Section: 乘数和分解复合体数 | 5.9 乘数和分解复合体数

Section outline

Mr. Marchez draws a triangle on the board. He labels the height (2 + 3 i ) and the base (2 - 4 i ). "Find the area of the triangle," he says. (Recall that the area of a triangle is $\begin{aligned} A = \frac{1}{2} b h \end{aligned}$ , b is the length of the base and h is the length of the height.)
::Marchez先生在棋盘上画了一个三角形。他标注了高度(2+3i)和基座(2-4i)。他说,“找到三角形的区域”。 (回顾三角形的区域是A=12bh,b是基底的长度,h是高度的长度。)

Multiplying and Dividing Complex Numbers
::乘数和分解复合体数

When multiplying , FOIL the two numbers together and then combine like terms. At the end, there will be an $\begin{aligned} i^{2} \end{aligned}$ term. Recall that $\begin{aligned} i^{2} = - 1 \end{aligned}$ and continue to simplify.
::当乘法时, FOIL 将两个数字相加, 然后将类似条件合并。结尾处将有一个 i2 术语。回顾 i2 @% 1, 并继续简化。

Simplify the following expression.
::简化以下表达式。

$\begin{aligned} 6 i (1 - 4 i) \end{aligned}$

::6i(1--4i)

Distribute the $\begin{aligned} 6 i \end{aligned}$ to both parts inside the parenthesis.
::将6i分布在括号内的两个部分。

$\begin{aligned} 6 i (1 - 4 i) = 6 i - 24 i^{2} \end{aligned}$

::6i( 1- 4i) =6i- 24i2

Substitute $\begin{aligned} i^{2} = - 1 \end{aligned}$ and simplify further.
::替换 i21, 进一步简化。

$\begin{aligned} = 6 i - 24 (- 1) \\ = 24 + 6 i \end{aligned}$

::=6i-24(-1)=24+6i

Remember to always put the real part first.
::记住永远把真正的部分放在第一位

$\begin{aligned} (5 - 2 i) (3 + 8 i) \end{aligned}$
:5-2i)(3+8i)

FOIL the two terms together.
::将这两个条件放在一起。

$\begin{aligned} (5 - 2 i) (3 + 8 i) & = 15 + 40 i - 6 i - 16 i^{2} \\ = 15 + 34 i - 16 i^{2} \end{aligned}$

:5-2i)(3+8i)=15+40i-6i-16i2=15+34i-16i)

Substitute $\begin{aligned} i^{2} = - 1 \end{aligned}$ and simplify further.
::替换 i21, 进一步简化。

$\begin{aligned} = 15 + 34 i - 16 (- 1) \\ = 15 + 34 i + 16 \\ = 31 + 34 i \end{aligned}$

::=15+34i-16(-1)=15+34i+16=31+34i

Dividing complex numbers is a bit more complicated. Similar to irrational numbers, complex numbers cannot be in the denominator of a fraction. To get rid of the complex number in the denominator, we need to multiply by the complex conjugate . If a complex number has the form $\begin{aligned} a + b i \end{aligned}$ , then its complex conjugate is $\begin{aligned} a - b i \end{aligned}$ . For example, the complex conjugate of $\begin{aligned} - 6 + 5 i \end{aligned}$ would be $\begin{aligned} - 6 - 5 i \end{aligned}$ . Therefore, rather than dividing complex numbers, we multiply by the complex conjugate.
::分解复杂数字比较复杂。与非理性数字相似, 复杂数字不能在分母的分母中。要摆脱分母中的复杂数字, 我们需要乘以复杂的同母体。如果一个复杂数字具有 a+Bi 的形式, 那么它的复杂共和就是 a- bi 。例如, 6+5i 的复杂共和体将是 -6 - 5i 。因此, 我们不是将复杂数字分开,而是通过复杂的同母体来乘以 6 - 5i 。

Simplify the following expression.
$\begin{aligned} \frac{8 - 3 i}{6 i} \end{aligned}$

::简化以下表达式.8-3i6i

In the case of dividing by a pure , you only need to multiply the top and bottom by that number. Then, use multiplication to simplify.
::在除法为纯净的情况下,只需将上下乘以该数字即可。然后,使用乘法简化。

