复数

基本概念

复数定义

由实数部分和虚数部分所组成的数，形如a＋bi 。

其中a、b为实数，i 为“虚数单位”，i² = -1，即虚数单位的平方等于-1。

a、b分别叫做复数a＋bi的实部和虚部。

当b=0时，a＋bi＝a 为实数；

当b≠0时，a＋bi 又称虚数；

当b≠0、a=0时，bi 称为纯虚数。

实数和虚数都是复数的子集。如同实数可以在数轴上表示一样复数也可以在平面上表示，复数x+yi以坐标点(x，y)来表示。表示复数的平面称为“复平面”。

复数相等

两个复数不能比较大小，但当个两个复数的实部和虚部分别相等时，即表示两个复数相等。

共轭复数

如果两个复数的实部相等，虚部互为相反数，那么这两个复数互为共轭复数。

复数的模

复数的实部与虚部的平方和的正的平方根的值称为该复数的模，数学上用与绝对值“|z|”相同的符号来表示。虽然从定义上是不相同的，但两者的物理意思都表示“到原点的距离”。

复数的四则运算

加法（减法）法则

复数的加法法则：设z1=a+bi,z2 =c+di是任意两个复数。两者和的实部是原来两个复数实部的和，它的虚部是原来两个虚部的和。两个复数的和依然是复数。

即(a+bi)±(c+di)=(a±c)+(b±d)

乘法法则

复数的乘法法则：把两个复数相乘，类似两个多项式相乘，结果中i²=-1，把实部与虚部分别合并。两个复数的积仍然是一个复数。

即(a+bi)(c+di)=(ac-bd)+(bc+ad)i

除法法则

复数除法法则：满足(c+di)(x+yi)=(a+bi)的复数x+yi(x,y∈R)叫复数a+bi除以复数c+di的商。

运算方法：可以把除法换算成乘法做，将分子分母同时乘上分母的共轭复数，再用乘法运算。

即(a+bi)/(c+di)=(a+bi)(c-di)/(c*c+d*d)=[(ac+bd)+(bc-ad)i]/(c*c+d*d)

复数的Rust代码实现

结构定义

Rust语言中，没有像python一样内置complex复数数据类型，我们可以用两个浮点数分别表示复数的实部和虚部，自定义一个结构数据类型，表示如下：

struct Complex {

    real: f64,

    imag: f64,

}

示例代码：

#[derive(Debug)]
struct Complex {
    real: f64,
    imag: f64,
}
impl Complex {  
    fn new(real: f64, imag: f64) -> Self {
        Complex { real, imag }  
    }
}
fn main() {  
    let z = Complex::new(3.0, 4.0);
    println!("{:?}", z);
    println!("{} + {}i", z.real, z.imag);
}

注意：#[derive(Debug)] 自动定义了复数结构的输出格式，如以上代码输出如下：

Complex { real: 3.0, imag: 4.0 }

3 + 4i

重载四则运算

复数数据结构不能直接用加减乘除来做复数运算，需要导入标准库ops的运算符：

use std::ops::{Add, Sub, Mul, Div, Neg};

Add, Sub, Mul, Div, Neg 分别表示加减乘除以及相反数，类似C++或者python语言中“重载运算符”的概念。

根据复数的运算法则，写出对应代码：

fn add(self, other: Complex) -> Complex {

    Complex {

        real: self.real + other.real,

        imag: self.imag + other.imag,

    }  

}  

fn sub(self, other: Complex) -> Complex {

    Complex {  

        real: self.real - other.real,

        imag: self.imag - other.imag,

    }  

} 

fn mul(self, other: Complex) -> Complex {  

    let real = self.real * other.real - self.imag * other.imag;

    let imag = self.real * other.imag + self.imag * other.real;

    Complex { real, imag }  

}  

fn div(self, other: Complex) -> Complex {

    let real = (self.real * other.real + self.imag * other.imag) / (other.real * other.real + other.imag * other.imag);

    let imag = (self.imag * other.real - self.real * other.imag) / (other.real * other.real + other.imag * other.imag);

