Course: 11.2 与复杂数字的相对性

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具有复杂数字的自解

The idea of a complex number can be hard to comprehend, especially when you start thinking about absolute value. In the past you may have thought of the absolute value of a number as just the number itself or its positive version. How should you think about the absolute value of a complex number?
::复杂数字的概念很难理解, 特别是当你开始思考绝对值的时候。过去你可能认为数字的绝对值只是数字本身或其正本。您应该如何看待复杂数字的绝对值 ?

Arithmetic Operations with Complex Numbers
::具有复杂数字的神学操作

Complex numbers follow all the same rules as real numbers for the operations of adding, subtracting, multiplying and dividing. There are a few important ideas to remember when working with complex numbers:
::复杂数字在添加、减、乘和分离操作中遵循与实际数字相同的规则。在使用复杂数字时,有一些重要的想法需要记住:

When simplifying, you must remember to combine imaginary parts with imaginary parts and real parts with real parts. For example, $\begin{align*}4+5i+2-3i=6+2i.\end{align*}$
::当简化时, 您必须记住要将想象中的部件与想象中的部件和真实部件结合起来。例如, 4+5i+2 - 2 - 3i=6+2i 。

If you end up with a complex number in the denominator of a fraction, eliminate it by multiplying both the numerator and denominator by the complex conjugate of the denominator.
::如果您最后在分数分母的分母中加上一个复数,则通过乘以分数的分母的复数和分母来消除分数。

The powers of $\begin{align*}i\end{align*}$ are:
::我的权力是:

$\begin{align*}i=\sqrt{-1}\end{align*}$
::i1

$\begin{align*}i^2=-1\end{align*}$
::i21

$\begin{align*}i^3=-\sqrt{-1}=-i\end{align*}$
::i31i

$\begin{align*}i^4=1\end{align*}$
::i4=1

$\begin{align*}i^5=i\end{align*}$
::i5=i

. . . and the pattern repeats
::和模式重复

Consider this complex expression:

\begin{aligned} (2 + 3 i) (1 - 5 i) - 3 i + 8 \end{aligned}

$\begin{align*}(2+3i)(1-5i)-3i+8\end{align*}$
::考虑这个复杂的表达式: (2+3i)(1-5i)--3i+8)

First, multiply the two binomials and then combine the imaginary parts with imaginary parts and real parts with real parts.
::首先,乘以两个二元体,然后将假想部分与假想部分结合起来,将真实部分与真实部分结合起来。

\begin{aligned} = 2 - 10 i + 3 i - 15 i^{2} - 3 i + 8 \\ = 10 - 10 i + 15 \\ = 25 - 10 i \end{aligned}

$\begin{align*}&= 2-10i+3i-15i^2-3i+8 \\ &= 10-10i+15 \\ &= 25-10i\end{align*}$
::=2-10i+3i-15i-2-3i+8=10-10i+15=25-10i

Note that a power higher than 1 of $\begin{align*}i\end{align*}$ can be simplified using the pattern above.
::请注意,高于i 1 的功率可以使用上述模式简化。

The complex plane is set up in the same way as the regular $\begin{align*}x, y\end{align*}$ plane, except that real numbers are counted horizontally and complex numbers are counted vertically. The following is the number $\begin{align*}4+3i\end{align*}$ plotted in the complex number plane. Notice how the point is four units over and three units up.
::复杂平面的设置与正常的xy平面相同,但实际数字是横向计算的,复杂数字是垂直计算的。以下是复杂数字平面中绘制的4+3i号。注意点是4个单位以上,3个单位以上。

The absolute value of a complex number like $\begin{align*}|4+3i|\end{align*}$ is defined as the distance from the complex number to the origin. You can use the Pythagorean Theorem to get the absolute value. In this case, $\begin{align*}|4+3i|=\sqrt{4^2+3^2}=\sqrt{25}=5.\end{align*}$
::%% 4+3i 等复数的绝对值被定义为从复数到来源的距离。您可以使用 Pytagorean 理论来获得绝对值。在这种情况下, +4+3i42+32=25=5 。

Examples
::实例

Example 1
::例1

Earlier, you were asked how to think about the absolute value of a complex number. A good way to think about the absolute value for all numbers is to define it as the distance from a number to zero. In the case of complex numbers where an individual number is actually a coordinate on a plane, zero is the origin.
::早些时候,有人问您如何思考一个复杂数字的绝对值。一个思考所有数字的绝对值的好方法就是将它定义为从数字到零的距离。在复杂的数字中,一个单个数字实际上是在平面上的坐标,在这种情况下,一个数字的源头是零。

