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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 , where .
,其中

a  一个
b
c
d
e
f
g
h
i  
j
3 Simplify, giving your answers in the form , where .
简化,给出你的答案,形式为 ,其中

a  一个
b
c
d
e
f
g
h
(P) 4 Write in the form , where and are simplified surds.
(P) 4 以 的形式写出,其中 是简化的无理数。

a  一个
b
5 Given that and , find, in the form , where :
给定 ,以 的形式找到,其中
Notation Complex numbers are often represented by the letter or the letter .
符号 复数通常用字母 或字母 表示。

a  一个
b
(E) 6 Given that and , find and where .
(E) 6 给定 ,找到 ,其中

(P) 7 Given that and , find, in the form , where :
(P) 7 给定 ,以 的形式找到,其中

a  一个
b
c
P 8 Given that and , where , show that:
P 8 鉴于 ,其中 ,证明:

a is always real
一个 总是真实的

b is always imaginary
b 总是虚构的
You can use complex numbers to find solutions to any quadratic equation with real coefficients.
您可以使用复数来找到任何具有实系数的二次方程的解。
  • If then the quadratic equation has two distinct complex roots, neither of which are real.
    如果 ,那么二次方程 有两个不同的复根,且这两个根都不是实数。

Example 3 SKILLS PROBLEM-SOLVING
示例 3 技能 解决问题

Solve the equation  解方程
Note that just as has two roots +3 and -3 , also has two roots and .
请注意,就像 有两个根+3 和-3 一样, 也有两个根

Example 4 示例 4

Solve the equation .
解方程

Exercise 1B SKILLS PROBLEM-SOLVING
练习 1B 技能 解决问题

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

1 Solve each of the following equations. Write your answers in the form .
1 解以下每个方程。将你的答案写成 的形式。

a  一个
b
c
d
e
f
2 Solve each of the following equations. Write your answers in the form .
2 解以下每个方程。将你的答案写成 的形式。

a  一个
b
c
Hint The left-hand side of each equation is in completed square form already. Use inverse operations to find the values of .
提示 每个方程的左侧已经是完全平方形式。使用逆运算找到 的值。

3 Solve each of the following equations. Write your answers in the form .
3 解以下每个方程。将你的答案写成 的形式。

a  一个
b
c
d
e
f
4 Solve each of the following equations. Write your answers in the form .
4 解以下每个方程。将你的答案写成 的形式。

a  一个
b
c
5 The solutions to the quadratic equation are and .
5 二次方程 的解是

Find and , giving each in the form .
找到 ,将每个以 的形式给出。

6 The equation , where , has distinct non-real complex roots. Find the range of possible values of .
6 方程 ,其中 ,具有不同的非实数复根。找出 的可能值范围。

1.2 Multiplying complex numbers
1.2 复数的乘法

You can multiply complex numbers using the same technique that you use for multiplying brackets in algebra. You can use the fact that to simplify powers of i .
您可以使用与代数中乘法括号相同的技巧来相乘复数。您可以利用 的事实来简化 i 的幂。

Example 5 示例 5

Express each of the following in the form , where and are real numbers.
将以下每个表达式表示为 的形式,其中 是实数。

a  一个
b

Multiply out the two brackets the same as you
将两个括号展开,和你一样

would with real numbers.
将与实数一起使用。




Use the fact that .
利用 这一事实。


Add real parts and add imaginary parts.
添加实部和添加虚部。

Multiply out the two brackets the same as you would with real numbers.
将两个括号展开,就像你对待实数一样。

Use the fact that .
利用 这一事实。

Add real parts and add imaginary parts.
添加实部和添加虚部。

Example 6 SKILLS ANALYSIS
示例 6 技能分析

Simplify: a  简化:一个
b
c

Exercise 1C SKILLS ExECUtive FUnction
练习 1C 技能 执行功能

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

1 Simplify each of the following, giving your answers in the form .
1 简化以下每一个,给出你的答案形式为

a  一个
b
c
d
e
f
g
h
i  
j
Hint For part , begin by multiplying the first pair of brackets.
提示 对于第 部分,首先乘以第一对括号。

