课程： 1.11 想象和复杂数字 | The Way To Learn

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想象和复杂数字
Introduction
::导言

How do you evaluate the square root of a negative number? The square roots of negative numbers are referred to as . Complex numbers are of the form $\begin{aligned} a + b i \end{aligned}$ , where $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ are real numbers and $\begin{aligned} i \end{aligned}$ is the imaginary uni t. What is $\begin{aligned} i \end{aligned}$ ? $\begin{aligned} i^{2} = - 1 \end{aligned}$ . Therefore, $\begin{aligned} \sqrt{- 1} = i \end{aligned}$ . This section will cover evaluating expressions and solving equations involving complex numbers.
::您如何评估负数的平方根 ? 负数的平方根被称为。复数为 a+bi 的表单 a+B, 其中a和b 是真实数字, i 是假想单位。 i 是什么 ? i2=-1。因此, - 1=i 。本节将包含对表达式的评价和解决包含复杂数字的方程式。

Imaginary Numbers
::想象数字

What is the square root of -1?
::-1的平方根是什么?

You may recall finding roots of negatives in algebra when attempting to solve equations like $\begin{aligned} x^{2} + 4 = 0 \end{aligned}$ .
::在试图解析 X2+4=0 等方程式时,您可能记得在代数中找到底部根。

Since there are no real numbers that can be squared to equal -4, this equation has no real solution. What is the imaginary constant "i"?
::因为没有真正数字可以平方到 4, 这个方程式没有真正的解决方案。什么是想象中的常数“ i ” ?

   The definition of "i" :
::“i”的定义:

$\begin{aligned} i = \sqrt{- 1} \end{aligned}$


The use of the word imaginary does not mean these numbers are not very useful. For a long period in the history of mathematics, it was thought that the square root of a negative number was in fact only within the mathematical imagination. That has changed. Mathematicians now consider imaginary numbers as another set of numbers that have real significance, but do not fit on the number line. Engineers, scientists, and others solve real-world problems using combinations of real and imaginary numbers—called complex numbers—every day.
::假想字的使用并不意味着这些数字不是非常有用的。在数学史的很长一段时间里,人们认为负数的平方根实际上只是在数学想象中。这一点已经改变。数学家现在将假想数视为另一组具有实际意义但并不符合数字线的数字。工程师、科学家和其他人利用真实数字和假想数字的组合 — — 所谓复杂数字 — — 每天解决现实世界的问题。

Imaginary values such as $\begin{aligned} \sqrt{- 16} \end{aligned}$ can be simplified by simplifying the radical into $\begin{aligned} \sqrt{16} \cdot \sqrt{- 1} \end{aligned}$ , yielding $\begin{aligned} 4 \sqrt{- 1} \end{aligned}$ or $\begin{aligned} 4 i \end{aligned}$ .
::将激进性简化为16+1, 产生4-1或4i, 可以简化象- 16这样的想象值。

The uses of i become more apparent when you begin working with increased powers of i , as you will see in the examples below.
::当您开始使用更大的i能力时,I的用途就更加明显了,如下文的例子所示。

Complex Numbers
::复数数

The video below provides an overview of how to identify and simplify expressions involving . It includes relevant vocabulary and examples.
::以下视频概述了如何识别和简化涉及......的表达方式,包括相关的词汇和实例。



When you combine imaginary numbers with real numbers, you get complex numbers.
::当将假想数字与实际数字结合起来时,就会得到复杂的数字。

   The definition of complex numbers :
::复杂数字的定义:

Complex numbers are of the form $\begin{aligned} a + b i \end{aligned}$ , where $\begin{aligned} a \end{aligned}$ is a real number, $\begin{aligned} b \end{aligned}$ is a real number, and $\begin{aligned} i \end{aligned}$ is the imaginary constant $\begin{aligned} \sqrt{- 1} \end{aligned}$ .

Plotting points was something you may have done in another mathematics course. For instance, plotting the point (4, 5) meant starting at the origin and moving 4 units to the right, the x direction, and 5 units up, the y direction.
::绘图点是你在另一个数学课程中可能做过的事情。例如,绘制点(4,5)意味着从起始点开始将4个单位移到右边,X方向和向上5个单位,y方向。

In this lesson, one of the things we will consider is the graphing of complex numbers such as $\begin{aligned} 4 + 3 i \end{aligned}$ .
::在这个教训中,我们将要考虑的一件事是绘制4+3i等复杂数字的图表。

In essence it doesn't sound that hard, but which direction do you only imagine moving 3 units?
::本质上,它听起来并不那么难, 但哪个方向你只想象移动3个单位?

a + bi is the standard or rectangular form of a complex number .
::a + 双是复数的标准或矩形。