Example 2
::例2

Compute the following power by hand and use your calculator to support your work.
::手工计算以下功率,并使用计算器支持您的工作。

$\begin{align*}\left(\sqrt{3}+2i\right)^3\end{align*}$
:3+2i)3

$\begin{align*}\left(\sqrt{3}+2i\right) \cdot \left(\sqrt{3}+2i\right) \cdot \left(\sqrt{3}+2i\right)\end{align*}$
:3+2i)(3+2i)(3+2i)(3+2i)(3+2i)

\begin{aligned} = (3 + 4 i \sqrt{3} - 4) (\sqrt{3} + 2 i) \\ = (- 1 + 4 i \sqrt{3}) (\sqrt{3} + 2 i) \\ = - \sqrt{3} - 2 i + 12 i - 8 \sqrt{3} \\ = - 9 \sqrt{3} + 10 i \end{aligned}

$\begin{align*}&= \left(3+4i \sqrt{3}-4\right)\left(\sqrt{3}+2i\right) \\ &= \left(-1+4i \sqrt{3}\right)\left(\sqrt{3}+2i\right) \\ &= -\sqrt{3}-2i+12i-8 \sqrt{3} \\ &= -9 \sqrt{3}+10i\end{align*}$
::=(3+4i3-4)(3+2i)=(-1+4i3)(3+2i)___3-2i+12i-83_93+10i)

A TI-84 can be switched to imaginary mode and then compute exactly what you just did. Note that the calculator will give a decimal approximation for $\begin{align*}-9\sqrt{3}\end{align*}$ .
::将 TI- 84 转换为假想模式, 然后精确计算您刚才所做的。注意计算器将给出- 93 的十进制近似值。

Example 3
::例3

Simplify the following complex expression.
::简化以下复杂表达式。

$\begin{align*}\frac{7-9i}{4-3i}+\frac{3-5i}{2i}\end{align*}$
::7-9i4-3i+3-5i2i

To add fractions you need to find a common denominator.
::要添加分数, 您需要找到一个共同的分母。

\begin{aligned} \frac{(7 - 9 i) \cdot 2 i}{(4 - 3 i) \cdot 2 i} + \frac{(3 - 5 i) \cdot (4 - 3 i)}{2 i \cdot (4 - 3 i)} \\ = \frac{14 i + 18}{8 i + 6} + \frac{12 - 20 i - 9 i - 15}{8 i + 6} \\ = \frac{15 - 15 i}{8 i + 6} \end{aligned}

$\begin{align*}& \frac{(7-9i) \cdot 2i}{(4-3i) \cdot 2i}+\frac{(3-5i) \cdot (4-3i)}{2i \cdot (4-3i)} \\ &= \frac{14i+18}{8i+6}+\frac{12-20i-9i-15}{8i+6} \\ &= \frac{15-15i}{8i+6}\end{align*}$
:

7-9i)(4-3i)(4-3i)(4-3-5i)(3-3-5i)(4-3i)(3-3i)(4-3i)(4-3i)(4-3i)=14i+1888i(4-3i)=14i+1888i+6+12-20i(12-20i)(9i)(158i)+6=15-15i-8i+6)

Lastly, eliminate the imaginary component from the denominator by using the conjugate.
::最后,用共产体从分母中除去假构成份。

\begin{aligned} = \frac{(15 - 15 i) \cdot (8 i - 6)}{(8 i + 6) \cdot (8 i - 6)} \\ = \frac{120 i - 90 + 120 + 90 i}{100} \\ = \frac{30 i + 30}{100} \\ = \frac{3 i + 3}{10} \end{aligned}

$\begin{align*}&=\frac{(15-15i) \cdot (8i-6)}{(8i+6) \cdot (8i-6)} \\ &= \frac{120i-90+120+90i}{100} \\ &= \frac{30i+30}{100} \\ &= \frac{3i+3}{10}\end{align*}$
::=(15-15i) =(8i-6)(8i+6) =(8i-6) =(120i-90+120+90i100=30i+30100=3i+310)

Example 4
::例4

Simplify the following complex number.
::简化下列复数。

$\begin{align*}i^{2013}\end{align*}$
::i2013

When simplifying complex numbers, $\begin{align*}i\end{align*}$ should not have a power greater than 1. The powers of $\begin{align*}i\end{align*}$ repeat in a four part cycle:
::在简化复杂数字时,我不应拥有大于1的功率。在四个部分周期里,我重复的功率:

\begin{aligned} i^{5} & = i = \sqrt{- 1} \\ i^{6} & = i^{2} = - 1 \\ i^{7} & = i^{3} = - \sqrt{- 1} = - i \\ i^{8} & = i^{4} = 1 \end{aligned}