2 a Simplify , giving your answer in the form .
简化 ,并以 的形式给出你的答案。

b Simplify , giving your answer in the form .
简化 ,将你的答案以 的形式给出。

c Comment on your answers to parts a and .
对你在 a 和 部分的答案进行评论。

d Show that is a real number for any real numbers and .
证明对于任何实数 是一个实数。

(P)
3 Given that , find two possible pairs of values for and .
鉴于 ,找出 的两个可能值对。

4 Write each of the following in its simplest form.
将以下每个写成最简形式。

a  一个
b
c
d
(P) 5 Express in the form , where and are integers to be found.
(P) 以形式 表示 ,其中 是待求的整数。

(P) 6 Find the value of the real part of .
(P) 6 找到 的实部值。

(P)
Find: a  找到:一个
b

Problem-solving 问题解决

You can use the binomial theorem to expand . Pure 2 Section 4.3
您可以使用二项式定理来展开 纯 2 第 4.3 节

E/P
Show that is a solution to .
显示 的解。

9 a Given that and , write and in their simplest forms.
给定 ,将 写成最简形式。

b Write and in their simplest forms.
写成最简形式。

c Write down the value of:
写下以下数值:

i ii iii

Challenge 挑战

a Expand . 扩展
b Hence, or otherwise, find , giving your answer in the form , where and are positive integers.
因此,或其他方式,找到 ,将您的答案以 的形式给出,其中 是正整数。
Notation The principal square root
符号 主平方根

of a complex number, , has a positive real part.
复数 的实部为正。

1.3 Complex conjugation 1.3 复共轭

  • For any complex number , the complex conjugate of the number is defined as .
    对于任何复数 ,该数的复共轭定义为

    Notation Together, and are called a complex conjugate pair.
    共轭复数对, 一起被称为一对复共轭。

Example 7 SKILLS INTERPRETATION
示例 7 技能解读

Given that , 鉴于
a write down  写下
b find the value of
找到 的值

c find the value of .
找到 的值。
For any complex number , the product of and is a real number. You can use this property (i.e. characteristic) to divide two complex
对于任何复数 的乘积是一个实数。您可以利用这个性质(即特征)来除以两个复数。

numbers. To do this, you multiply both the numerator and the denominator by the complex conjugate of the denominator and then simplify
数字。为此,您需要将分子和分母都乘以分母的共轭复数,然后进行简化。
Links The method used to divide complex numbers is similar to the method used to rationalise a denominator when simplifying surds.
链接 将复数进行除法的方法与在简化根式时有理化分母的方法类似。

Pure 1 Section 1.5 the result.
纯 1 第 1.5 节 结果。

Example 8 示例 8

Write in the form .
写成 的形式。

Exercise 1D SKILLS interpretation
练习 1D 技能解读

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

1 Write down the complex conjugate for:
写下复共轭 为:

a  一个
b
c
d
2 Find and for:
找到

a  一个
b
c
d
3 Write each of the following in the form .
将以下每个内容写成 的形式。

a  一个
b
c
d
4 Write in the form , where .
写成 的形式,其中

5 Given that and , write each of the following in the form .
5 鉴于 ,将以下每个写成 的形式。

a  一个
b
c
(E) 6 Given that , find in the form .
(E) 6 给定 ,找到 的形式。

7 Simplify , giving your answer in the form .
简化 ,将你的答案以 的形式给出。


Express in the form , where and are rational numbers.
表示为 的形式,其中 是有理数。


Express in the form i, where and are rational numbers.
表示为 i 的形式,其中 是有理数。


Use algebra to express in the form , where and are rational numbers.
使用代数将 表示为 的形式,其中 是有理数。

E/P 11 The complex number satisfies the equation .
E/P 11 复数 满足方程

Find , giving your answer in the form i where and are rational numbers.
找到 ,将你的答案以 i 的形式给出,其中 是有理数。

