3.3 科学说明

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  科学符号

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  In science, measurements are often extremely small or extremely large.  It is inefficient to write the many zeroes in very small numbers like 0.00000000000000523.  Usually, the order of magnitude and the first few digits of the number are what people are interested in.  How should you represent these extreme numbers?
  ::在科学方面,测量往往非常小或非常大。 写出数量极小的零位数,比如00000000000523. 通常情况下,数量大小和数字的前几位是人们感兴趣的。 您应该如何代表这些极端数字呢?

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  Scientific Notation
  ::科学符号

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  Scientific notation is a means of representing very large and very small numbers in a more efficient way.  The general form of scientific notation is $\begin{array}{r}a\cdot {10}^b\end{array}$
  ::科学标记是一种以更有效率的方式代表非常大和非常小数量的手段。

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  The $\begin{array}{r}a\end{array}$  is a number between 1 and 10 and most often includes a decimal.  The integer $\begin{array}{r}b\end{array}$  is called the order of magnitude and is a measure of the general size of the number.  If $\begin{array}{r}b\end{array}$  is negative then the number is small and if $\begin{array}{r}b\end{array}$  is positive then the number is large.
  ::a 是一个介于 1 和 10 之间的数字,且通常包含小数。整数 b 称为数量级,是数字一般大小的量度。如果 b 为负数,则数字小,如果 b 为正数,则数字大。

  $$\begin{array}{rl}1,240,000& =1.24\cdot {10}^6\\ 0.0000354& =3.54\cdot {10}^{-5}\end{array}$$

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  Note that when switching to and from scientific notation the sign of $\begin{array}{r}b\end{array}$  indicates which direction and how many places to move the decimal point .
  ::请注意,在切换到科学符号和从科学符号中取出时,b的符号表示哪个方向和有多少位位位位可以移动小数点。

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  Take the number 0.000 000 000 000 000 000 000 000 000 000 910 938 22 kg. This is about the mass of an electron. The number is too long to write out all the time so it is best to write it in scientific notation.
  ::数字为 000 000 000 000 000 000 000 000 000 000 000 000 000 000 910 938 22 公斤。 这是关于电子质量的。 数字太长, 无法随时写出, 所以最好写在科学符号中 。

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  To write this number in scientific notation, count the number of decimal places that you have to move to determine the exponent. Since the decimal is moving to the right, it should be a negative exponent. An electron's mass written in scientific notation is:
  ::要在科学符号中写入此数字, 请计算您为确定指数而必须移动的小数位数。 由于小数点向右移动, 它应该是一个负指数。 以科学符号书写的电子质量是 :

    $\begin{array}{r}9.1093822\cdot {10}^{-31}\end{array}$

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  numbers that are in scientific notation is just an exercise in exponent rules:
  ::在科学符号中的数字只是对引言规则的一种练习:

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  $$\begin{array}{rl}(a\cdot {10}^x)\cdot (b\cdot {10}^y)& =a\cdot b\cdot {10}^{x+y}\\ (a\cdot {10}^x)÷(b\cdot {10}^y)& =\frac{a}{b}\cdot {10}^{x-y}\end{array}$$

  
  :a10x) (b10y) = ab(10)10x+y (a10x) (b10y) = 10x-y

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  Addition and subtraction require the numbers to have identical order of magnitudes.
  ::加和减要求数字在数量上必须具有相同的数量级。

  $\begin{array}{r}1.2\cdot {10}^6-5.5\cdot {10}^5=12\cdot {10}^5-5.5\cdot {10}^5=6.5\cdot {10}^5\end{array}$

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked how to represent numbers with a large number of zeroes either before or after the decimal point. In order to represent an extremely large or small number you should count the number of moves necessary for the decimal point to be directly after the first non- zero digit.  This count will be the order of magnitude and will be used as the exponent of 10 as a means of representing how large or small the number is. 
  ::早些时候,有人询问您如何在小数点之前或之后代表数字中有大量零的数字。为了代表一个非常大或小的数字,您应该计算小数点在第一个非零位数之后直接为第一个非零位数之后所需的移动次数。这个数字将是音量的顺序,并将用作10的提示,作为代表数字大小的手段。

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  Example 2
  ::例2

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  The Earth’s circumference is approximately 40,000,000 meters.  What is the radius of the earth in scientific notation? 
  ::地球环绕约4000万米。 地球在科学符号中的半径是多少?

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  The relationship between circumference and radius is $\begin{array}{r}C=2\pi r\end{array}$ . 
  ::环绕和半径之间的关系是C=2°r。

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  $$\begin{array}{rl}4.0\cdot {10}^7& =2\pi r\\ r& =\frac{4.0}{2\pi }\cdot {10}^7\approx 0.6366\dots \cdot {10}^7=6.366\cdot {10}^6\end{array}$$

  
  ::4 0107=2rr=4.021070.6366... 107=6.366106

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  Note that the number of significant digits required depends on the number of significant digits (or significant figures)  in the original measurements . This example is an approximation, therefore the number of significant digits aren't necessarily accurate.
  ::请注意, 所需的重要数字数取决于原始测量中的重要数字数( 或重要数字) 。 这个示例是一个近似值, 因此, 重要数字数不一定准确 。

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  Example 3
  ::例3

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  Order the following numbers from least to greatest. 
  ::排序以下数字,从最小到最大。

  $\begin{array}{r}5.411\cdot {10}^{-3}7.837\cdot {10}^{-4}9.999\cdot {10}^{3}9.5983\cdot {10}^{-7}8.0984\cdot {10}^3\end{array}$

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  First consider the order of magnitude of each number.  Small numbers have negative .  If two numbers have the same order of magnitude, then compare the actual digits.
  ::首先考虑每个数字的大小。 小数字为负数。 如果两个数字的大小相同, 那么比较实际的数字 。

