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高中数学 第三章 第2课 3.2复数的四则运算(1)教学案苏教版选修1-2

第2课
教 学目标:

3. 2 复数的四则运算(1)
)班 姓名:__________ __

班级:高二(

1.掌握复数的加减法及乘法运算法则及意义;理解共轭复数的概 念. 2.理解并掌握实数进行四则运算的规律. 教学重点:复数乘法运算. 教学难点:复数运算法则在计算中的熟练应用. 教学方法:类比探究法. 教学过程: 复习复数的定义,复数的分类及复数相等的充要条件等上节课所学内容. 一、问题情境

1+x) ,类比你能计算 (2+3i)+(- 1+i) 吗? 问题 1 化简: (2+3x)+(-
问题 2 化简:多项式 (2 ? 3x)(?1 ? x) ,类比你能计算 (2 ? 3i)(?1 ? i) 吗? 问题 3 两个复数 a+b i , a-b i 有什 么联系? 二、学生活动

1+i)=+ 1 4i ,进而猜想 1.由多项式的加法类比猜想 (2+3i)+(- (a+b i)+(c+d i)=(a+c)+(b+d )i . 若 ? x+y i ?+(c+d i)=a+b i ,根据复数相等的定义 x+y i=(a-c)+(b-d )i . 1+i)=-5-i ,进而猜想 2.由多项式的乘法类比猜想 (2+3i)(- (a+b i)(c+d i)=(ac-bd )+(bc+ad )i . 3.两个复数 a+b i , a-b i 实部相 等,虚部互为相反数.
三、建构 数学 1.复数和的定义: z1+z2 ? (a+c)+(b+d )i . 2.复数差的定义: z1-z2=(a-c)+(b-d )i . 3.复数积的定义: z1 z2=(ac-bd )+(bc+ad )i . 性质: z2 z1=z1 z2 ; 复数 z1=a+b i , z2=c+d i .

( z1 z2 ) z3=z1 ( z2 z3 ) ; z1 ( z2+z3 )=z1 z2+z1 z3 . 4.共轭复数: z=a+b i 与 z=a? 复数;实数的共轭复数是它本身. -b i 互为共轭 ? ? 2 2 共 轭复数的简单性 质: z+ z =2a ; z- z =2b i ; z ? z =a +b .
四、数学应用

-3i)-(2+5i)+(-4+9i) . 例 1 计算 (1

1+3i) . 例 2 计算 (-2-i)(3-2i)(-

例 3 计算 (a+b i)(a-b i) .

思考 1 当 a>0 时,方程 x +a=0 的根是什么?

2

思考 2 设 x,y∈ R,在复数 集内,能将 x 2+y 2 分解因式吗?

例 4 已知复数 z 满足: z ? z +2i? z=4+2i ,求复数 z.

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