4.4 想象数字

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  想象数字

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  Have you ever had an imaginary pet? Some  people have, particularly as young children.
  ::你有过想象中的宠物吗?

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  Wouldn't you be surprised if you and your real friend left your two imaginary puppies  alone together, and you came back to find a real puppy ?
  ::难道你不惊讶吗? 如果你和你真正的朋友 把你两个想象中的小狗单独留在一起 你回来找一只真正的小狗?

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  It's a silly thought, so what does it have to do with imaginary numbers?
  ::这是一个愚蠢的想法, 所以它有什么 与想象的数字有什么关系?

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  Imaginary Numbers
  ::想象数字

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

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  You may recall running into roots of negatives in algebra, when attempting to solve equations like $\begin{align*}x^2 + 4 = 0\end{align*}$ .
  ::当您试图解析 X2+4=0 等方程式时, 可能记得在代数中的底部根部正在冲入根部 。

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  Since there are no real numbers that can be squared to equal -4, this equation has no real solution. Enter the imaginary constant: "i".
  ::因为没有真正数字可以平方到 4 。 此方程式没有真正的解决方案。 输入假想的常数 : “ i ” 。

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  The definition of "i" : $\begin{align*}i = \sqrt{-1}\end{align*}$
  ::“i”的定义:i1

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  The use of the word imaginary does not mean these numbers are useless. 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, without real-world significance hence, imaginary. That has changed. Mathematicians now consider the imaginary numbers as another set of numbers that have real significance, but do not fit on what is called the number line, and 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是无用的。 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,没有真实世界的意义,因此是虚想的。 这已经改变了。 数学家现在认为虚想数字是另一组具有真正意义的数字,但并不符合所谓的数字线,工程师、科学家和其他人每天使用真实数字和假想数字(所谓的复杂数字)的组合解决真实世界的问题。

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

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  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。

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  Complex Numbers
  ::复数数

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  When you combine imaginary numbers with real numbers, you get :
  ::当将假想数字与实际数字结合起来时,你就会得到:

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  Complex numbers are of the form $\begin{align*}a + bi\end{align*}$ , where $\begin{align*}a\end{align*}$ is a real number, $\begin{align*}b\end{align*}$ is a real number, and $\begin{align*}i\end{align*}$ is the imaginary constant $\begin{align*}\sqrt{-1}\end{align*}$ .
  ::复数为窗体 a+Bi 的复数,其中a为实际数字,b为实际数字,i为想象中的常数-1。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given an analogy about imaginary pet s .
  ::早些时候,你被给了一个假想宠物的类比。

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  Do you see its application?
  ::你看到它的应用吗?

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  Two imaginary puppies creating a real puppy  is an oddly effective metaphor for the behavior of the powers of i .
  ::两只想象中的小狗创造出一只真正的小狗 是一个奇怪的有效比喻 代表我的力量行为

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  One i is imaginary, but two i's multiply to be a real number. In fact, every even power of i results in a real number!
  ::一个我想象中,但两个我乘以一个真正的数字。事实上,我的每一股力量都产生了一个真正的数字!

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  Example 2
  ::例2

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  Simplify $\begin{align*}\sqrt{-5}\end{align*}$ .
  ::简化 - 5 。

  $\begin{align*}\sqrt{-5} = \sqrt{(-1)\cdot (5)}\end{align*}$

  $\begin{align*}= \sqrt{-1}\sqrt{5}\end{align*}$

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

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  Example 3
  ::例3

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  Simplify $\begin{align*}\sqrt{-72}\end{align*}$ .
  ::简化-72。

  $\begin{align*}\sqrt{-72} = \sqrt{(-1)\cdot (72)}\end{align*}$

  $\begin{align*}= \sqrt{-1}\sqrt{72}\end{align*}$

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  $\begin{align*}= i\sqrt{72}\end{align*}$
  ::= i72

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  But, we’re not done yet! Since $\begin{align*}72 = 36 \cdot 2\end{align*}$
  ::但我们还没有完成! 自72=362

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  $\begin{align*}i\sqrt{72} = i\sqrt{36} \sqrt{2}\end{align*}$
  ::i72=i362

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  $\begin{align*}= i(6)\sqrt{2}\end{align*}$
  ::=i(6)2

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  $\begin{align*}= 6i\sqrt{2}\end{align*}$
  ::=6i2

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  Example 4
  ::例4

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  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。

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  1. What is i ² ?
     ::什么是i2?

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  $\begin{align*}i^2\end{align*}$ is the same as $\begin{align*}(\sqrt{-1})^2\end{align*}$ . When you square a square root, they cancel and you are left with the number originally inside the radical, in this case $\begin{align*}-1\end{align*}$
  ::i2 与 (- 1) 相同 。 当平方方根时, 它们取消, 并留给您原在基底中的数字, 在此情况下为-1 。

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  $\begin{align*}\therefore i^2 = -1\end{align*}$
  ::21

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  2. What is i ³ ?
     ::什么是i3?

