2.16科学计数法

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  科学计数法

  

  [Figure 1]

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  Brian is doing research on the Sun for science class. He finds an article that states that the average distance from Earth to the Sun is  $\begin{array}{r}1.496\times {10}^8\end{array}$  kilometers. Brian is confused because 1.496 doesn't seem like a very big number, and he knows that Earth is far away from the Sun. How can Brian correctly interpret the distance from Earth to the Sun that he found in his research?
  ::布莱恩正在研究太阳科学课。 他发现一篇文章指出从地球到太阳的平均距离是1.496×10 8公里。 布赖恩很困惑, 因为1.496看起来并不很大, 而且他知道地球离太阳很远。 布莱恩如何正确解释他在研究中发现的从地球到太阳的距离?

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  In this concept, you will learn how to write very large and very small numbers in scientific notation.
  ::在这个概念中,你将学会如何在科学符号中写出非常大和非常小的数字。

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  Writing Numbers in Scientific Notation
  ::科学标注中的写作编号

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  Very large and very small numbers are used frequently in science. Here are some examples.
  ::科学中经常使用大量和极小的数量,例如。

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  • The distance between Earth and Jupiter is about 595,000,000 kilometers.
    ::地球和木星之间的距离约为595,000,000公里。
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  • The diameter of an insect's cell is about 0.000000000017 meters.
    ::昆虫细胞的直径约为0.0000000017米。

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  To make it easier to read, write, and calculate these extreme numbers, scientists use scientific notation.  Scientific notation  is a way of representing a very large or very small number without having to write all of the zeros at the beginning or end of the number.
  ::为了方便阅读、写作和计算这些极端数字,科学家们使用科学符号。 科学符号是一种代表非常大或非常小数字的方式,不必在数字的开始或结束处写所有的零。

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  When a number is written in  scientific notation  it is written as a product of a number that is at least 1 but less than 10 multiplied by a power of 10. Large numbers (numbers greater than 1) are written with a  positive  power of ten. Small numbers (numbers between 0 and 1) are written with a negative power of ten. The specific power of 10 indicates just how big or how small the number is.
  ::当一个数字以科学符号写成时,它作为数字的产物写成,该数至少为1,但小于10乘以10的功率。大数(数字大于1)写成,正数为10。小数(数字介于0和1之间)写成,负数为10。10的具体功率为10,表示数字有多大或多小。

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  Here are the same quantities from before written in scientific notation. 
  ::这是以前以科学符号写成的相同数量。

  • $\begin{array}{r}595,000,000=5.95\times {10}^8\end{array}$  
  • $\begin{array}{r}0.000000000017=1.7\times {10}^{\text{-}11}\end{array}$

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  Notice that the first number is very large and it has a positive exponent on the 10. The second number is very small and it has a negative exponent on the 10. Also notice that when written in scientific notation, both numbers are the product of a decimal number less than 10 and a power of 10.
  ::注意第一个数字非常大,在10个数字上有一个正数字,第二个数字非常小,在10个数字上有一个负数字,在10个数字上有一个负数字,还注意,在以科学符号书写时,这两个数字都是小数数小于10和10的功率的产物。

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  Here are the steps for writing a number in scientific notation.
  ::以下是在科学符号中写数的步骤。

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  1. Move the  decimal point  so that it is to the right of the first non- zero  digit of the number. The result should be a number that is at least 1 but less than 10. This will be the first part of your number in scientific notation.
     ::将小数点移到数字的第一个非零位数右侧。结果应该是至少一个数字,但小于10。这将是您数字中科学符号的第一部分。
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  2. Count how many spaces you needed to move your decimal point in step 1. The number of spaces will be your power of 10. If you moved the decimal point to the left, your exponent will be positive. If you moved your decimal point to the right, your exponent will be negative.
     ::计算您需要多少空格才能将小数点移动到第一步 1 。 空格数将是您10 的功率 。 如果您将小数点移到左边, 您的表情将是正数 。 如果您将小数点移到右边, 您的表情将是负数 。
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  3. The number in scientific notation is the decimal number from step 1 multiplied by 10 to the power from step 2.
     ::科学标记中的数字是从第1步乘以10小数乘以第2步乘以第10小数乘以第2步的功率。

