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基本概念
复数定义
由实数部分和虚数部分所组成的数#xff0c;形如a#xff0b;bi 。
其中a、b为实数#xff0c;i 为“虚数单位”#xff0c;i -1#xff0c;即虚数单位的平方等于-1。
a、b分别叫做复数a#xff0b;bi的实部和虚部。
当b0时#xff0c;a形如abi 。
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abi的实部和虚部。
当b0时abia 为实数 当b≠0时abi 又称虚数 当b≠0、a0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xyi以坐标点(xy)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z1abi,z2 cdi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(abi)±(cdi)(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(abi)(cdi)(ac-bd)(bcad)i
除法法则
复数除法法则满足(cdi)(xyi)(abi)的复数xyi(x,y∈R)叫复数abi除以复数cdi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(abi)/(cdi)(abi)(c-di)/(c*cd*d)[(acbd)(bc-ad)i]/(c*cd*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 AddComplex for Complex {type Output Complex;fn add(self, other: Complex) - Complex {Complex {real: self.real other.real,imag: self.imag other.imag,} }
} impl SubComplex for Complex {type Output Complex;fn sub(self, other: Complex) - Complex {Complex { real: self.real - other.real,imag: self.imag - other.imag,} }
} impl MulComplex 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 DivComplex 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 AddComplex for Complex {type Output Complex;fn add(self, other: Complex) - Complex {Complex {real: self.real other.real,imag: self.imag other.imag,} }
} impl SubComplex for Complex {type Output Complex;fn sub(self, other: Complex) - Complex {Complex { real: self.real - other.real,imag: self.imag - other.imag,} }
} impl MulComplex 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 DivComplex 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