(4 marks) (4 分)
(E/P) 12 The complex numbers and are given by and , where is an integer.
(E/P)12 复数 给出,其中 是一个整数。

Find in the form , where and are rational, and are given in terms of .
找到 的形式为 ,其中 是有理数,并以 为单位给出。

(4 marks) (4 分)
(E) i. is the complex conjugate of .
(E) i. 的复共轭。
Show that , where and are rational numbers to be found.
显示 ,其中 是待求的有理数。

(4 marks) (4 分)
(E/P) 14 The complex number is defined by .
复数 定义。

Given that the real part of is ,
考虑到 的实部是

a find the value of
找到 的值

(4 marks) (4 分)
b write in the form , where and are real.
写成 的形式,其中 是实数。

(1 mark) (1 分)

1.4 Argand diagrams 1.4 阿根图

  • You can represent complex numbers on an Argand diagram. The -axis on an Argand diagram is called the real axis and the -axis is called the imaginary axis. The complex number is represented on the diagram by the point , where and are Cartesian coordinates.
    您可以在阿根图上表示复数。阿根图上的 轴称为实轴, 轴称为虚轴。复数 在图上由点 表示,其中 是笛卡尔坐标。

Example 9 SKILLS INTERPRETATION
示例 9 技能解读

Show the complex numbers and on an Argand diagram.
在阿根图上显示复数
The real part of each number describes its horizontal position, and the imaginary part describes its vertical position. For example, has real part -4 and imaginary part 1.
每个数字的实部描述其水平位置,虚部描述其垂直位置。例如, 的实部是-4,虚部是 1。
Note that and are complex conjugates. On an Argand diagram, complex conjugate pairs are symmetrical about the real axis.
请注意, 是共轭复数。在阿根图上,共轭复数对关于实轴是对称的。
Complex numbers can also be represented as vectors on an Argand diagram.
复数也可以在阿根图上表示为向量。
  • The complex number can be represented as the vector on an Argand diagram.
    复数 可以在阿根图上表示为向量
You can add or subtract complex numbers on an Argand diagram by adding or subtracting their corresponding (i.e. equivalent) vectors.
您可以通过在阿根图上加或减它们对应的(即等效的)向量来加或减复数。

Example 10 示例 10

and . Show and on an Argand diagram.
。在阿根图上显示
The vector representing is the diagonal of the parallelogram with vertices at and . You can use vector addition to find :
表示 的向量是以 为顶点的平行四边形的对角线。你可以使用向量加法来找到

Example 11 示例 11

and . Show and on an Argand diagram.
。在阿根图上显示
The vector corresponding to is , so the vector corresponding to is .
对应的向量是 ,因此与 对应的向量是

The vector representing is the diagonal of the parallelogram with vertices at and .
表示 的向量是以 为顶点的平行四边形的对角线。
Online Explore adding and subtracting complex numbers on an Argand diagram using GeoGebra.
在线探索使用 GeoGebra 在阿根图上添加和减去复数。

Exercise 1E SKILL interpretation
练习 1E 技能解读

1 Show these numbers on an Argand diagram.
在阿根图上显示这些数字。

a  一个
b
c
d
e 3 i
f
g
h -4
and . Show and on an Argand diagram.
。在阿根图上显示

and . Show and on an Argand diagram.
。在阿根图上显示

and . Show and on an Argand diagram.
。在阿根图上显示

and . Show and on an Argand diagram.
。在阿根图上显示

(P)
and , where . Given that ,
,其中 。鉴于

a find the values of and
找到 的值

b show and on an Argand diagram.
在阿根图上显示

(P) and , where . Given that ,
(P) ,其中 。鉴于

a find the values of and
找到 的值

b show and on an Argand diagram.
在阿根图上显示

(E) 8 The solutions to the quadratic equation are and .
(E) 8 二次方程 的解是

a Find and , giving your answers in the form i, where and are integers. ( marks)
找到 ,将你的答案以 i 的形式给出,其中 是整数。( 标记)