A complex number is a number that has a real part (in this case, a ), and an imaginary part—that is, the i with a coefficient b .
::复数是一个具有真实部分(在此情况下,a)和假想部分(即具有系数b的i)的数字。

are a superset of the real numbers, meaning that all of the real numbers are part of the set of complex numbers.
::是一个真实数字的超级, 这意味着所有真实数字都是一组复杂数字的一部分。

Given a + bi , if b = 0 (meaning there is no imaginary part to the complex number), then all you have remaining is a real number. Viewed this way, every real number can be written as a complex number (just let it equal a ), but there are many more complex numbers than real numbers. Hence, the complex numbers are a superset of the real numbers:
::以 + 双表示, 如果 b = 0 (意思是复数中没有虚构的部分) , 那么您所剩的只是一个真实的数字。从这个角度看, 每个真实的数字都可以写成一个复杂的数字( 让它等于 a ) , 但数字比实际数字复杂得多。因此, 复杂的数字是真实数字的超集 :

Graphing complex numbers
::绘制复杂数字的图表

In standard form, a + bi , a complex number can be graphed using rectangular coordinates ( a , b ). The $\begin{aligned} x \end{aligned}$ -coordinate represents “real number” values, while the $\begin{aligned} y \end{aligned}$ -coordinate represents the “imaginary” values.
::在标准格式, + 双, 复数可以使用矩形坐标(a, b) 绘制图表。 x 坐标代表“实际数字”值, Y 坐标代表“想象”值。

Examples
::实例

Example 1
::例1

Simplify $\begin{aligned} \sqrt{- 5} \end{aligned}$
::简化 - 5

Solution:
::解决方案 :

Step 1: Factor out the -1: $\begin{aligned} \sqrt{- 5} = \sqrt{(- 1) \cdot (5)} \end{aligned}$
::第1步:将-1:-5=(-1)(5)乘以-1:-5=(-1)(5)

Step 2: Apply the multiplication rule for radicals: $\begin{aligned} = \sqrt{- 1} \sqrt{5} \end{aligned}$
::第2步:对激进物适用乘法规则:%15

Step 3: Convert to $\begin{aligned} i \end{aligned}$ : $\begin{aligned} = i \sqrt{5} \end{aligned}$
::第3步:转换为i:=i5

Example 2
::例2

Simplify $\begin{aligned} \sqrt{- 72} \end{aligned}$
::简化 - 72

Solution:
::解决方案 :

Step 1: Factor out the -1: $\begin{aligned} \sqrt{- 72} = \sqrt{(- 1) \cdot (72)} \end{aligned}$
::第1步:将-1:-72=(-1)-(72)-(72)乘以第1步

Step 2: Apply the multiplication rule for radicals: $\begin{aligned} = \sqrt{- 1} \sqrt{72} \end{aligned}$
::第2步:对激进物适用乘法规则:172

Step 3: Convert to $\begin{aligned} i \end{aligned}$ : $\begin{aligned} = i \sqrt{72} \end{aligned}$
::第3步:转换为i:=i72

Step 4: We’re not done yet! Since $\begin{aligned} 72 = 36 \cdot 2 \end{aligned}$
::步骤4:我们还没有完成!

$\begin{aligned} i \sqrt{72} & = i \sqrt{36} \sqrt{2} \\ = i (6) \sqrt{2} \\ = 6 i \sqrt{2} \end{aligned}$

::i72=i362=i(6)2=6i2

Example 3
::例3

Strange things happen when the imaginary constant i is multiplied by itself different numbers of times.
::当想象中的常数i 本身乘以不同次数时,奇怪的事情就会发生。

a) What is i ² ?
:a) 什么是i2?

Solution:
::解决方案 :

$\begin{aligned} i^{2} \end{aligned}$ is the same as $\begin{aligned} (\sqrt{- 1})^{2} \end{aligned}$ . When you square a square root, the two numbers cancel and you are left with the number originally inside the radical, in this case, $\begin{aligned} - 1 \end{aligned}$ .
::i2 与 (- 1) 2 相同。当平方方根时, 两个数字取消, 然后留给您原在基数内的数字, 此处为-1 。

$\begin{aligned} ∴ i^{2} = - 1 \end{aligned}$

b) What is i ³ ?
:b) 什么是i3?

Solution:
::解决方案 :

$\begin{aligned} i^{3} \end{aligned}$ is the same thing as $\begin{aligned} i^{2} \cdot i \end{aligned}$ , which is $\begin{aligned} - 1 \cdot i \end{aligned}$ or $\begin{aligned} - i \end{aligned}$ .
::i3与i2i相同,即-1i或-i。

$\begin{aligned} ∴ i^{3} = - i . \end{aligned}$
::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}是的 {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}是的 {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}是的 {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}是的 {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}是的

c) What is i ⁴ ?
:c) 什么是i4?

