这是用户在 2024-9-10 15:19 为 https://app.immersivetranslate.com/pdf-pro/f691b4ac-b403-4650-aecd-2b86da474b05 保存的双语快照页面,由 沉浸式翻译 提供双语支持。了解如何保存?
2024_09_10_b2aa537d6f75512c5d1cg

1 COMPLEX NUMBERS  1 复数

Learning objectives 学习目标

After completing this chapter you should be able to:
完成本章后,您应该能够:
  • Understand and use the definitions of imaginary and complex numbers
    理解并使用虚数和复数的定义

    page 2  第 2 页
  • Add and subtract complex numbers
    加法和减法复数

    pages  
  • Find solutions to any quadratic equation with real coefficients
    找到任何具有实系数的二次方程的解
  • Multiply complex numbers 乘法复数
  • Understand the definition of a complex conjugate
    理解复共轭的定义
  • Divide complex numbers 除复数
  • Show complex numbers on an Argand diagram
    在阿根图上显示复数
  • Find the modulus and argument of a complex number
    找到复数的模和幅角
  • Write a complex number in modulus-argument form
    将复数写成模-幅角形式
  • Solve quadratic equations that have complex roots
    求解具有复根的二次方程
  • Solve cubic or quartic equations that have complex roots
    求解具有复根的三次或四次方程

    pages  
    pages 5-6  页 5-6
    pages 7-8  页 7-8
    pages  
    pages  
    pages 11-14  页 11-14
    pages 15-16  页 15-16
    pages 16-18  页 16-18
    pages  

Prior knowledge check 先前知识检查

1 Simplify each of the following:
1 简化以下每一项:

a  一个
b
c
Pure 1 Section 1.5
纯 1 第 1.5 节

2 In each case, determine the number of distinct real roots of the equation .
在每种情况下,确定方程 的不同实根的数量。

a  一个
b
c
Pure 1 Section 2.3
纯 1 第 2.3 节

3 For the triangle shown, find the values of:
对于所示的三角形,找出以下值:

a  一个
b

International GCSE Mathematics
国际 GCSE 数学

4 Find the solutions of , giving your answers in the form where and are integers.
找到 的解,答案以 的形式给出,其中 是整数。

Pure 1 Section 2.1
纯 1 第 2.1 节

5 Write in the form where and are rational numbers.
将 5 写成 的形式,其中 是有理数。

Pure 1 Section 1.6
纯 1 第 1.6 节

1.1 Imaginary and complex numbers
1.1 虚数和复数

The quadratic equation has solutions given by
二次方程 的解为
If the expression under the square root is negative, there are no real solutions.
如果平方根下的表达式为负,则没有实数解。
the discriminant is .
  • If , there are two distinct real roots.
    如果 ,则有两个不同的实根。
  • If , there are two equal real roots.
    如果 ,则有两个相等的实根。
  • If , there are no real roots.
    如果 ,则没有实根。
You can find solutions to the equation in all cases by extending the number system to include . Since there is no real number that squares to produce -1 , the number is called an imaginary number, and is represented using the letter . Complex numbers have a real part and an imaginary part, for example .
您可以通过扩展数字系统以包括 来找到方程的所有解。由于没有实数的平方等于-1,因此数字 被称为虚数,并用字母 表示。复数具有实部和虚部,例如
  • An imaginary number is a number of the form , where .
    虚数是形如 的数,其中
  • A complex number is written in the form , where .
    复数以 的形式表示,其中

    Notation The set of all complex numbers is written as .
    符号 所有复数的集合写作

    For the complex number :
    对于复数 :
  • is the real part
    是实部
  • is the imaginary part
    是虚部

Example 1 SKILLS interPRetation
示例 1 技能 解释

Write each of the following in terms of i.
将以下每个表达式用 i 表示。

a  一个
a  一个
b
You can use the rules of surds to manipulate imaginary numbers.
你可以使用无理数的规则来操作虚数。

Watch out An alternative way of writing
注意 一种替代写法

is . Avoid writing as this can easily be confused with .
In a complex number, the real part and the imaginary part cannot be combined to form a single term.
在复数中,实部和虚部不能合并成一个单一的项。
  • Complex numbers can be added or subtracted by adding or subtracting their real parts and adding or subtracting their imaginary parts.
    复数可以通过加或减它们的实部和加或减它们的虚部来进行加法或减法。
  • You can multiply a real number by a complex number by multiplying out the brackets in the usual way.
    您可以通过以通常的方式展开括号,将实数与复数相乘。

Example 2 示例 2

Simplify each of the following, giving your answers in the form , where .
简化以下每一个,答案以 的形式给出,其中

a  一个
b
c
d

Add the real parts and add the imaginary parts.
将实部相加,将虚部相加。

Subtract the real parts and subtract the imaginary parts.
减去实部并减去虚部。

Exercise (1A SKILLS interpRetation
练习 (1A 技能 解释)

Do not use your calculator in this exercise.
在这个练习中请不要使用计算器。

1 Write each of the following in the form , where is a real number.
将以下每个写成 的形式,其中 是一个实数。

a  一个
b
c
d
e
f
g
h
i  
j
2 Simplify, giving your answers in the form