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

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  $\begin{align*}\therefore i^3 = -i\end{align*}$
  ::

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  3. What is i ⁴ ?
     ::什么是i4?

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  $\begin{align*}i^4 = i^2 \cdot i^2\end{align*}$ which is $\begin{align*}-1 \cdot -1\end{align*}$
  ::i4=i2i2, 即-11

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

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  Example 5
  ::例5

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  Simplify the following radical:  $\begin{align*}\sqrt{108-140}\end{align*}$ .
  ::简化以下激进:108-140。

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  $\begin{align*}\sqrt{-32}\end{align*}$ : Subtract within the parenthesis
  ::- 32:括号内减法

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  $\begin{align*}\sqrt{32 \cdot -1}\end{align*}$ : Rewrite $\begin{align*}-32\end{align*}$ as $\begin{align*}-1 \cdot 32\end{align*}$
  ::321: 重写-32为-132

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  $\begin{align*}\sqrt{32} \cdot \sqrt{-1}\end{align*}$ : Rewrite as a product of radicals
  ::321:重写是激进的产物

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  $\begin{align*}\sqrt{32} \cdot i\end{align*}$ : Substitute $\begin{align*}\sqrt{-1} \to i\end{align*}$
  ::32i:替代物-1i

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  $\begin{align*}\sqrt{16 \cdot 2} \cdot i\end{align*}$ : Factor $\begin{align*}32\end{align*}$
  ::162i:因素32

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  $\begin{align*}4i\sqrt{2}\end{align*}$ : Simplify $\begin{align*}\sqrt{16}\end{align*}$
  ::4i2 : 简化 16

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  Example 6
  ::例6

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  Multiply the imaginary numbers:  $\begin{align*}4i \cdot 3i\end{align*}$ .
  ::乘以假想数字: 4i3i。

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  $\begin{align*}4 \cdot 3 \cdot i \cdot i\end{align*}$ : Using the commutative law for multiplication
  ::43ii:使用通货法进行乘法

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  $\begin{align*}12 \cdot i^2\end{align*}$ : Simplify
  ::12i2 : 简化

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  $\begin{align*}12 \cdot -1\end{align*}$ : Recall $\begin{align*}i^2 = -1\end{align*}$
  ::121:回顾i21

  $\begin{align*}-12\end{align*}$

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  Example 7
  ::例7

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  Multiply the imaginary numbers:  $\begin{align*}\sqrt{4i^2} \cdot \sqrt{12}i\end{align*}$ .
  ::乘以假想数字: 4i212i。

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  $\begin{align*}\sqrt{4} \cdot \sqrt{i^2} \cdot \sqrt{4} \cdot \sqrt{3} \cdot i\end{align*}$ : Factor
  ::4243i:因数

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  $\begin{align*}2 \cdot i \cdot 2 \cdot \sqrt{3} \cdot i\end{align*}$ : Simplify the roots
  ::223i:简化根

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  $\begin{align*}4\sqrt{3} \cdot i^2\end{align*}$ : Collect terms and simplify
  ::43i2:收集术语和简化

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  $\begin{align*}4\sqrt{3} \cdot -1\end{align*}$ : Recall $\begin{align*}i^2 = -1\end{align*}$
  ::校对:Portnoy

  $\begin{align*}-4\sqrt{3}\end{align*}$

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  Review
  ::回顾

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  Simplify:
  ::简化 :

  1. $\begin{align*}\sqrt{-49}\end{align*}$
  2. $\begin{align*}\sqrt{-81}\end{align*}$
  3. $\begin{align*}\sqrt{-324}\end{align*}$
  4. $\begin{align*}\sqrt{-121}\end{align*}$
  5. $\begin{align*}-\sqrt{-16}\end{align*}$
  6. $\begin{align*}-\sqrt{-1}\end{align*}$
  7. $\begin{align*}\sqrt{-1.21}\end{align*}$

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  Simplify:
  ::简化 :

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  8. $\begin{align*}i^8\end{align*}$
     ::i8 i8
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  9. $\begin{align*}i^{12}\end{align*}$
     ::i12 i12
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  10. $\begin{align*}i^3\end{align*}$
      ::i3 i3
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  11. $\begin{align*}24i^{20}\end{align*}$
      ::24i20 24i20
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  12. $\begin{align*}i^{225}\end{align*}$
      ::i225
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  13. $\begin{align*}i^{1024}\end{align*}$
      ::i1024

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  Multiply:
  ::乘数 :

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  14. $\begin{align*}i^4 \cdot i^{11}\end{align*}$
      ::i4i11
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  15. $\begin{align*}5i^6 \cdot 5i^8\end{align*}$
      ::5i6×5i8
  16. $\begin{align*}3\sqrt{-75} \cdot 5\sqrt{-3}\end{align*}$
  17. $\begin{align*}2\sqrt{-12} \cdot 6\sqrt{-27}\end{align*}$
  18. $\begin{align*}-4 \sqrt{-10} \cdot 5 \sqrt{-3} \cdot 6\sqrt {-18}\end{align*}$

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  Review (Answers)
  ::回顾(答复)

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  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 。