Solution:
::解决方案 :

$\begin{aligned} i^{4} = i^{2} \cdot i^{2} \end{aligned}$ which is $\begin{aligned} (- 1) \cdot (- 1) \end{aligned}$
::i4=i2-i2, 即 (- 1)_(- 1)_( - 1)_BAR_( - 1)_BAR_( - 1)_BAR_( - 1)_BAR_( - 1)_BAR_

$\begin{aligned} ∴ i^{4} = 1 \end{aligned}$

Example 4
::例4

Solve for $\begin{aligned} x \end{aligned}$ : $\begin{aligned} (x - 1)^{2} + 4 = 0 \end{aligned}$
::解决 x: (x- 1) 2+4=0

Solution:
::解决方案 :

Subtract 4 from both sides of the equation:
::从等式两侧减4:

$\begin{aligned} (x - 1)^{2} = - 4 \end{aligned}$

:x-1)24

Take the square root of both sides of the equation. Since the square root of a squared number is equal to the absolute value, you must include the positive and negative version of the solution:
::选择方程式两侧的平方根。由于平方数字的平方根等于绝对值, 您必须包含正反解决方案版本 :

$\begin{aligned} \sqrt{(x - 1)^{2}} = \pm \sqrt{- 4} \end{aligned}$

:x-1)24

$\begin{aligned} x - 1 = \pm \sqrt{- 4} \end{aligned}$

::x-14

Convert $\begin{aligned} \sqrt{- 1} \end{aligned}$ to $\begin{aligned} i \end{aligned}$ :
::转换 -1 至 i :

$\begin{aligned} x - 1 = \pm \sqrt{- 1} \sqrt{4} \end{aligned}$

::x-114

$\begin{aligned} x - 1 = \pm 2 i \end{aligned}$

::x-12i

Solve for $\begin{aligned} x \end{aligned}$ :
::解决 x:

$\begin{aligned} x = 1 \pm 2 i \end{aligned}$

::x=12i

Therefore, $\begin{aligned} x = 1 + 2 i \end{aligned}$ or $\begin{aligned} 1 - 2 i \end{aligned}$ .
::因此,x=1+2i或1-2i。

Example 5
::例5

Graph the complex number $\begin{aligned} z = 2 + 2 i \end{aligned}$ in rectangular form . Note that $\begin{aligned} z \end{aligned}$ is often used to denote complex numbers.
::矩形形的复数 z=2+2i 图形。请注意, z 通常用于表示复数。

Solution:
::解决方案 :

The coordinate (2, 2) is graphed as shown below. First, move along the real (or horizontal) axis the number of units for the real part (2), and then go up or down along the imaginary (or vertical) axis the number of units for the imaginary part (the coefficient in front of i ).
::坐标 (2, 2) 如下图所示。首先, 沿着实际( 水平) 轴移动实际部分(2) 的单位数量, 然后沿着想象( 垂直) 轴上下移动, 沿着想象部分的单位数量( i 前的系数) 上下移动。

Example 6
::例6

Solve each equation and express it as a complex number. (Note: If the imaginary part is 0, express the solution as $\begin{aligned} a + 0 i \end{aligned}$ ).
::解析每个方程式并将其表达为一个复数。 (注: 如果假想部分为 0, 则表示解析为+0i) 。

a) $\begin{aligned} x^{2} + 24 = 0 \end{aligned}$
::a) x2+24=0

Solution:
::解决方案 :

$\begin{aligned} x^{2} & = - 24 \\ \sqrt{x^{2}} & = \pm \sqrt{- 24} \\ x & = \pm \sqrt{- 24} \\ x & = \pm 2 i \sqrt{6} \end{aligned}$

::x224x224x24x}2i6

b) $\begin{aligned} 2 x^{2} - 4 x + 7 = 0 \end{aligned}$
::b) 2x2-4x+7=0

Solution:
::解决方案 :

To solve the equation $\begin{aligned} 2 x^{2} - 4 x + 7 = 0 \end{aligned}$ , we need to use the quadratic formula.
::要解析方程 2x2 -4x+7=0, 我们需要使用二次方程公式。

Given $\begin{aligned} a x^{2} + b x + c = 0 \end{aligned}$ , $\begin{aligned} x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \end{aligned}$ . In this example, $\begin{aligned} a = 2 \end{aligned}$ , $\begin{aligned} b = - 4 \end{aligned}$ , and $\begin{aligned} c = 7 \end{aligned}$ . Start by substituting the values into the formula:
::给定 ax2+bx+c=0, xbb2-4ac2a。在此示例中, a=2, b=4, c=7. 开始将值替换为公式 :