$\begin{aligned} \frac{8 - 3 i}{6 i} \cdot \frac{6 i}{6 i} & = \frac{48 i - 18 i^{2}}{36 i^{2}} \\ = \frac{18 + 48 i}{- 36} \\ = \frac{18}{- 36} + \frac{48}{- 36} i \\ = - \frac{1}{2} - \frac{4}{3} i \end{aligned}$

::8-3i6i6i6i=48i-18i236i2=18+48i-36=18-36-36+48-36i12-43i

When the complex number contains fractions, write the number in standard form, keeping the real and imaginary parts separate. Reduce both fractions separately.
::当复数包含分数时, 请以标准格式写入数字, 将真实部分和想象部分分开。将两个分数分开。

Simplify the following expression.
::简化以下表达式。

$\begin{aligned} \frac{3 - 5 i}{2 + 9 i} \end{aligned}$

::3-5i2+9i 3-5i2+9i

Now we are dividing by $\begin{aligned} 2 + 9 i \end{aligned}$ , so we will need to multiply the top and bottom by the complex conjugate, $\begin{aligned} 2 - 9 i \end{aligned}$ .
::现在我们除以 2+9i, 所以我们需要乘以上层和下层复杂的共和体, 2 - 9i 。

$\begin{aligned} \frac{3 - 5 i}{2 + 9 i} \cdot \frac{2 - 9 i}{2 - 9 i} & = \frac{6 - 27 i - 10 i + 45 i^{2}}{4 - 18 i + 18 i - 81 i^{2}} \\ = \frac{6 - 37 i - 45}{4 + 81} \\ = \frac{- 39 - 37 i}{85} \\ = - \frac{39}{85} - \frac{37}{85} i \end{aligned}$

::3-5i2+9i2-9i2-9i-2-9i=6-27i-10i+45i24-18i+18i-81i2=6-37i-454+8139-37i853985-3785i

Notice, by multiplying by the complex conjugate, the denominator becomes a real number and you can split the fraction into its real and imaginary parts.
::注意,通过乘以复杂的共鸣, 分母变成了一个真实的数字, 你可以将分母分割成其真实的和想象的部位。

In the previous three problems above , we substituted $\begin{aligned} i^{2} = - 1 \end{aligned}$ to simplify the fraction further. Your final answer should never have any power of $\begin{aligned} i \end{aligned}$ greater than 1.
::在以上前三个问题中, 我们替换了i2 @% 1 来进一步简化分数。您的最后答案绝对不能比 1 强。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the area of the triangle.
::早些时候,你被要求找到三角形的区域。

The area of the triangle is $\begin{aligned} \frac{(2 + 3 i) (2 - 4 i)}{2} \end{aligned}$ so FOIL the two terms together and divide by 2.
::三角形区域为(2+3i)(2-4i)2,因此,FOIL两个词相加,除以2。

$\begin{aligned} (2 + 3 i) (2 - 4 i) = 4 - 8 i + 6 i - 12 i^{2} \\ = 4 - 2 i - 12 i^{2} \end{aligned}$

:2+3i)(2-4i)=4-8i+6i-12i2=4-2i-12i2)

Substitute $\begin{aligned} i^{2} = - 1 \end{aligned}$ and simplify further.
::替换 i21, 进一步简化。

$\begin{aligned} = 4 - 2 i - 12 (- 1) \\ = 4 - 2 i + 12 \\ = 16 - 2 i \end{aligned}$

::=4-2i-12(-1)=4-2i+12=16-2i

Now divide this product by 2.
::现在将这一产品除以2。

$\begin{aligned} \frac{16 - 2 i}{2} = 8 - i \end{aligned}$
::16-2i2=8-i

Therefore the area of the triangle is $\begin{aligned} 8 - i \end{aligned}$ .
::因此三角形的面积是8-i。

Example 2
::例2

What is the complex conjugate of $\begin{aligned} 7 - 5 i \end{aligned}$ ?
::7 -5i的复杂结合是什么?