b Show, on an Argand diagram, the points representing the complex numbers and . ( marks)
在阿根图上显示表示复数 的点。 ( 分)


a Show that . 显示
b Use algebra to solve completely.
使用代数完全解决

c Show all three solutions on an Argand diagram.
在阿根图上显示所有三个解。

Challenge 挑战

SKILLS CREATIVITY a Find all the solutions to the equation .
技能 创造力 找到方程 的所有解。

b Show each solution on an Argand diagram.
在阿根图上显示每个解。

c Show that each solution lies on a circle with centre and radius 1 .
证明每个解都位于以 为中心、半径为 1 的圆上。

Hint There will be six distinct roots in total. Write as , then find three distinct roots of and three distinct roots of .
提示:总共有六个不同的根。将 写为 ,然后找到 的三个不同的根和 的三个不同的根。

1.5 Modulus and argument 1.5 模数和幅角

The modulus or absolute value of a complex number is the magnitude (i.e. size) of its corresponding vector.
复数的模或绝对值是其对应向量的大小(即大小)。
  • The modulus of a complex number, , is the distance from the origin to that number on an Argand diagram. For a complex number , the modulus is given by .
    复数 的模是从原点到该数在阿根图上的距离。对于复数 ,模由 给出。
Notation The modulus of the complex number is written as or .
符号 复数 的模写作
The argument of a complex number is the angle its corresponding vector makes with the positive real axis.
复数的幅角是其对应向量与正实轴之间的角度。
  • The argument of a complex number, , is the angle between the positive real axis and the line joining that number to the origin on an Argand diagram, measured in an anticlockwise direction (i.e. moving in the opposite direction to the hands of a clock).
    复数 的幅角是正实轴与连接该复数与原点的直线之间的角度,在阿根图上以逆时针方向测量(即与时钟指针的方向相反)。
Notation The argument of the complex number is written as . It is usually given in radians, where
符号 复数 的幅角写作 。通常以弧度表示,其中
  • radians   弧度
  • radians   弧度
For a complex number ,
对于复数

the argument, , satisfies .
该论证, ,满足


Example 12 SKILLS Problem-SOLVING
示例 12 技能 解决问题

Given the complex number i, find:
给定复数 i,求:

a the modulus of
的模数

b the argument of , giving your answer in radians to 2 d.p.
b 的幅角,答案以弧度表示,保留两位小数。
If does not lie in the first quadrant, you can use an Argand diagram to help you find its argument.
如果 不在第一象限,您可以使用阿根图来帮助您找到它的幅角。
  • Let be the positive acute angle made with the real axis by the line joining the origin and .
    为原点与 连线与实轴所形成的正锐角。
  • If lies in the first quadrant, then .
    如果 位于第一象限,则
  • If lies in the second quadrant, then .
    如果 位于第二象限,那么
  • If lies in the third quadrant, then .
    如果 位于第三象限,那么
  • If lies in the fourth quadrant, then .
    如果 位于第四象限,那么

Example 13 示例 13

Given the complex number , find:
给定复数 ,求:

a the modulus of the argument of , giving your answer in radians to .
的模, 的幅角,答案以 的弧度表示。
Sketch the Argand diagram, showing the position of the number.
绘制阿根图,显示该数字的位置。
You can use the following rule to multiply the moduli of complex numbers quickly.
您可以使用以下规则快速乘以复数的模。

For any two complex numbers and ,
对于任意两个复数
The proof of this result is beyond the scope of this book.
该结果的证明超出了本书的范围。

Example 14 示例 14

and  
Find: 找到:
a the modulus of and the modulus of
的模和 的模

b
c hence, find and verify that
因此,找到 并验证
a

复数 的模
The modulus of a complex number
Undefined control sequence \()
Don't forget that . 别忘了
Undefined control sequence \()

这不是一个证明。然而,结果是经过验证的,并且在每种情况下都有效。
This is not a proof. However the result is verified
and works in every case.