    Complex { real, imag }

}

fn neg(self) -> Complex {

    Complex {

        real: -self.real,

        imag: -self.imag,

    }

}

Rust 重载运算的格式，请见如下示例代码：

use std::ops::{Add, Sub, Mul, Div, Neg};
#[derive(Clone, Debug, PartialEq)]
struct Complex {
    real: f64,
    imag: f64,
}
impl Complex {  
    fn new(real: f64, imag: f64) -> Self {
        Complex { real, imag }  
    }
  
    fn conj(&self) -> Self {
        Complex { real: self.real, imag: -self.imag }
    }
    fn abs(&self) -> f64 {
        (self.real * self.real + self.imag * self.imag).sqrt()
    }
}
fn abs(z: Complex) -> f64 {
    (z.real * z.real + z.imag * z.imag).sqrt()
}
impl Add for Complex {
    type Output = Complex;
    fn add(self, other: Complex) -> Complex {
        Complex {
            real: self.real + other.real,
            imag: self.imag + other.imag,
        }  
    }  
}  
  
impl Sub for Complex {
    type Output = Complex;
  
    fn sub(self, other: Complex) -> Complex {
        Complex {  
            real: self.real - other.real,
            imag: self.imag - other.imag,
        }  
    } 
}  
  
impl Mul for Complex {
    type Output = Complex;  
  
    fn mul(self, other: Complex) -> Complex {  
        let real = self.real * other.real - self.imag * other.imag;
        let imag = self.real * other.imag + self.imag * other.real;
        Complex { real, imag }  
    }  
}
impl Div for Complex {
    type Output = Complex;
  
    fn div(self, other: Complex) -> Complex {
        let real = (self.real * other.real + self.imag * other.imag) / (other.real * other.real + other.imag * other.imag);
        let imag = (self.imag * other.real - self.real * other.imag) / (other.real * other.real + other.imag * other.imag);
        Complex { real, imag }
    }
}  
  
impl Neg for Complex {
    type Output = Complex;
  
    fn neg(self) -> Complex {
        Complex {
            real: -self.real,
            imag: -self.imag,
        }
    }
}
fn main() {  
    let z1 = Complex::new(2.0, 3.0);
    let z2 = Complex::new(3.0, 4.0);
    let z3 = Complex::new(3.0, -4.0);
    // 复数的四则运算
    let complex_add = z1.clone() + z2.clone();
    println!("{:?} + {:?} = {:?}", z1, z2, complex_add);
    let complex_sub = z1.clone() - z2.clone();
    println!("{:?} - {:?} = {:?}", z1, z2, complex_sub);
    let complex_mul = z1.clone() * z2.clone();
    println!("{:?} * {:?} = {:?}", z1, z2, complex_mul);
    let complex_div = z2.clone() / z3.clone();
    println!("{:?} / {:?} = {:?}", z1, z2, complex_div);
    // 对比两个复数是否相等
    println!("{:?}", z1 == z2);
    // 共轭复数
    println!("{:?}", z2 == z3.conj());
    // 复数的相反数
    println!("{:?}", z2 == -z3.clone() + Complex::new(6.0,0.0));
    // 复数的模
    println!("{}", z1.abs());
    println!("{}", z2.abs());
    println!("{}", abs(z3));
}

输出：

Complex { real: 2.0, imag: 3.0 } + Complex { real: 3.0, imag: 4.0 } = Complex { real: 5.0, imag: 7.0 }

Complex { real: 2.0, imag: 3.0 } - Complex { real: 3.0, imag: 4.0 } = Complex { real: -1.0, imag: -1.0 }

Complex { real: 2.0, imag: 3.0 } * Complex { real: 3.0, imag: 4.0 } = Complex { real: -6.0, imag: 17.0 }

Complex { real: 2.0, imag: 3.0 } / Complex { real: 3.0, imag: 4.0 } = Complex { real: -0.28, imag: 0.96 }

false

true

true

3.605551275463989

5

5

示例代码中，同时还定义了复数的模 abs()，共轭复数 conj()。

两个复数的相等比较 z1 == z2，需要 #[derive(PartialEq)] 支持。

自定义 trait Display

复数结构的原始 Debug trait 表达的输出格式比较繁复，如：

Complex { real: 2.0, imag: 3.0 } + Complex { real: 3.0, imag: 4.0 } = Complex { real: 5.0, imag: 7.0 }

想要输出和数学中相同的表达(如 a + bi)，需要自定义一个 Display trait，代码如下：

impl std::fmt::Display for Complex {

    fn fmt(&self, formatter: &mut std::fmt::Formatter) -> std::fmt::Result {

        if self.imag == 0.0 {

            formatter.write_str(&format!("{}", self.real))

        } else {

            let (abs, sign) = if self.imag > 0.0 {  

                (self.imag, "+" )

            } else {

                (-self.imag, "-" )

            };

            if abs == 1.0 {

                formatter.write_str(&format!("({} {} i)", self.real, sign))

            } else {

                formatter.write_str(&format!("({} {} {}i)", self.real, sign, abs))