$\begin{aligned} 7 + 5 i \end{aligned}$
::7+5i 7+5i

Example 3
::例3

Simplify the following complex expression: $\begin{aligned} (7 - 4 i) (6 + 2 i) \end{aligned}$ .
::简化以下复杂表达式: (7-4i)(6+2i)

FOIL the two expressions.
::发现两种表达方式。

$\begin{aligned} (7 - 4 i) (6 + 2 i) & = 42 + 14 i - 24 i - 8 i^{2} \\ = 42 - 10 i + 8 \\ = 50 - 10 i \end{aligned}$

:7-4i)(6+2i)=42+14i-24i-8i2=42-10i+8=50-10i)

Example 4
::例4

Simplify the following complex expression: $\begin{aligned} \frac{10 - i}{5 i} \end{aligned}$ .
::简化以下复杂表达式: 10- i5i。

Multiply the numerator and denominator by $\begin{aligned} 5 i \end{aligned}$ .
::乘以 5i 乘以分子和分母。

$\begin{aligned} \frac{10 - i}{5 i} \cdot \frac{5 i}{5 i} & = \frac{50 i - 5 i^{2}}{25 i^{2}} \\ = \frac{5 + 50 i}{- 25} \\ = \frac{5}{- 25} + \frac{50}{- 25} i \\ = - \frac{1}{5} - 2 i \end{aligned}$

::10 - i5i5i5i=50i-5i225i2=5+50i-225=5 - 25=5 - 25+50-25i15-2i

Example 5
::例5

Simplify the following complex expression: $\begin{aligned} \frac{8 + i}{6 - 4 i} \end{aligned}$ .
::简化以下复杂表达式: 8+i6-4i。

Multiply the numerator and denominator by the complex conjugate, $\begin{aligned} 6 + 4 i \end{aligned}$ .
::乘以 6+4i 的复合二次曲线的分子和分母。

$\begin{aligned} \frac{8 + i}{6 - 4 i} \cdot \frac{6 + 4 i}{6 + 4 i} & = \frac{48 + 32 i + 6 i + 4 i^{2}}{36 + 24 i - 24 i - 16 i^{2}} \\ = \frac{48 + 38 i - 4}{36 + 16} \\ = \frac{44 + 38 i}{52} \\ = \frac{44}{52} + \frac{38}{52} i \\ = \frac{11}{13} + \frac{19}{26} i \end{aligned}$

::8+i6-4i6+4i6+4i6+4i6+4i=48+32i+6i+6i+4i+4i236+24i-24i-166i2=48+38i-436+16=44+38i52=4452+3852i=1113+1926i

Review
::回顾

Simplify the following expressions. Write your answers in standard form.
::简化以下表达式。以标准格式写入您的答复。

$\begin{aligned} i (2 - 7 i) \end{aligned}$
::i(2-7i) i(2-7i)

$\begin{aligned} 8 i (6 + 3 i) \end{aligned}$
::8i(6+3i)

$\begin{aligned} - 2 i (11 - 4 i) \end{aligned}$
::--2i(11-4i)

$\begin{aligned} (9 + i) (8 - 12 i) \end{aligned}$
:9+一)(8-12i)

$\begin{aligned} (4 + 5 i) (3 + 16 i) \end{aligned}$
:4+5i)(3+16i)

$\begin{aligned} (1 - i) (2 - 4 i) \end{aligned}$
:1-(一)(2)-4(一))

$\begin{aligned} 4 i (2 - 3 i) (7 + 3 i) \end{aligned}$
::4i(2-3i)(7+3i)

$\begin{aligned} (8 - 5 i) (8 + 5 i) \end{aligned}$
:8-5i)(8+5i)

$\begin{aligned} \frac{4 + 9 i}{3 i} \end{aligned}$
::4+9i3i 4+9i3i

$\begin{aligned} \frac{6 - i}{12 i} \end{aligned}$
::6-i-i12i

$\begin{aligned} \frac{7 + 12 i}{- 5 i} \end{aligned}$
::7+12i-5i

$\begin{aligned} \frac{4 - 2 i}{6 - 6 i} \end{aligned}$
::4-26-6-6i 4-26-6-6i

$\begin{aligned} \frac{2 - i}{2 + i} \end{aligned}$
::2-一2+一

$\begin{aligned} \frac{10 + 8 i}{2 + 4 i} \end{aligned}$
::10+8i2+4i

$\begin{aligned} \frac{14 + 9 i}{7 - 20 i} \end{aligned}$
::14+9i7-20i

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Section outline

Multiplying and Dividing Complex Numbers ::乘数和分解复合体数

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Multiplying and Dividing Complex Numbers
::乘数和分解复合体数

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)