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www.958508.com:3.2复数的运算导学案 文

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3.2 复数的运算 3.2.1 复数的加法与减法
一、学习目标: (1)掌握复数加法与减法运算法则,能熟练地进行加、减法运算; (2)理解并掌握复数加法与减法的几何意义,会用平行四边形法则和三角形法则解决一 些简单的问题; (3)能初步运用复平面两点间的距离公式解决有关问题; (4)通过学习平行四边形法则和三角形法,培养学生的数形结合的数学思想; (5)通过本节内容的学习,培养学生良好思维品质(思维的严谨性,深刻性,灵活性) 二、教学重点、难点 本节的重点是复数加法法则.难点是复数加减法的几何意义.复数加法法则是教材首先 规定的法则,它是复数加减法运算的基础,对于这个规定的合理性,在学习过程中要加以重 视.复数加减法的几何意义的难点在于复数加减法转化为向量加减法,以它为根据来解决某 些平面图形的问题,学生对这一点不容易接受. 三、学习过程 (一)新课引入 建立了复数的概念以后,很重要的一个问题就是建立复数集里的各种运算.由于实数是 复数的一部分,所以建立复数运算时,应当遵循的一个原则是,作为复数的实数,在复数集 里的运算和在实数集里的运算应当是一致的. (二)课程讲解 复数的加法运算满足交换律、结合律. 1.复数的加法法则:设 z1 ? a ? bi, z2 ? c ? di, a, b, c, d ? R ,定义

z1 ? z2 ? (a ? bi) ? (c ? di) ? (a ? c) ? (b ? d )i
当然,两个复数的和仍为复数. 2. 复数的相反数:? a ? bi 叫做 a ? bi 的相反数.在复平面内,互为相反数的两个复数关 于原点对称. 3.根据相反数的概念,我们规定两个复数的减法法则如下:

z1 ? z2 ? (a ? bi) ? (c ? di) ? (a ? c) ? (b ? d )i
-1-

两个复数的差仍是复数.是唯一确定的复数. 复数的加(减)法与多项式加(减)法是类似的.就是把复数的实部与实部,虚部与虚 部分别相加(减) ,即( (三)典例分析 例 1 计算: ?1 ? 3i ? ? ? 2 ? 5i ? ? ? ?4 ? 9i ? +

i )±(

+

i )=(

±

)+(

±

)i .

例 2 计算:

(1-2i) ? (-2 ? 3i) ? (3-4i) ? (-4 ? 5i) ??? (-2002 ? 2003i) ? (2003-2004i)

例 3.在复平面内,满足下列复数形式方程的动点 Z 的轨迹是什么. (1) z ?1 ? i ? z ? 2 ? i ;

-2-

(2) z ? i ? z ? i ? 4;

(3) z ? 2 ? z ? 2 ? 1.

(四)变式训练:

4) i3 (? 4) ? i ? (1) (2?

; ;

(2) 5 ? 6i ? (?2 ? i) ? (3 ? 4i) ? (3) 5 ? (3 ? 2i) ? (4) (?3 ? 4i) ? (2 ? i) ? (1 ? 5i) ? ;

(5) 已知 z1 ? a ? bi(a, b ? R), z2 ? 3 ? i 且 z1 ? z2 与 z3 ? ?2 ? i 在复平面内对应的点关于原点 对称,试求 a , b 的值;

(6)已知复数 z1 ? ?2 ? i, z2 ? 4 ? 2i ,试求 z1 ? z2 对应的点关于虚轴对称点的复数.

-3-

(7)设 z1 ? x ? 2i, z2 ? ?3 ? yi ( x, y ? R) 且 z1 ? z2 ? 5 ? 6i ,求 z1 ? z2 ;

(8)设 m ? R ,复数 z1 ? 的取值范围.

m2 ? m ? (m ? 15)i, z2 ? ?2 ? m(m ? 3)i ,如 z1 ? z2 是虚数,求 m m?2

(五)课时小结 我们通过推导得到复数减法法则,并进一步得到了复数减法几何意义,应用复数减法几 何意义和复平面内两点间距离公式,可以用复数研究解析几何问题,不等式以及最值问题. 四、课后反思

-4-


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