笛卡尔形式
虚数是什么?
- 到目前为止,当我们遇到这样的方程
" class="Wirisformula" role="math" alt="X的2次方空间端指数等于空间- 1" style="vertical-align:-6px;height:23px;width:68px">我们会说“没有真正的解决方案”x 2 = - 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 解决方案是
" class="Wirisformula" role="math" alt="X等于正负- 1的平方根" style="vertical-align:-6px;height:26px;width:84px">哪些不是实数x = ± - 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 解决方案是
- 为了解决这个问题,数学家们将- 1的平方根之一定义为
" class="Wirisformula" role="math" alt="我直" style="vertical-align:-6px;height:22px;width:6px">;虚数i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="√(- 1)根号等于I" style="vertical-align:-6px;height:26px;width:63px">- 1 = i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="I的平方等于- 1" style="vertical-align:-6px;height:23px;width:56px">i 2 = - 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 其他负数的平方根可以通过重写为的倍数来求
" class="Wirisformula" role="math" alt="根号下- 1的平方根" style="vertical-align:-6px;height:26px;width:41px">- 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 使用
" class="Wirisformula" role="math" alt="根号下a b的平方根等于根号下a乘以根号下b" style="vertical-align:-6px;height:26px;width:126px">a b = a × b {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 使用
什么是复数?
- 复数有实部和虚部
- 例如:
" class="Wirisformula" role="math" alt="3加4个I" style="vertical-align:-6px;height:22px;width:41px">3 + 4 i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 实部是3虚部是4i
- 例如:
- 复数通常表示为
" class="Wirisformula" role="math" alt="z" style="vertical-align:-6px;height:22px;width:10px">z {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 我们分别用实部和虚部
" class="Wirisformula" role="math" alt="Re左括号z右括号" style="vertical-align:-6px;height:22px;width:42px">而且Re ( z ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="我是左括号z右括号" style="vertical-align:-6px;height:22px;width:42px">Im ( z ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 我们分别用实部和虚部
- 两个复数相等当且仅当实部和虚部相等。
- 例如,
" class="Wirisformula" role="math" alt="3加2个I" style="vertical-align:-6px;height:22px;width:41px">而且3 + 2 i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="3加3个I" style="vertical-align:-6px;height:22px;width:41px">是不平等的3 + 3 i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 例如,
- 所有复数的集合给出了这个符号
" class="Wirisformula" role="math" alt="直线复数" style="vertical-align:-3px;height:19px;width:13px">ℂ {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
什么是笛卡尔式?
- 复数有很多种不同的形式
- 表单z=一个+bI被称为笛卡尔形式
- a、b∈
" class="Wirisformula" role="math" alt="直接实数" style="vertical-align:-3px;height:19px;width:14px" loading="lazy">ℝ {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 这是公式小册子中给出的第一种形式
- 一般来说,对于z=一个+b我
- (z) =一个
- 我(z) =b
- 复数用笛卡尔形式可以很容易地用几何表示出来
- 你的GDC可能会这么叫矩形形式
- 当你的GDC设置为矩形时,它将以笛卡尔形式给出答案
- 如果你的GDC是不设置在复杂模式下,它将不会给出任何复数形式的输出
- 确保你能找到笛卡尔式中使用复数的设置,并练习输入问题
- 笛卡尔式是最简单的复数加减法
考试技巧
- 记住复数有实部和虚部
- 1是纯实数(它的虚部为零)
- I是纯虚数(实部为零)
- 1 + I是一个复数(实部和虚部都等于1)
工作的例子
一)
解方程x 2 = - 9 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="X方等于- 9" style="vertical-align:-6px;height:23px;width:61px" loading="lazy">
b)
解方程x + 7 2 = - 16 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="左括号x + 7右括号的平方等于- 16" style="vertical-align:-6px;height:23px;width:108px" loading="lazy">,用笛卡尔的形式给出答案。
复杂的加法,减法和乘法
如何在笛卡尔式中加减复数?
- 复数的加减法应该在它们在的时候做笛卡尔形式
- 复数加减时,分别化简实部和虚部
- 就像你在代数和数学中收集类似的项,或者处理向量中的不同分量
" class="Wirisformula" role="math" alt="开括号a + b直I闭括号加上开括号c + d直I闭括号等于开括号a + c闭括号加上开括号b + d闭括号直I" style="vertical-align:-6px;height:22px;width:259px" loading="lazy">a + b i + c + d i = a + c + b + d i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="开括号a + b直I闭括号减去开括号c + d直I闭括号等于开括号a - c闭括号加上开括号b - d直I" style="vertical-align:-6px;height:22px;width:259px" loading="lazy">a + b i - c + d i = a - c + b - d i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
如何用笛卡尔式乘复数?