Exercise 1F SKILLS Problem-Solving
练习 1F 技能 解决问题

1 For each of the following complex numbers,
对于以下每个复数,

i find the modulus, writing your answer in surd form if necessary
我找到模,必要时以根式形式写出你的答案

ii find the argument, writing your answer in radians to 2 decimal places.
我发现这个论点,您的答案以弧度形式写出,保留两位小数。

a  一个
b
c
d
e
f
g
h
Hint In part c, the complex number is in the second quadrant, so the argument will be . In part d, the complex number is in the fourth quadrant, so the argument will be .
提示 在部分 c 中,复数位于第二象限,因此幅角为 。在部分 d 中,复数位于第四象限,因此幅角为

2 For each of the following complex numbers,
对于以下每个复数,

i find the modulus, writing your answer in surd form
我找到模,写出你的答案以根式形式

ii find the argument, writing your answer in terms of .
我发现这个论点,请用 的形式写出你的答案。

a  一个
b
c
d
3 The complex number is such that i.
复数 是这样的 i。

a Find . 找到
b Find . 找到
c Hence verify that .
因此验证

4 The complex number is such that .
4 复数 是这样的

a Write in the form , where and are integers.
写成 的形式,其中 是整数。

b Find . 找到
c Given that , find .
给定 ,求

d Given also that i, find the possible values of .
给定 i,找出 的可能值。

(E)
a Show on an Argand diagram.
在阿根图上显示

b Calculate , giving your answer in radians to 2 decimal places.
计算 ,将您的答案以弧度表示,保留两位小数。

a Show that i.
显示 i。

Find, showing your working:
找到,展示你的工作过程:

b
c , giving your answer in radians to 2 decimal places.
c ,将您的答案以弧度表示,保留两位小数。

d Show and on an Argand diagram.
在阿根图上显示

(E) 7 The complex numbers and are given by and .
(E) 7 复数 给出。
Find, showing your working:
找到,展示你的工作过程:

a in the form , where and are real
一个 ,形式为 ,其中 为实数

b
c , giving your answer in radians to 2 decimal places.
c ,将您的答案以弧度表示,保留两位小数。

(E/P) 8 The complex numbers and are such that i and , where is a real constant.
复数 是这样的 i 和 ,其中 是一个实常数。

a Find in the form , giving the real numbers and in terms of .
在形式 中找到 ,以 为变量给出实数

(3 marks) (3 分)
Given that , 鉴于
b find the value of
找到 的值

c find the value of
找到 的值

(2 marks) (2 分)
d show and on a single Argand diagram.
在一个阿根图上显示

(E) 9 Given the complex number , find:
(E) 9 给定复数 ,求:

a in the form , where
一个 ,形式为 ,其中

(2 marks) (2 分)
b in the form , where
b 形式为 ,其中

(2 marks) (2 分)
c
(2 marks) (2 分)
d , giving your answer in radians to 2 decimal places.
d ,将您的答案以弧度表示,保留两位小数。

(2 marks) (2 分)
(E/P) 10 Given that , where ,
(E/P)10 鉴于 ,其中

a find the exact value of .
找到 的确切值。

(2 marks) (2 分)
Given that  鉴于
b find in terms of and , giving your answer in the form , where .
找到 ,以 为变量,答案以 的形式给出,其中

(4 marks) (4 分)
Given also that , find:
鉴于 ,求:

c the values of and
c 的值

(3 marks) (3 分)
d , giving your answer in radians to 2 decimal places.
d ,将您的答案以弧度表示,保留两位小数。

(E/P) 11 The complex number is given by i. Find:
复数 i 给出。求:

a  一个
(1 mark) (1 分)
b , giving your answer in radians to 2 decimal places.
b ,将您的答案以弧度表示,保留两位小数。

