如何将指数复数转换为矩形？

复数是以 a+ib 形式表示的数值表达式，其中 a 和 b 是实数，i 代表虚数。考虑一个复数 2 + 3i，在这个表达式中，2 和 3 是整数或实数，而“i”表示虚数。

复数的形式

根据表示的方式，这些复数分为三种不同的形式。它们是矩形形式、极坐标形式和指数形式。

Forms	Formula
Rectangular form	z = a + ib
Polar form	z = r(cosθ + isinθ)
exponential form	z = reiθ

复数的类型

复数的标准表示以 a + ib 的形式给出。并且，根据这种表示，复数分为四种不同的类型。这四种类型是零复数、纯实数、纯虚数和虚数。

Types of the complex number	Requirement	Example
Zero complex number	a = 0 and b = 0	0 (zero)
Purely real number	a ≠ 0 and b = 0	2, 3, 4, 5, 6, 7
Purely imaginary number	a = 0 and b ≠ 0	-3i, 4i, 7i
Imaginary number	a ≠ 0 and b ≠ 0	(-2 + i)(1 – i)

如何将指数复数转换为矩形？

解决方案：

Exponential complex numbers are given in the form of re^iθ where r stands for the amplitude of the expression and θ is the phase that is expressed in unit radian.

At the same time, rectangular forms of a complex number are expressed in form of x + iy where x and y are the real integers and ‘i’ represents an imaginary number.

Derivation of rectangular form

There is an exponential form

= re^iθ

Where θ is the phase expressed in unit radians.

Now, firstly determine the value of x and y from the given expression to convert it in rectangular form x + iy

rcos(θ) = x

rsin(θ) = y

Hence, the rectangular form would be

x + iy

示例问题

问题1：将复数20e ^1.95i转换成矩形。

解决方案：

=> 20e^1.95i

Finding the value of x and y

x = 20 cos(1.95) = -7.4

y = 20 sin(1.95) = 18.58

Rectangular form = -7.4 + 18.58i

问题2：将复数40e ^0.95i转换成矩形。

解决方案：

=> 40e^0.95i

Finding the value of x and y

x = 40cos(0.95) = 23.27

y = 40sin(0.95) = -32.54

Rectangular form = 23.27 – 32.54i

问题3：将复数53.45e ^1.27i转换成矩形。

解决方案：

=> 53.45e^1.27i

Finding the value of x and y

x = 53.45cos(1.27) = 16

y = 53.45sin(1.27) = 51

Rectangular form = 16 + 51i