- 复数可以用与代数表达式相同的方法乘以常数:
" class="Wirisformula" role="math" alt="K开括号a + b I闭括号等于K a + kb I" style="vertical-align:-6px;height:22px;width:140px" loading="lazy">k a + b i = k a + k b i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 用笛卡尔形式乘以两个复数的方法与两个线性表达式相乘的方法相同:
" class="Wirisformula" role="math" alt="开括号a加b直I闭括号开括号c加d直I闭括号等于a c加开括号a d加b c闭括号直I加b d直I的平方等于空白a c加开括号a d加b c闭括号直I减b d" style="vertical-align:-6px;height:23px;width:437px" loading="lazy">a + b i c + d i = a c + a d + b c i + b d i 2 = a c + a d + b c i - b d {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 这是一个实数部的复数
" class="Wirisformula" role="math" alt="ac - bd空白" style="vertical-align:-6px;height:22px;width:58px" loading="lazy">虚部a c - b d {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="A d加上bc" style="vertical-align:-6px;height:22px;width:54px" loading="lazy">a d + b c {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 复数相乘最重要的是
" class="Wirisformula" role="math" alt="I的平方等于- 1" style="vertical-align:-6px;height:23px;width:56px" loading="lazy">i 2 = - 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 你的GDC将能够以笛卡尔形式将复数相乘
- 练习这样做,并用它来检查你的答案
- 很容易看出,用笛卡尔式将两个以上的复数相乘是一个容易出错的漫长过程
- 当复数的形式不同时,相乘更容易,通常先将它们从笛卡尔形式转换为极坐标形式或欧拉形式是有意义的
- 有时,当一个问题描述多个复数时,符号
" class="Wirisformula" role="math" alt="Z下标1逗号空白Z下标2逗号空白水平省略号" style="vertical-align:-12px;height:28px;width:64px" loading="lazy">是用来表示每个复数的z 1 , z 2 , … {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
如何处理I的更高次幂呢?
- 因为
" class="Wirisformula" role="math" alt="I的平方等于- 1" style="vertical-align:-6px;height:23px;width:56px" loading="lazy">这可以导致一些有趣的结果对于更高的I次方i 2 = - 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="加粗I的立方等于加粗I的平方乘以加粗I等于空白减去加粗I" style="vertical-align:-6px;height:23px;width:112px" loading="lazy">i 3 = i 2 × i = - i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="加粗I的4次方等于左括号加粗I的平方右括号的平方等于开括号- 1闭括号的平方等于1" style="vertical-align:-6px;height:23px;width:147px" loading="lazy">i 4 = ( i 2 ) 2 = - 1 2 = 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="加粗I的5次方等于左括号加粗I平方右括号平方空白叉乘加粗I等于加粗I" style="vertical-align:-6px;height:23px;width:114px" loading="lazy">i 5 = ( i 2 ) 2 × i = i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="黑体I的6次方等于开括号黑体I的平方闭括号的立方等于开括号- 1闭括号的立方等于空白- 1" style="vertical-align:-6px;height:23px;width:169px" loading="lazy">i 6 = i 2 3 = - 1 3 = - 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 我们可以用同样的方法来使用i2与更高的权力打交道
" class="Wirisformula" role="math" alt="黑体I的23次方等于开括号黑体I的平方闭括号的11叉乘加粗I的次方等于开括号- 1闭括号的11叉乘加粗I的次方等于空白减去加粗I" style="vertical-align:-6px;height:23px;width:233px" loading="lazy">i 23 = i 2 11 × i = - 1 11 × i = - i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 记住-1的偶数次方是1的奇数次方是-1
考试技巧
- 当你复习考试时,练习使用你的GDC来检查你用手计算的复数
- 这将加快你使用矩形形式的GDC的速度,同时也为你提供大量手工计算的练习机会
工作的例子
一)
简化表达式
2 8 - 6 i - 5 3 + 4 i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="2个开括号8减去6个直的I闭括号减去5个直的I 3加上4个直的I闭括号" style="vertical-align:-6px;height:22px;width:140px" loading="lazy">。
b)
给定两个复数z 1 = 3 + 4 i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="Z下标1等于3加4个I" style="vertical-align:-12px;height:28px;width:74px" loading="lazy">而且z 2 = 6 + 7 i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="Z下标2等于6加7个I" style="vertical-align:-12px;height:28px;width:74px" loading="lazy">,找z 1 × z 2 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="Z下标1叉乘空白Z下标2" style="vertical-align:-12px;height:28px;width:54px" loading="lazy">。
复数共轭和除法
当分复数,复共轭用于将分母更改为实数。
什么是复共轭?