$\begin{align*}i^5 &=i= \sqrt{-1} \\ i^6 &=i^2=-1 \\ i^7 &=i^3=-\sqrt{-1}=-i \\ i^8 &=i^4=1\end{align*}$
::i5=i*%1i6=i2*1i7=i3*1*1*i8=i4=1

Therefore, you just need to determine where 2013 is in the cycle. To do this, determine the remainder when you divide 2013 by 4. The remainder is 1 so $\begin{align*}i^{2013}=i\end{align*}$ .
::因此,您只需要确定2013年的周期位置。为此, 当您将2013年除以4时, 确定剩余部分。剩余部分为 1 so i2013=i 。

Example 5
::例5

Plot the following complex number on the complex coordinate plane and determine its absolute value.
::在复杂的坐标平面上绘制以下复杂数字并确定其绝对值。

$\begin{align*}-12+5i\end{align*}$
::- 12+5i

The sides of the right triangle are 5 and 12, which you should recognize as a Pythagorean triple with a hypotenuse of 13. $\begin{align*}|-12+5i|=13\end{align*}$ .
::右三角形的两边是5和12, 你应该承认它为匹达哥伦的三联赛, 时值为13, 12+5, 13。

Summary

Complex numbers follow the same rules as real numbers for adding, subtracting, multiplying, and dividing.
::复杂数字在添加、减、乘和分隔时遵循与实际数字相同的规则。

When simplifying complex numbers, combine imaginary parts with imaginary parts and real parts with real parts.
::当简化复杂数字时,将想象中的部件与想象中的部件结合起来,将真实部分与真实部分结合起来。

When a complex number is in the denominator of a fraction, multiply both the numerator and denominator by the complex conjugate of the denominator.
::当一个复杂数字在分数分母中时,将分子和分母乘以分母的复杂组合。

The powers of $\begin{align*}i\end{align*}$ are
- $\begin{align*}i = \sqrt{-1}\end{align*}$
  ::i1
- $\begin{align*}i^2 = -1\end{align*}$
  ::i21
- $\begin{align*}i^3 = - \sqrt{-1} = -1\end{align*}$
  ::i3111
- $\begin{align*}i^4 = 1\end{align*}$
  ::i4=1
::我的权力是i1 i21 i31 i4=1

The complex plane is set up with real numbers counted horizontally and complex numbers counted vertically.
::复杂的平面是用实际数字横向计算的,复杂的平面是垂直计算的。

The absolute value of a complex number is defined as the distance from the complex number to the origin, which can be calculated using the Pythagorean Theorem.
::复数的绝对值定义为从复数到原数的距离,可以使用毕达哥伦理论来计算。

Review
::回顾

Simplify the following complex numbers.
::简化下列复杂数字。

1. $\begin{align*}i^{252}\end{align*}$
::1.252

2. $\begin{align*}i^{312}\end{align*}$
::2. i312

3. $\begin{align*}i^{411}\end{align*}$
::3. i411

4. $\begin{align*}i^{2345}\end{align*}$
::4. i2345

For each of the following, plot the complex number on the complex coordinate plane and determine its absolute value.
::对于以下每一种,在复杂的坐标平面上绘制复合数字并确定其绝对值。

5. $\begin{align*}6-8i\end{align*}$
::6-8i 6-8i

6. $\begin{align*}2+i\end{align*}$
::6. 2+1

7. $\begin{align*}4-2i\end{align*}$
::7. 4-2i

8. $\begin{align*}-5i+1\end{align*}$
::8.--5i+1

Let $\begin{align*}c=2+7i\end{align*}$ and $\begin{align*}d=3-5i\end{align*}$ .
::c=2+7i和d=3-5i。

9. What is $\begin{align*}c+d\end{align*}$ ?
::9. 什么是C+D?

10. What is $\begin{align*}c-d\end{align*}$ ?
::10. 什么是C-D?

11. What is $\begin{align*}c \cdot d\end{align*}$ ?
::11. 什么是Cd?

12. What is $\begin{align*}2c-4d\end{align*}$ ?
::12. 什么是2c-4d?

13. What is $\begin{align*}2c \cdot 4d\end{align*}$ ?
::13. 什么是2c4d?

14. What is $\begin{align*}\frac{c}{d}\end{align*}$ ?
::14. 什么是CD?

15. What is $\begin{align*}c^2 - d^2\end{align*}$ ?
::15. 什么是C2 - d2?

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

General

具有复杂数字的自解

Arithmetic Operations with Complex Numbers ::具有复杂数字的神学操作

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)