$\begin{aligned} x & = \frac{4 \pm \sqrt{(- 4)^{2} - 4 \cdot 2 \cdot 7}}{2 \cdot 2} \\ = \frac{4 \pm \sqrt{16 - 56}}{4} \\ = \frac{4 \pm \sqrt{- 40}}{4} \\ = \frac{4 \pm 2 i \sqrt{10}}{4} \\ = 1 \pm \frac{\sqrt{10}}{2} i \end{aligned}$

::x=4(-4)2-2-42722=416-564=4404=42i104=1102i

Example 7
::例7

Plot each of the following complex numbers in rectangular form:
::以矩形形式绘制下列复杂数字的每一个:

a) (4 + 2 i )
:a) (4+2i)

b) (-3 + i )
:b) (-3+一)

c) (3 - 4 i )
:c) (3-4)

d) 3 i
:d) 3i

Solutions:
::解决办法:

Your graph should look like this:
::您的图表应该像这个样子 :



Review
::回顾

Simplify the following radicals:
::简化以下激进:

1. $\begin{aligned} \sqrt{- 9} \end{aligned}$

2. $\begin{aligned} \sqrt{- 17} \end{aligned}$

3. $\begin{aligned} \sqrt{108 - 140} \end{aligned}$

Multiply the imaginary numbers:
::乘以假想的数字 :

4. $\begin{aligned} 4 i \cdot 3 i \end{aligned}$
::4. 4i3i

5. $\begin{aligned} \sqrt{16} i \cdot 3 \end{aligned}$
::5. 16i3

6. $\begin{aligned} \sqrt{4 i^{2}} \cdot \sqrt{12} i \end{aligned}$
::6. 4i212i

7. Simplify and express as a complex number: $\begin{aligned} - \sqrt{60} + \sqrt{- 121} \end{aligned}$
::7. 简化和表示复杂数字:-60121

8. Solve the equation and express the answer as a simplified complex number: $\begin{aligned} x (4 x) + 4 = 0 \end{aligned}$
::8. 解析方程,以简化的复合号表示答案:x(4x)+4=0

9. Graph the complex numbers:
a) 3 + 2i
b) 2 - 3i
c) -2 + 2i
::9. 图解复合数字:a) 3+2ib) 2 - 3ic)-2+2i

Simplify:
::简化 :

10. $\begin{aligned} \sqrt{- 324} \end{aligned}$

11. $\begin{aligned} \sqrt{- 121} \end{aligned}$

12. $\begin{aligned} - \sqrt{- 16} \end{aligned}$

13. $\begin{aligned} - \sqrt{- 1} \end{aligned}$

14. $\begin{aligned} \sqrt{- 1.21} \end{aligned}$

Simplify:
::简化 :

15. $\begin{aligned} i^{3} \end{aligned}$
::15. i3

16. $\begin{aligned} 24 i^{20} \end{aligned}$
::16. 24i20

17. $\begin{aligned} i^{225} \end{aligned}$
::17. i225

18. $\begin{aligned} i^{1024} \end{aligned}$
::18. i1024

Multiply:
::乘数 :

19. $\begin{aligned} i^{4} \cdot i^{11} \end{aligned}$
::19.4i11

20. $\begin{aligned} 5 i^{6} \cdot 5 i^{8} \end{aligned}$
::20. 5i6__5i8

21. $\begin{aligned} 3 \sqrt{- 75} \cdot 5 \sqrt{- 3} \end{aligned}$

22. $\begin{aligned} - 4 \sqrt{- 10} \cdot 5 \sqrt{- 3} \cdot 6 \sqrt{- 18} \end{aligned}$

23. What are complex numbers technically the sum of?
::23. 从技术上讲,什么是复杂数字的总和?

Express in simplest form in terms of $\begin{aligned} i \end{aligned}$ :
::以最简单的形式以一表示:

24. $\begin{aligned} 13 - \sqrt{- 49} \end{aligned}$

25. $\begin{aligned} 10 - \sqrt{\frac{- 4}{36}} \end{aligned}$

26. $\begin{aligned} 4 + \sqrt{- 250} \end{aligned}$

27. $\begin{aligned} 3^{2} \sqrt{- 0.0009} \end{aligned}$

28. $\begin{aligned} \sqrt{- 0.16} - (- \sqrt{27}) \end{aligned}$

29. $\begin{aligned} 9 \sqrt{- 8 i^{9}} + 3 \sqrt{25} \end{aligned}$
::29. 9-89+325

30. Two complex numbers are graphed below. What are the numbers expressed in standard complex number form?
::30. 以下图表列出了两个复杂数字,标准复杂数字表格中的数字是多少?

Review (Answers)
::回顾(答复)

Please see the Appendix.
::请参看附录。

Resource
::资源资源资源资源资源资源资源资源资源资源资源资源

Note: For a very detailed explanation of i and the complex numbers, visit:
::注:关于i和复杂数字的非常详细解释,访问:

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