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  Here is an example.
  ::举一个例子。

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  Write 595,000,000 in scientific notation.
  ::在科学符号中写下595,000,000

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  Start by finding the first non-zero digit and put a decimal point to its right. Here, the 5 at the beginning of the number is the first non-zero digit.
  ::首先找到第一个非零位数, 然后将小数点放在右侧。 这里, 数字开头的 5 是第一个非零位数 。

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  5.95000000 which is equal to 5.95
  ::等于5.95000的5.95000

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  Notice that you don't need to write the zeros at the end of the number anymore because they are to the right of a decimal point.
  ::请注意,您不再需要在数字结尾处写零, 因为它们位于小数点右侧。

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  Next, count how many spaces you needed to move the decimal point to get from 595,000,000 to 5.95. Remember that in 595,000,000 the decimal point is at the very end.
  ::下一步, 计数您需要多少空格才能将小数点从595,000,000 移动到5.95, 记住, 在595,000,000 中, 小数点在结尾处。

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  595,000,000 going to 5.95: Move the decimal point 8 spaces to the left.
  ::595 000 000 将移动到5.95:将小数点8空格向左移。

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  Now, put everything together. Your number in scientific notation is 5.95 multiplied by 10 to the power of 8. Because you have a very large number and you moved the decimal point to the left in the first step, your exponent will be positive.
  ::现在,把一切组合起来。你在科学符号中的编号是5.95乘以10乘以8的功率8, 因为你有一个非常大的数字, 在第一步将小数点移到左边, 你的表情将是正数 。

  $$\begin{array}{r}5.95\times {10}^8\end{array}$$

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  The answer is  $\begin{array}{r}595,000,000=5.95\times {10}^8.\end{array}$
  ::答案是595,000,000=5.95x10,8。

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  Sometimes you will be given a number in scientific notation and you will want to write it as a regular number not in scientific notation. To do this, just follow the steps in reverse.
  ::有时候,科学符号会给您一个数字,而您会把它写成一个普通数字,而不是科学符号。要做到这一点,请遵循倒数步骤。

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  Here is an example.
  ::举一个例子。

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  Write  $\begin{array}{r}3.24\times {10}^{\text{-}5}\end{array}$  as a number not in scientific notation.
  ::将3.24 × 10 - 5的编号写成不是科学符号的编号。

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  First, look at the exponent on the 10. The exponent is -5. Because the exponent is negative, your number is a very small number less than 1 and you will be moving the decimal point to the left to get back to the original number.
  ::首先,请看10点的指数 -5. 因为指数为负值, 您的数值小于 1, 您将会将小数点移到左边, 以便回到原来的数字 。

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  Next, move the decimal point on the 3.24. You will move the decimal point 5 spaces to the left. Insert zeros into any blank spaces.
  ::下一步,移动 3. 24 上的小数点。您将把小数点 5 空格向左移动。在任何空格中插入零。

  $$\begin{array}{r}3.24\text{goes to}0.0000324\end{array}$$

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  Notice that the result is a small number less than 1. This is exactly what you wanted since the number in scientific notation had a negative exponent.
  ::注意结果小于1, 这正是你想要的, 因为科学符号中的数字有一个负数字。

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  The answer is  $\begin{array}{r}3.24\times {10}^{\text{-}5}=\mathrm{0.0000324.}\end{array}$
  ::答案是3.24×10 -5=0.000324。

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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 Brian and his research on the Sun.
  ::早些时候,你得到一个问题 关于布莱恩 和他研究太阳。

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  He found that the average distance from Earth to the Sun is  $\begin{array}{r}1.496\times {10}^8\end{array}$  kilometers and he wasn't sure what that meant.
  ::他发现从地球到太阳的平均距离是 1.496×10 8公里 他不知道这意味着什么

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  First, Brian needs to realize that the distance is given in scientific notation. The distance isn't 1.496 kilometers, it's a lot more than that!
  ::距离不是1.496公里,远不止这些!

