Complex(x)).
Stacktrace:
[...]

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

The [literal numeric coefficient notation](@ref man-numeric-literal-coefficients) does not work when constructing a 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. Instead, use the more efficient complex function to construct
a complex value directly from its real and imaginary parts:

julia> a = 1; b = 2; 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
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 // 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 numerator and denominator
functions:

julia> numerator(2//3)
2

julia> denominator(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 easily be 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> a = 1; b = 2;

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 NaN rational value, however, is invalid:

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

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