4.15科学记数法值

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  科学记数法值

  

  [Figure 1]

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  David is working on his science homework. The assignment is to find the mass of each planet in the solar system and write it using scientific notation. While researching the assignment, he finds that the mass of the Earth is 5,973,600,000,000,000,000,000,000 kg. How can David express the mass of the Earth in scientific notation?
  ::大卫正在做科学作业,任务是在太阳系中找到每个行星的质量,并用科学符号写出来。在研究这一任务时,他发现地球的质量是5,973,600,000,000,000,000公斤。大卫如何用科学符号表达地球的质量?

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  In this concept, you will learn how to write numbers using scientific notation.
  ::在此概念中,您将学会如何使用科学符号写数字。

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  Scientific Notation
  ::科学符号

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  Scientific notation  is a shortcut for writing very small and very large numbers. It is very useful for scientists, mathematicians, and engineers. It is useful in careers where people work with very large or very small numbers. For example, the distance from the Earth to the Sun is 96,960,000 miles. Instead of writing out the number every time, you can use scientific notation. Numbers in scientific notation follow the form
  ::科学标记是写出非常小和非常大的数字的捷径。 它对于科学家、数学家和工程师非常有用。 它对于人们以非常大或非常小的数字工作的职业非常有用。 例如, 从地球到太阳的距离是96,960,000英里。 您可以使用科学标记, 而不是每次写出数字。 科学标记中的数字可以跟随形式。

  $$\begin{array}{r}a\times {10}^n,\end{array}$$

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  where  $\begin{array}{r}a\end{array}$ , a number between 1 and 10, is multiplied by a  power of ten .
  ::a 的数值在1至10之间,乘以10的功率。

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  Remember that a power of ten is written as  $\begin{array}{r}{10}^n\end{array}$ , where  $\begin{array}{r}n\end{array}$  is the exponent that tells you how many times 10 is multiplied by itself.
  ::记住10的功率是10n,n是表示多少乘以10的指数。

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  Here is 96,960,000 written in scientific notation.
  ::这是96,960,000 写在科学符号上。

  $$\begin{array}{r}96,960,000=9.696\times {10}^7\end{array}$$

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  To change 96,960,000 into a number between 1 and 10, you must move the  decimal point  7 spaces to the left. So to get 96,960,000 from 9.696, you must do the opposite and move the decimal point 7 spaces to the right. Moving the decimal point to the right requires   a total of 7 times or  $\begin{array}{r}{10}^7\end{array}$ . Note that large numbers written in scientific notation will use  positive   .
  ::要将96,960,000个位数改变为1至10之间的数字,您必须将小数点7的空格移到左边。因此,要从9.696中取出96,960,000个空格,您必须反转,将小数点7的空格移到右边。将小数点移到右边总共需要7倍或10.7倍。请注意,大量以科学符号写成的数字将使用正值 。

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  Here is a very small number written in scientific notation.
  ::这是在科学符号中写下的非常小的数字。

  $$\begin{array}{r}0.000000023=2.3\times {10}^{-8}\end{array}$$

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  To change 0.000000023 into a number between 1 and 10, you must move the decimal point 8 spaces to the right. So to convert 2.3 to 0.000000023, move the decimal 8 spaces to the left. Moving the decimal to the left requires multiplying by a negative power of 10 a total of 8 times or  $\begin{array}{r}{10}^{-8}\end{array}$ . Remember that multiplying by a  negative power of ten  is the same as dividing by a power of ten.
  ::将 0.000023 转换为 1 到 10 之间的数字, 您必须将小数点 8 空格向右移动。 因此, 要将小数点 2.3 转换为 0.000023, 将小数点 8 空格向左移动。 要将小数点 8 空格向左移动, 需要以负功率乘以 10 乘以 8 次或 10 - 8 次。 记住, 负功率乘以 10 乘以 10 等于 10 次。

  $$\begin{array}{r}{10}^{-8}=\frac{1}{{10}^8}\end{array}$$

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  Note that decimal numbers less than 1 will use negative  powers of ten  when written in scientific notation.
  ::请注意,小数小于1的小数点在科学符号书写时将使用负十分的负功率。