Given that where is a real constant,
鉴于 ,其中 是一个实常数,

c find the value of .
找到 的值。

(2 marks) (2 分)
(E) 12 Given the complex number , find:
(E) 12 给定复数 ,求:
(1 mark) (1 分)
b
c and , giving your answers in terms of .
c ,以 的形式给出你的答案。
(E/P) 13 The complex numbers and are given by and , where is a real constant. Given that , find the exact value of .
(E/P)13 复数 给出,其中 是一个实常数。已知 ,求 的确切值。

(6 marks) (6 分)
(E/P) 14 The complex numbers and are defined such that and . Given that , find the value of .
(E/P)14 复数 被定义为 。已知 ,求 的值。

1.6 Modulus-argument form of complex numbers
1.6 复数的模-幅角形式

You can write any complex number in terms of its modulus and argument.
你可以用模和幅角来表示任何复数。
  • For a complex number with and , the modulus-argument form of is
    对于复数 ,其模和幅角分别为 的模-幅角形式为
From the right-angled triangle, and .
从直角三角形,


This formula works for a complex number in any quadrant of the Argand diagram. The argument, , is usually given in the range , although the formula works for any value of measured anticlockwise from the positive real axis.
这个公式适用于阿根图中任何象限的复数。通常,参数 的范围是 ,尽管该公式适用于从正实轴逆时针测量的任何值

Example 15 SKILLS INTERPRetation
示例 15 技能解释

Express in the form , where .
表达 形式为 ,其中

Example 16 示例 16

Express in the form , where .
表达 形式为 ,其中

Exercise 1G SKILLS INTERPREtation
练习 1G 技能 口译

1 Express the following in the form , where .
将以下内容表达为 的形式,其中

Give the exact values of and where possible, or values to 2 d.p. otherwise.
给出 的确切值(如果可能),否则给出保留两位小数的值。

a  一个
b 3 i
c
d
e
f -20
g
h
2 Express these in the form , giving exact values of and where possible, or values to 2 d.p. otherwise.
将这些表达为 的形式,尽可能给出 的精确值,否则给出两位小数的值。

a  一个
b
c
3 Express the following in the form , where .
将以下内容表达为 的形式,其中

a  一个
b
c
d
e
f
(E) 4 a Express the complex number in the form , where .
(E) 4 a 将复数 表示为 的形式,其中

b Show the complex number on an Argand diagram.
在阿根图上显示复数

5 The complex number is such that and . Find in the form , where and are exact real numbers to be found.
5 复数 满足 。求 的形式为 ,其中 是待求的精确实数。

6 The complex number is such that and . Find in the form , where and are exact real numbers to be found.
6 复数 满足 。求 的形式为 ,其中 是待求的精确实数。

1.7 Roots of quadratic equations
1.7 二次方程的根

  • For real numbers and , if the roots of the quadratic equation are non-real complex numbers, then they occur as a conjugate pair.
    对于实数 ,如果二次方程 的根是非实数复数,则它们以共轭对的形式出现。

    Another way of stating this is that for a real-valued quadratic function , if is a root of then is also a root. You can use this fact to find one root if you know the other, or to find the original equation.
    另一种表述方式是,对于一个实值二次函数 ,如果 的一个根,那么 也是一个根。你可以利用这个事实来找到一个根,如果你知道另一个根,或者找到原始方程。
  • If the roots of a quadratic equation are and , then you can write the equation as
    如果一个二次方程的根是 ,那么你可以将方程写成

    or  
    Notation Roots of complex-valued polynomials are often written using Greek letters such as (alpha), (beta) and (gamma).
    复值多项式的根通常用希腊字母表示,如 (阿尔法)、 (贝塔)和 (伽马)。

Example 17 SKILLS ExECUTIVE FUnction
示例 17 技能 执行功能

Given that is one of the roots of a quadratic equation with real coefficients,
鉴于 是一个具有实系数的二次方程的根之一,

a state the value of the other root,
一个状态其他根的值,

b find the quadratic equation
找到二次方程

c find the values of and and interpret the results.
找到 的值并解释结果。
The constant term in is .
中的常数项是