            }

        }

    }

}

输出格式分三种情况：虚部为0，正数和负数。另外当虚部绝对值为1时省略1仅输出i虚数单位。

完整代码如下：

use std::ops::{Add, Sub, Mul, Div, Neg};
#[derive(Clone, PartialEq)]
struct Complex {
    real: f64,
    imag: f64,
}
impl std::fmt::Display for Complex {
    fn fmt(&self, formatter: &mut std::fmt::Formatter) -> std::fmt::Result {
        if self.imag == 0.0 {
            formatter.write_str(&format!("{}", self.real))
        } else {
            let (abs, sign) = if self.imag > 0.0 {  
                (self.imag, "+" )
            } else {
                (-self.imag, "-" )
            };
            if abs == 1.0 {
                formatter.write_str(&format!("({} {} i)", self.real, sign))
            } else {
                formatter.write_str(&format!("({} {} {}i)", self.real, sign, abs))
            }
        }
    }
}
impl Complex {  
    fn new(real: f64, imag: f64) -> Self {
        Complex { real, imag }  
    }
  
    fn conj(&self) -> Self {
        Complex { real: self.real, imag: -self.imag }
    }
    fn abs(&self) -> f64 {
        (self.real * self.real + self.imag * self.imag).sqrt()
    }
}
fn abs(z: Complex) -> f64 {
    (z.real * z.real + z.imag * z.imag).sqrt()
}
impl Add for Complex {
    type Output = Complex;
    fn add(self, other: Complex) -> Complex {
        Complex {
            real: self.real + other.real,
            imag: self.imag + other.imag,
        }  
    }  
}  
  
impl Sub for Complex {
    type Output = Complex;
  
    fn sub(self, other: Complex) -> Complex {
        Complex {  
            real: self.real - other.real,
            imag: self.imag - other.imag,
        }  
    } 
}  
  
impl Mul for Complex {
    type Output = Complex;  
  
    fn mul(self, other: Complex) -> Complex {  
        let real = self.real * other.real - self.imag * other.imag;
        let imag = self.real * other.imag + self.imag * other.real;
        Complex { real, imag }  
    }  
}
impl Div for Complex {
    type Output = Complex;
  
    fn div(self, other: Complex) -> Complex {
        let real = (self.real * other.real + self.imag * other.imag) / (other.real * other.real + other.imag * other.imag);
        let imag = (self.imag * other.real - self.real * other.imag) / (other.real * other.real + other.imag * other.imag);
        Complex { real, imag }
    }
}  
  
impl Neg for Complex {
    type Output = Complex;
  
    fn neg(self) -> Complex {
        Complex {
            real: -self.real,
            imag: -self.imag,
        }
    }
}
fn main() {
    let z1 = Complex::new(2.0, 3.0);
    let z2 = Complex::new(3.0, 4.0);
    let z3 = Complex::new(3.0, -4.0);
    // 复数的四则运算
    let complex_add = z1.clone() + z2.clone();
    println!("{} + {} = {}", z1, z2, complex_add);
    let z = Complex::new(1.5, 0.5);
    println!("{} + {} = {}", z, z, z.clone() + z.clone());
    let complex_sub = z1.clone() - z2.clone();
    println!("{} - {} = {}", z1, z2, complex_sub);
    let complex_sub = z1.clone() - z1.clone();
    println!("{} - {} = {}", z1, z1, complex_sub);
    let complex_mul = z1.clone() * z2.clone();
    println!("{} * {} = {}", z1, z2, complex_mul);
    let complex_mul = z2.clone() * z3.clone();
    println!("{} * {} = {}", z2, z3, complex_mul);
    let complex_div = z2.clone() / z3.clone();
    println!("{} / {} = {}", z1, z2, complex_div);
    let complex_div = Complex::new(1.0,0.0) / z2.clone();
    println!("1 / {} = {}", z2, complex_div);
    // 对比两个复数是否相等
    println!("{:?}", z1 == z2);
    // 共轭复数
    println!("{:?}", z2 == z3.conj());
    // 复数的相反数
    println!("{:?}", z2 == -z3.clone() + Complex::new(6.0,0.0));
    // 复数的模
    println!("{}", z1.abs());
    println!("{}", z2.abs());
    println!("{}", abs(z3));
}

输出：

(2 + 3i) + (3 + 4i) = (5 + 7i)

(1.5 + 0.5i) + (1.5 + 0.5i) = (3 + i)

(2 + 3i) - (3 + 4i) = (-1 - i)

(2 + 3i) - (2 + 3i) = 0

(2 + 3i) * (3 + 4i) = (-6 + 17i)

(3 + 4i) * (3 - 4i) = 25

(2 + 3i) / (3 + 4i) = (-0.28 + 0.96i)

1 / (3 + 4i) = (0.12 - 0.16i)

false

true

true

3.605551275463989

5

5

---

小结

如此，复数的四则运算基本都实现了，当然复数还有三角表示式和指数表示式，根据它们的数学定义写出相当代码应该不是很难。有了复数三角式，就能方便地定义出复数的开方运算，有空可以写写这方面的代码。

本文完