- 对于一个给定的复数
" class="Wirisformula" role="math" alt="Z = a + b I" style="vertical-align:-6px;height:22px;width:68px" loading="lazy">的复共轭z = a + b i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="z" style="vertical-align:-6px;height:22px;width:10px" loading="lazy">表示为z {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="Z的星号次幂" style="vertical-align:-6px;height:23px;width:18px" loading="lazy">,在那里z * {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="Z的星号次方乘以等于(a - b) I" style="vertical-align:-6px;height:23px;width:76px" loading="lazy">z * = a - b i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 如果
" class="Wirisformula" role="math" alt="Z = a - b I" style="vertical-align:-6px;height:22px;width:68px" loading="lazy">然后z = a - b i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="Z的星号次方等于(a + b) I" style="vertical-align:-6px;height:23px;width:76px" loading="lazy">z * = a + b i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 你会发现:
" class="Wirisformula" role="math" alt="Z + Z的星号次幂" style="vertical-align:-6px;height:23px;width:44px" loading="lazy">总是真实的,因为z + z * {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="左括号a + b I右括号加上左括号a - b I右括号等于2a" style="vertical-align:-6px;height:22px;width:159px" loading="lazy">( a + b i ) + ( a - b i ) = 2 a {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 例如:
" class="Wirisformula" role="math" alt="左括号6加上5个直I右括号空格加上空格左括号6减去5个直I右括号空格等于空格6加上6加上5个直I减去5个直I空格等于空格12" style="vertical-align:-6px;height:22px;width:295px" loading="lazy">( 6 + 5 i ) + ( 6 - 5 i ) = 6 + 6 + 5 i - 5 i = 12 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 例如:
" class="Wirisformula" role="math" alt="Z - Z的星号次幂" style="vertical-align:-6px;height:23px;width:44px" loading="lazy">总是虚的,因为z - z * {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="左括号a + b I右括号减去左括号a - b I右括号等于2b I" style="vertical-align:-6px;height:22px;width:165px" loading="lazy">a + b i - ( a - b i ) = 2 b i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 例如:
" class="Wirisformula" role="math" alt="左括号6 + 5直I右括号空格减去空格左括号6 - 5直I右括号空格等于空格6 - 6 + 5直I减去左括号- 5直I右括号空格等于空格10直I" style="vertical-align:-6px;height:22px;width:329px" loading="lazy">( 6 + 5 i ) - ( 6 - 5 i ) = 6 - 6 + 5 i - ( - 5 i ) = 10 i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 例如:
" class="Wirisformula" role="math" alt="Z叉乘Z的星号次幂" style="vertical-align:-6px;height:23px;width:44px" loading="lazy">总是真实的,因为z × z * {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="开括号a加b直I闭括号开括号a减b直I闭括号等于a方加a b直I减a b直I减b方直I方等于a方加b方" style="vertical-align:-6px;height:23px;width:333px" loading="lazy">(如a + b i a - b i = a 2 + a b i - a b i - b 2 i 2 = a 2 + b 2 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="I的平方等于- 1" style="vertical-align:-6px;height:23px;width:56px" loading="lazy">)i 2 = - 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 例如:
" class="Wirisformula" role="math" alt="左括号6加上5个直I右括号6减去5个直I右括号空格等于空格36空格加上30个直I空格-空格30个直I空格减去25个直I平方空格等于空格36空格-空格25左括号- 1右括号空格等于空格61" style="vertical-align:-6px;height:23px;width:450px" loading="lazy">( 6 + 5 i ) ( 6 - 5 i ) = 36 + 30 i – 30 i - 25 i 2 = 36 – 25 ( - 1 ) = 61 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 例如:
复数怎么除?
- 要除两个复数:
- 步骤1:用分数的形式表示计算结果
- 2 .相乘分母的共轭得到上下:
" class="Wirisformula" role="math" alt="分数分子a + b直I /分母c + d直I端分数等于空白分数分子a + b直I /分母c + d直I端分数空白交叉乘以空白分数分子c - d直I /分母c - d直I端分数" style="vertical-align:-17px;height:47px;width:197px" loading="lazy">a + b i c + d i = a + b i c + d i × c - d i c - d i {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 这确保我们乘以了1;所以不会影响整体价值
- 第三步:将答案乘出来并简化
- 分母应该是实数
- 第四步:把你的答案用笛卡尔式写成两项,如果需要,每项都可以简化
- 或转换为所需的形式,如有需要
- 你的GDC将能够以笛卡尔形式除两个复数
- 练习这样做,如果可以的话用它来检查你的答案
考试技巧
- 我们可以加快寻找的过程
" class="Wirisformula" role="math" alt="zz *号乘以" style="vertical-align:-6px;height:22px;width:29px" loading="lazy">的基本模式z z * {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="开括号x加上一个闭括号开括号x减去一个闭括号等于x方减去a方" style="vertical-align:-6px;height:23px;width:164px" loading="lazy">x + a x - a = x 2 - a 2 {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} - 我们可以把这个应用到复数上:
" class="Wirisformula" role="math" alt="左括号a + b I右括号a - b I右括号等于a方- b方I方等于a方+ b方" style="vertical-align:-6px;height:23px;width:253px" loading="lazy">a + b i a - b i = a 2 - b 2 i 2 = a 2 + b 2 {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"}
(利用事实是 " class="Wirisformula" role="math" alt="I的平方等于- 1" style="vertical-align:-6px;height:23px;width:56px" loading="lazy">)i 2 = - 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} - 所以
" class="Wirisformula" role="math" alt="3加4个I" style="vertical-align:-6px;height:22px;width:41px" loading="lazy">乘以它的共轭3 + 4 i {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="3的平方加上4的平方等于25" style="vertical-align:-6px;height:23px;width:85px" loading="lazy">3 2 + 4 2 = 25 {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"}
- 所以
工作的例子
找到…的价值