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：
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 页 - 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
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pages 5-6 页 5-6
pages 7-8 页 7-8
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pages 11-14 页 11-14
pages 15-16 页 15-16
pages 16-18 页 16-18
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Prior knowledge check 先前知识检查
1 Simplify each of the following:
1 简化以下每一项：
a
b
c
2 In each case, determine the number of distinct real roots of the equation
在每种情况下，确定方程
a
b
c
3 For the triangle shown, find the values of:
对于所示的三角形，找出以下值：
a
b
1 简化以下每一项：
a
b
c
2 In each case, determine the number of distinct real roots of the equation
在每种情况下，确定方程
a
b
c
3 For the triangle shown, find the values of:
对于所示的三角形，找出以下值：
a
b
4 Find the solutions of
找到
5 Write
将 5 写成
1.1 Imaginary and complex numbers
1.1 虚数和复数
The quadratic equation
二次方程
二次方程
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。
如果平方根下的表达式为负，则没有实数解。
Links For the equation
方程
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
您可以通过扩展数字系统以包括
您可以通过扩展数字系统以包括
-
- 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
将以下每个表达式用 i 表示。
a
a
b
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。
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
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。
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
简化以下每一个，答案以
a
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。
简化以下每一个，答案以
a
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
将以下每个写成
a
b
c
d
e
f
g
h
i
j
2 Simplify, giving your answers in the form
将以下每个写成
a
b
c
d
e
f
g
h
i
j
2 Simplify, giving your answers in the form