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  Here is a small decimal number. Write the number using scientific notation.
  ::这是小小数数。 使用科学符号写入数字 。

  $$\begin{array}{r}0.00056\end{array}$$

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  First, use the scientific notation form. Change the number to be a number between 1 and 10. This number is 5.6.
  ::首先,使用科学符号表。将数字改为1至10之间的数字。这个数字是5.6。

  $$\begin{array}{r}5.6\times {10}^n\end{array}$$

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  Then, identify the power of ten,  $\begin{array}{r}n\end{array}$ . Since 0.00056 is a number less than 1, the power of ten will be negative. Notice that to go from 0.00056 to 5.6, you must move the decimal point four places to the right. This means the exponent will be −4.
  ::那么,请标明10, n. 的功率。 因为 0. 0 056 的功率小于 1, 10 的功率是负的。 请注意, 要从 0. 0 0 056 到 5. 6, 您必须将小数点 4 移到右边。 这意味着推算符将是- 4 。

  $$\begin{array}{r}5.6\times {10}^{-4}\end{array}$$

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  To check if this is correct, multiply 5.6 times  $\begin{array}{r}{10}^{-4}\end{array}$ . When multiplying by a negative power of ten, move the decimal point to the left 4 times.
  ::要检查是否正确, 请乘以 5.6 乘以 10 至 4。 当负功率乘以 10 时, 将小数点移到左边 4 次 。

  $$\begin{array}{r}0.\underset{\leftarrow }{0005}6\end{array}$$

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  Here are some charts that might help you remember how to convert numbers to scientific notation and scientific notation to numbers.
  ::以下是一些图表,可以帮助您记住如何将数字转换为科学符号,将科学符号转换为数字。

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  Converting a Number to Scientific Notation
  ::将数字转换为科学符号

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  Large Numbers → Positive Power of Ten
  ::10国的正权 10国的正权 10国的正权

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  Small Decimal Numbers → Negative Power of Ten
  ::小十进制数字 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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  Converting Scientific Notation to a Number
  ::将科学符号转换为数字

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  Positive Power of Ten → Move the Decimal to the Right
  ::将十进制向右移

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  Negative Power of Ten → Move the Decimal to the Left
  ::10 的负功率 + + + 移动小数向左

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about David’s science homework.
  ::更早之前, 大卫的科学作业给您带来了问题。

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  David needs to express the mass on the Earth, 5,973,600,000,000,000,000,000,000 kg, using scientific notation.
  ::David需要用科学标记来表达地球上的重量,即5,973,600,000,000,000,000公斤。

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  First, use the scientific notation form. Change the number to be a number between 1 and 10. This number is 5.9736.
  ::首先,使用科学符号表。将数字改为1至10之间的数字。这个数字是5.9736。

  $$\begin{array}{r}5.9736\times {10}^n\end{array}$$

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  Then, identify the power of ten,  $\begin{array}{r}n\end{array}$ . Since the mass of the Earth is a large number, the power of ten will be positive. The decimal moves 24 places to get from 5,973,600,000,000,000,000,000,000 to 5.9736, so  $\begin{array}{r}n\end{array}$  is 24.
  ::然后,确定10,n的功率。由于地球质量是巨大的,10的功率将是正数。小数点将24个位移到5,973,600,000,000,000,000到5.9736,所以是24个位。

  $$\begin{array}{r}5.9736\times {10}^{24}\end{array}$$

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  The scientific notation form of 5,973,600,000,000,000,000,000,000 kg is  $\begin{array}{r}5.9736\times {10}^{24}kg\end{array}$ .
  ::科学标记表5,973,600,000,000,000,000公斤为5.9736×10,24千克。

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  Example 2
  ::例2

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  Write the number in scientific notation.
  ::在科学符号中写下编号 。

  0.0000000034

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  First, use the scientific notation form. Change the number to be a number between 1 and 10. This number is 3.4.
  ::首先,使用科学符号表。将数字改为1至10之间的数字。这个数字是3.4。

  $$\begin{array}{r}3.4\times {10}^n\end{array}$$

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  Then, identify the power of ten,  $\begin{array}{r}n\end{array}$ . Since 0.0000000034 is a small number, the power of ten will be negative. The decimal moves 9 places to get from 0.0000000034 to 3.4, so  $\begin{array}{r}n\end{array}$  is -9.
  ::那么,请标明10, n 的功率。 因为 0000000034 是小数, 10 的功率将是负数。 小数点后将 9 位移到 3.4 位, 所以 n 是 - 9 位 。

  $$\begin{array}{r}3.4\times {10}^{-9}\end{array}$$

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  The scientific notation form of 0.0000000034 is  $\begin{array}{r}3.4\times {10}^{-9}\end{array}$ .
  ::0000000034的科学标记形式为3.4×10-9。