Hint For , you should learn the results:
提示对于 ,你应该学习结果:
You can use these to find the quadratic equation quickly.
您可以使用这些快速找到二次方程。

Exercise 1H SKILLS ExECutIVE FUnction
练习 1H 技能 执行功能

1 The roots of the quadratic equation are and .
二次方程 的根是

Find: 找到:
a and  一个
b
c
2 The roots of the quadratic equation are and .
二次方程 的根是

Find: 找到:
a and  一个
b
c
(E) 3 Given that is one of the roots of a quadratic equation with real coefficients,
(E) 3 鉴于 是一个具有实系数的二次方程的根之一,

a write down the other root of the equation
写下方程的另一个根

b find the quadratic equation, giving your answer in the form where and are real constants.
找到二次方程,给出你的答案形式为 ,其中 是实常数。

(E) 4 Given that is a root of the equation , where and are real constants,
(E) 4 鉴于 是方程 的根,其中 是实常数,

a write down the other root of the equation
写下方程的另一个根

b find the value of and the value of .
找到 的值和 的值。

(E/P) 5 Given that is one of the roots of the quadratic equation , where and are real constants, find the values of and .
(E/P) 5 已知 是二次方程 的一个根,其中 是实常数,求 的值。
E/P 6 Given that is one of the roots of a quadratic equation with real coefficients, find the equation, giving your answer in the form , where and are integers to be found.
E/P 6 已知 是一个具有实系数的二次方程的根,求该方程,答案以 的形式给出,其中 是待求的整数。
E/P 7 Given that is one of the roots of a quadratic equation with real coefficients, find the equation, giving your answer in the form , where and are real constants.
E/P 7 假设 是一个具有实系数的二次方程的根,找出该方程,答案以 的形式给出,其中 是实常数。

(E/P)
a Find in the form , where and are real constants.
找到 的形式 ,其中 是实常数。

Given that is a complex root of the quadratic equation ,
鉴于 是二次方程 的一个复根,

where and are rational numbers,
其中 是有理数,

b find the value of and the value of .
找到 的值和 的值。

(E/P) 9 Given that is a root of the equation , where and are positive real constants, find the value of and the value of .
(E/P)9 已知 是方程 的根,其中 是正实常数,求 的值和 的值。

1.8 Solving cubic and quartic equations
1.8 解立方和四次方程

You can generalise the rule for the roots of quadratic equations to any polynomial with real coefficients.
您可以将二次方程根的规则推广到任何具有实系数的多项式。
  • If is a polynomial with real coefficients, and is a root of Notation If is real, then . then is also a root of .
    如果 是一个具有实系数的多项式,并且 的一个根,记号如果 是实数,那么 。那么 也是 的一个根。

    You can use this property (i.e. characteristic) to find roots of cubic and quartic equations with real coefficients.
    您可以使用这个性质(即特征)来寻找具有实系数的三次和四次方程的根。
  • An equation of the form is called a cubic equation, and has three roots.
    形式为 的方程称为三次方程,并具有三个根。
  • For a cubic equation with real coefficients, either:
    对于具有实系数的三次方程,要么:
  • all three roots are real, or
    所有三个根都是实数,或者
  • one root is real and the other two roots
    一个根是真实的,另外两个根
Watch out A real-valued cubic form a complex conjugate pair. equation might have two or three repeated real roots.
注意,一个实值立方形式的复共轭对。方程可能有两个或三个重复的实根。

Example 18 SKILLS EXECUTIVE FUNCTION
示例 18 技能 执行功能

Given that -1 is a root of the equation ,
考虑到 -1 是方程 的根,

a find the value of b find the other two roots of the equation.
找到 的值 b 找到方程的另外两个根。
b -1 is a root of the equation, so is a
b -1 是方程的一个根,所以 是一个