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  To write the distance not in scientific notation Brian should look at the exponent on the 10. The exponent is 8. Because the exponent is positive, he will be moving the decimal point to the right to get back to the original number.
  ::要在科学标记中写出距离,Brian应该看看10号上的表情,表情是8号,因为表情是正数,所以他将把小数点移到右侧,回到原来的数字。

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  Next, he should move the decimal point on the 1.496. He needs to move the decimal point 8 spaces to the right. He should insert zeros into any blank spaces.
  ::接下来, 他应该移动1.496 上的小数点。 他需要将小数点 8 空格移到右边。 他应该在任何空白空格中插入零 。

  $$\begin{array}{r}1.496\text{goes to}149,600,000\end{array}$$

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  Notice that the result is a large number. This makes sense because Brian knows that Earth and the Sun are far apart.
  ::注意结果是一个很大的数字。 这很有道理, 因为布莱恩知道地球和太阳是相距甚远的。

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  The answer is the average distance from Earth to the Sun is 149,600,000 kilometers.
  ::答案是从地球到太阳的平均距离是149,600,000公里。

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  Example 2
  ::例2

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  Write $\begin{array}{r}4.5\times {10}^{\text{-}6}\end{array}$  as a number not in scientific notation.
  ::将4.5 × 10 - 6的编号写成不是科学符号的编号。

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  First, look at the exponent on the 10. The exponent is -6. Because the exponent is negative, your number is a small number less than 1 and you will be moving the decimal point to the left to get back to the original number.
  ::首先,请看10点的指数。 指数是 -6. 因为指数是负数, 您的数值小于 1, 您将会将小数点移到左边, 以便回到原来的数字 。

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  Next, move the decimal point on the 4.5. You will move the decimal point 6 spaces to the left. Insert zeros into any blank spaces.
  ::下一步,移动4.5上的小数点。您将把小数点 6 空格向左移动。在任何空白空格中插入零。

  $$\begin{array}{r}4.5\text{goes to}0.0000045\end{array}$$

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  Notice that the result is a small number less than 1. This is exactly what you wanted since the number in scientific notation had a negative exponent.
  ::注意结果小于1, 这正是你想要的, 因为科学符号中的数字有一个负数字。

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  The answer is  $\begin{array}{r}4.5\times {10}^{\text{-}6}=\mathrm{0.0000045.}\end{array}$
  ::答案是4.5×10-6=0.000045。

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  Example 3
  ::例3

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  Write 450,000,000 in scientific notation.
  ::在科学符号上写450 000 000美元。

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  Start by finding the first non-zero digit and put a decimal point to its right. Here, the 4 at the beginning of the number is the first non-zero digit.
  ::首先找到第一个非零位数, 然后将小数点放在右侧。 这里, 数字开头的 4 是第一个非零位数 。

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  4.50000000 which is equal to 4.5
  ::4.50000 000,等于4.5

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  Next, count how many spaces you needed to move the decimal point to get from 450,000,000 to 4.5.
  ::接下来, 计数您需要多少空格才能移动小数点, 从450,000,000 移动到4.5 。

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  450,000,000. going to 4.5: Move the decimal point 8 spaces to the left.
  ::4.5:将小数点8空格向左移动。

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  Now, put everything together. Your number in scientific notation is 4.5 multiplied by 10 to the power of 8. Because you have a very large number and you moved the decimal point to the left in the first step, your exponent will be positive.
  ::现在,把一切组合起来。你在科学符号中的数字是4.5乘以10乘以8 的功率。因为你的数非常大,在第一步将小数点移到左边,你的表率将是正数。

  $$\begin{array}{r}4.5\times {10}^8\end{array}$$

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  The answer is  $\begin{array}{r}450,000,000=4.5\times {10}^8.\end{array}$
  ::答案是450,000,000=4.5×10,8。