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  Write the numbers in scientific notation.
  ::在科学符号中写下数字。

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  Example 3
  ::例3

  $$\begin{array}{r}0.0012\end{array}$$

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  First, use the scientific notation form. Change the number to be a number between 1 and 10. This number is 1.2.
  ::首先,使用科学符号表。将数字改为1至10之间的数字。这个数字是1.2。

  $$\begin{array}{r}1.2\times {10}^n\end{array}$$

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  Then, identify the power of ten,  $\begin{array}{r}n\end{array}$ . Since 0.0012 is a small number, the power of ten will be negative. The decimal moves 3 places to get from 0.0012 to 1.2, so  $\begin{array}{r}n\end{array}$  is -3.
  ::然后,请标明10 n 的功率。 因为0.0012是一个小数字, 10 的功率将是负数。 小数点后将3 位移到 0.0012 到 1.2, 所以 n 是 - 3 。

  $$\begin{array}{r}1.2\times {10}^{-3}\end{array}$$

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  The scientific notation form of 0.0012 is  $\begin{array}{r}1.2\times {10}^{-3}\end{array}$ .
  ::0.0012的科学编号表为1.2×10-3。

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  Example 4
  ::例4

  $$\begin{array}{r}78,000,000\end{array}$$

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  First, use the scientific notation form. Change the number to be a number between 1 and 10. This number is 7.8.
  ::首先,使用科学符号表。将数字改为1至10之间的数字。这个数字是7.8。

  $$\begin{array}{r}7.8\times {10}^n\end{array}$$

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  Then, identify the power of ten,  $\begin{array}{r}n\end{array}$ . Since 78,000,000 is a large number, the power of ten will be positive. The decimal moves 7 places to get from 78,000,000 to 7.8, so  $\begin{array}{r}n\end{array}$  is 7.
  ::那么,请指明10,n. 的功率。由于78,000,000是一个大数,10的功率将是正数。小数点将7位移到7.8位,从78,000,000移到7.8位,所以n是7位。

  $$\begin{array}{r}7.8\times {10}^7\end{array}$$

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  The scientific notation form of 78,000,000 is  $\begin{array}{r}7.8\times {10}^7\end{array}$ .
  ::科学标记表78 000 000是7.8×10 7。

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  Example 5
  ::例5

  $$\begin{array}{r}345,102,000,000\end{array}$$

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  First, use the scientific notation form. Change the number to be a number between 1 and 10. This number is 3.45102.
  ::首先,使用科学符号表。将数字改为1至10之间的数字。这个数字是3.45102。

  $$\begin{array}{r}3.45102\times {10}^n\end{array}$$

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  Then, identify the power of ten,  $\begin{array}{r}n\end{array}$ . Since 345,102,000,000 is a large number, the power of ten will be positive. The decimal moves 11 places to get from 345,102,000,000 to 3.45102, so  $\begin{array}{r}n\end{array}$  is 11.
  ::那么,请确认10 n. 的功率。 因为 345, 102, 000,000是一个大数, 10 的功率将是正数。 小数点后将11个位数从 345, 102 000, 000 移动到 3.45102, 所以 n是 11。

  $$\begin{array}{r}3.45102\times {10}^{11}\end{array}$$

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  The scientific notation form of 345,102,000,000 is  $\begin{array}{r}3.45102\times {10}^{11}\end{array}$ .
  ::科学标记表345,102,000,000为3.45102×10 11。

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  Review
  ::回顾

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  Write each decimal in scientific notation.
  ::在科学符号中写每个小数。

  1. 0.00045
  2. 0.098
  3. 30,000,000
  4. 0.000987
  5. 3,400,000
  6. 0.0000021
  7. 1,230,000,000,000
  8. 0.00000000345
  9. 0.00056
  10. 0.0098
  11. 0.024
  12. 0.000023
  13. 4,300
  14. 0.0000000000128
  15. 980
  16. 0.00000045

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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 。

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  Resources
  ::资源

   

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  ::注/亮

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