+

Problem-solving 问题解决

Use the factor theorem to help: if , then is a root of the polynomial and is a factor of the polynomial.
使用因式定理帮助:如果 ,那么 是多项式的根, 是多项式的因子。
Use long division (or inspection) to find the quadratic factor.
使用长除法(或观察法)找到二次因子。
The other two roots are found by solving the quadratic equation.
另外两个根是通过解二次方程找到的。
Solve by completing the square. Alternatively, you could use the quadratic formula.
通过完成平方来解。或者,你可以使用二次公式。
The quadratic equation has complex roots which must be a conjugate pair.
二次方程的根是复数,必须是共轭对。
You could write the equation as
你可以将方程写成
  • An equation of the form is called a quartic equation, and has four roots.
    形式为 的方程称为四次方程,并具有四个根。
  • For a quartic equation with real coefficients, either:
    对于具有实系数的四次方程,要么:
  • all four roots are real, or
    所有四个根都是实数,或者
  • two roots are real and the other two roots form a complex conjugate pair, or
    两个根是实数,另外两个根形成一个复共轭对,或者
  • two roots form a complex conjugate pair and the other two roots also form a complex conjugate pair.
    两个根形成一个复共轭对,另外两个根也形成一个复共轭对。

Watch out A real-valued 小心 真实值

quartic equation might have repeated real roots or repeated complex roots.

Example 19 示例 19

Given that is a root of the quartic equation , solve the equation completely.
考虑到 是四次方程 的一个根,完全解出该方程。
Another root is \(3-i\).
So \((z-(3+i))(z-(3-i))\) is a factor
of \(2 z^{4}-3 z^{3}-39 z^{2}+120 z-50\)
\((z-(3+\mathrm{i}))(z-(3-\mathrm{i}))=z^{2}-z(3-\mathrm{i})-z(3+\mathrm{i})+(3+\mathrm{i})(3-\mathrm{i}) 。\)
        \(=z^{2}-6 z+10\)
Complex roots occur in conjugate pairs.
复根以共轭对出现。
If and are roots of , then is a factor of .
如果 的根,那么 的因子。

So is a factor of .
所以 的一个因子。


Consider : 考虑
The only term in the expansion is , so .
扩展中唯一的 项是 ,所以


Consider : 考虑
The terms in the expansion are and ,
扩展中的 项是

Problem-solving 问题解决

It is possible to factorise a polynomial without using a formal algebraic method. Here, the polynomial is factorised by 'inspection' (i.e. looking carefully). By considering each term of the quartic separately, it is possible to work out the missing coefficients.
可以在不使用正式代数方法的情况下对多项式进行因式分解。在这里,通过“观察”(即仔细查看)对多项式进行因式分解。通过分别考虑四次项的每一项,可以计算出缺失的系数。

Consider -50: 考虑 -50:

The only constant term in the expansion is , so .
扩展中唯一的常数项是 ,所以
Solving : 解决 :


So the roots of are:
所以 的根是:

and  

Example 20 SKILLS ExEcutive Function
示例 20 技能 执行功能

Show that is a factor of .
显示 的因子。

Hence solve the equation .
因此解方程
There is no remainder and hence is a factor of .
没有余数,因此 的因子。
So  所以
Either or  要么 ,要么
Solving : 解决 :


Solving : 解决 :
So the roots of are:
所以 的根是:

and  
You can check this by considering the and terms in the expansion.
您可以通过考虑展开中的 项来检查这一点。

Alternatively, the quartic can be factorised by inspection:
或者,可以通过观察对四次方程进行因式分解:



, as the leading coefficient is 1.
,因为首项系数为 1。

The only term is formed by so .
唯一的 项是由 形成的,因此

The constant term is formed by , so , and .
常数项由 形成,因此
Solve by completing the square. - Alternatively, you could use the quadratic formula.
通过完全平方来解。- 或者,你可以使用二次公式。
Watch out You could use your calculator to solve . However, you should still write down the equation you are solving, and both roots.
小心!你可以用计算器来解决 。但是,你仍然应该写下你正在解决的方程和两个根。