  $\begin{array}{r}9.5983\cdot {10}^{-7}<7.837\cdot {10}^{-4}<5.411\cdot {10}^{-3}<8.0984\cdot {10}^3<9.999\cdot {10}^3\end{array}$

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  Example 4
  ::例4

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  Compute the following number and use scientific notation.
  ::计算以下数字并使用科学符号。

  $\begin{array}{r}2,000,{000}^3\cdot 3,{000}^4\end{array}$

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  First convert each number to scientific notation individually, then process the exponent and multiplication.
  ::首先将每个数字分别转换为科学标记,然后处理引号与乘法。

   

  $$\begin{array}{rl}2,000,{000}^3\cdot 3,{000}^4& =(2\cdot {10}^6{)}^3\cdot (3\cdot {10}^3{)}^4\\ & =8\cdot {10}^{18}\cdot 81\cdot {10}^{12}\\ & =648\cdot {10}^{30}\\ & =6.48\cdot {10}^{32}\end{array}$$

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  Example 5
  ::例5

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  Simplify the following expression.
  ::简化以下表达式 。

  $\begin{array}{r}(4.713\cdot {10}^7)+(8.985\cdot {10}^5)-(4.987\cdot {10}^2)\cdot (7.3\cdot {10}^{-6})÷(6.74\cdot {10}^{-9})\end{array}$

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  Resolve in order of standard order of operations
  ::按标准行动次序排列

  $\begin{array}{r}(4.713\cdot {10}^7)+(8.985\cdot {10}^5)-(4.987\cdot {10}^2)\cdot (7.3\cdot {10}^{-6})÷(6.74\cdot {10}^{-9})\end{array}$

  $$\begin{array}{rl}& =(4.713\cdot {10}^7)+(8.985\cdot {10}^5)-(5.40135\cdot {10}^5)\\ & =(471.3\cdot {10}^5)+(8.985\cdot {10}^5)-(5.40135\cdot {10}^5)\\ & =474.8836499\cdot {10}^5\\ & =4.748836499\cdot {10}^7\end{array}$$

  Summary

  播放段落 Scientific notation is a way of efficiently representing a very large or very small number, using the general form: $\begin{array}{r}a\times {10}^b.\end{array}$   ::科学标记是一种有效代表非常大或非常小数量的方法,使用一般形式: ax10b。 播放段落 The number $\begin{array}{r}a\end{array}$  is between 1 and 10, often including a decimal, while the integer $\begin{array}{r}b\end{array}$  is the order of magnitude. 播放段落 If $\begin{array}{r}b\end{array}$  is positive, the number is large. ::b 如果为正数,则数字很大。 播放段落 If $\begin{array}{r}b\end{array}$  is negative, the number is small. ::b 如为负数,数字较小。 ::数字a介于1至10之间,通常包括小数点,而整数b为数量级。如果b为正数,数字是大数。如果b为负数,数字是小数。 播放段落 When converting to and from scientific notation, the sign of $\begin{array}{r}b\end{array}$  indicates which direction and how many places to move the decimal point. ::当转换为科学符号或从科学符号转换时,b的符号表示哪个方向和有多少位位位位可以移动小数点。

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  Review
  ::回顾

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

  1. 152,780

  2. 0.00003256

  3. 56, 320

  4. 0.0821

  5. 1, 000, 000, 000, 000, 000, 000, 000

  6. 7.32

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  7. If the federal budget is $1.5 trillion, how much does it cost each individual, on average, if there are 300,000,000 people?
  ::7. 如果联邦预算为1.5万亿美元,那么如果有30万人,每人平均要花多少钱?

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  8. The Library of Congress has about 60,000,000 items.  How could you express this number in scientific notation?
  ::8. 国会图书馆有大约6 000万件物品,你怎么能在科学符号中表示这个数字?

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  9. The sun develops $\begin{array}{r}5\times {10}^{23}\end{array}$  horsepower per second.  How much horsepower is developed in a day? In a year with 365 days?
  ::9. 太阳每秒能发展5×1023马力,一天能发展多少马力?一年365天?

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  10. A light-year is about 5,869,713,600 miles.  A spacecraft travels $\begin{array}{r}8.23\times {10}^4\end{array}$  miles per hour.  How long will it take the spacecraft to travel a light–year?
  ::10. 光年约为5,869,713,600英里,航天器每小时飞行8.23×104英里,航天器每小时飞行8.23×104英里。

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  11. Compute the following number and use scientific notation: $\begin{array}{r}324,000\cdot 30,{000}^3\end{array}$ .
  ::11. 计算下列数字并使用科学标记: 324 000-130 0003。

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  12. Compute the following number and use scientific notation: $\begin{array}{r}14,300\cdot 20,{200}^2\end{array}$ .
  ::12. 计算下列数字并使用科学标记:14,30020,2002年。

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  Simplify the following expressions. 
  ::简化下列表达式。

  13. $\begin{array}{r}(3.29\cdot {10}^4)-(3.295\cdot {10}^5)+(1.25\cdot {10}^2)\cdot (3.97\cdot {10}^{15})\cdot (5.8\cdot {10}^{-6})\end{array}$

  14. $\begin{array}{r}(1.95\cdot {10}^2)+(6.798\cdot {10}^6)+(2.896\cdot {10}^3)\cdot (5.6\cdot {10}^{-3})÷(2.89\cdot {10}^4)\end{array}$

  15. $\begin{array}{r}(2.158\cdot {10}^7)\cdot (1.679\cdot {10}^6)-(9.98\cdot {10}^4)\cdot (3.4\cdot {10}^{-2})\end{array}$

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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 。