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  Example 4
  ::例4

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  Write $\begin{array}{r}3.4\times {10}^5\end{array}$  as a number not in scientific notation.
  ::将3.4 × 10 5 编号写成不是科学符号的编号。

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  First, look at the exponent on the 10. The exponent is 5. Because the exponent is positive, your number is a large number greater than 1 and you will be moving the decimal point to the right to get back to the original number.
  ::首先,请看10号指数的指数5。 因为指数是正数, 您的数字大大大于1, 您将会将小数点移到右侧, 以便回到原来的数字 。

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  Next, move the decimal point on the 3.4. You will move the decimal point 5 spaces to the right. Insert zeros into any blank spaces.
  ::下一步,移动3.4 上的小数点。您将把小数点 5 空格向右移动。在任何空白空格中插入零。

  $$\begin{array}{r}3.4\text{goes to}340,000\end{array}$$

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  Notice that the result is a large number. This is exactly what you wanted since the number in scientific notation had a positive exponent.
  ::注意结果是一个大数字。 这正是你想要的, 因为科学符号中的数字有一个积极的缩写 。

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  The answer is  $\begin{array}{r}3.4\times {10}^5=340,000.\end{array}$
  ::答案是3.4 × 10 5 = 340,000。

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  Example 5
  ::例5

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  Write 0.0000000067 in scientific notation.
  ::写500000067 在科学标记。

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  Start by finding the first non-zero digit and put a decimal point to its right. Here, the 6 is the first non-zero digit.
  ::首先找到第一个非零位数, 然后将小数点放在右侧。 在这里, 6 是第一个非零位数 。

  $\begin{array}{r}6.7\end{array}$
  

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  Next, count how many spaces you needed to move the decimal point to get from 0.0000000067 to 6.7.
  ::下一步, 计数您需要多少空格才能移动小数点, 从 00000000067 到 6. 7 。

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  0.0000000067 going to 6.7: Move the decimal point 9 spaces to the right.
  ::0000067 将移动到6.7:将小数点9空格向右移动。

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  Now, put everything together. Your number in scientific notation is 6.7 multiplied by 10 to the power of -9. Because you have a very small number and you moved the decimal point to the right in the first step, your exponent will be negative.
  ::现在,将一切组合起来。你在科学符号中的编号是6.7乘以10乘以 -9. 的功率。 因为你有一个非常小的数字, 第一步将小数点移到右边, 您的表情将是负的 。

  $$\begin{array}{r}6.7\times {10}^{\text{-}9}\end{array}$$

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  The answer is  $\begin{array}{r}0.0000000067=6.7\times {10}^{\text{-}9}.\end{array}$
  ::答案是0000000067=6.7×10 -9。

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  Review
  ::回顾

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  Write each number in scientific notation.
  ::在科学符号中写下每个编号。

  1. 0.0000000056731
  2. 24,010,000,000
  3. 960,000,000,000,000,000
  4. 0.0000001245
  5. 36,000,000
  6. 0.00098
  7. 0.000000034
  8. 345,000,000

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  Write each number as a number not in scientific notation.
  ::将每个数字写成数字,而不是在科学符号中。

  9. $\begin{array}{r}3.808\times {10}^{11}\end{array}$  
  10. $\begin{array}{r}2.1\times {10}^6\end{array}$  
  11. $\begin{array}{r}5.912\times {10}^8\end{array}$  
  12. $\begin{array}{r}6.78\times {10}^{-6}\end{array}$  
  13. $\begin{array}{r}5.7\times {10}^9\end{array}$  
  14. $\begin{array}{r}4.5\times {10}^{-5}\end{array}$  
  15. $\begin{array}{r}3.21\times {10}^7\end{array}$  

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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