8.8 科学说明

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  科学符号

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  Scientific Notation 
  ::科学符号

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  Consider the number six hundred and forty three thousand, two hundred and ninety seven. It would be written  as 643,297, with  each digit’s position having  a “value” assigned to it. You may have seen a table like this before:
  ::考虑一下数字643,330,297。它将写成643,297,每个数字的位置都配有“价值 ” ( value) 。 您可能以前见过这样的表格:

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  $$\begin{array}{rl}& \text{hundred-thousands}\text{ten-thousands}\text{thousands}\text{hundreds}\text{tens}\text{units}\\ & 643297\end{array}$$

  
  ::千千人单位 64329 7

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  We’ve seen that when we write an exponent above a number, it means that we have to multiply a certain number of copies of that number together. You may also know already  that any number with a zero exponent equals  1, and negative represent  fractional values .
  ::我们已经看到,当我们写出一个高于数字的指数时,这意味着我们必须将数字的一定份数相加。 你也许已经知道,任何数字,零份数等于1,负数代表分数值。

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  Look carefully at the table above. Do you notice that all the column headings are powers of ten ? Here they are listed from greatest to least :
  ::仔细查看上面的表格。 您注意到所有列标题都是 十 的功率吗 ? 这里列出它们从最大到最小 :

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  $$\begin{array}{rlrl}\text{Hundred thousands}& & 100,000& ={10}^5\\ \text{Ten thousands}& & 10,000& ={10}^4\\ \text{Thousands}& & 1,000& ={10}^3\\ \text{Hundreds}& & 100& ={10}^2\\ \text{Tens}& & 10& ={10}^1\end{array}$$

  
  ::10万至10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万 10万

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  Even the “units” column is really just a power of ten. Unit means 1, and 1 is $\begin{array}{r}{10}^0.\end{array}$
  ::即使“单位”一栏实际上也只是十倍的功率。单位是指1,1是100。

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  If we divide 643,297 by 100,000 we get 6.43297; if we multiply 6.43297 by 100,000 we get 643, 297. But we have just seen that 100,000 is the same as $\begin{array}{r}{10}^5\end{array}$ , so if we multiply 6.43297 by $\begin{array}{r}{10}^5\end{array}$ we should also get 643,297. In other words,
  ::如果我们把643 297除以10万,我们就会得到6 43297;如果我们乘以6 43297乘以10万,我们就会得到643 297。 但我们刚刚看到10万和105一样,所以如果我们乘以6 43297乘以105,我们也应该得到643 297。 换句话说,

  $$\begin{array}{r}643,297=6.43297\times {10}^5\end{array}$$

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

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  In scientific notation, numbers are always written in the form $\begin{array}{r}a\times {10}^b\end{array}$ , where $\begin{array}{r}b\end{array}$ is an integer and $\begin{array}{r}a\end{array}$ is between 1 and 10 ( meaning it has exactly 1 nonzero digit before the decimal). This notation is especially useful for numbers that are either very small or very large.
  ::在科学符号中,数字总是以 ax10b 的形式写成,其中 b 是一个整数, a 介于 1 和 10 之间(意指小数点之前有 精确的 1 个非 零位数 ) 。 这个符号对于非常小或非常大的数字特别有用 。

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  Here’s a set of examples:
  ::以下是一系列例子:

  $$\begin{array}{rl}1.07\times {10}^4& =10,700\\ 1.07\times {10}^3& =1,070\\ 1.07\times {10}^2& =107\\ 1.07\times {10}^1& =10.7\\ 1.07\times {10}^0& =1.07\\ 1.07\times {10}^{\text{-}1}& =0.107\\ 1.07\times {10}^{\text{-}2}& =0.0107\\ 1.07\times {10}^{\text{-}3}& =0.00107\\ 1.07\times {10}^{\text{-}4}& =0.000107\end{array}$$

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  Look at the first example and notice where the decimal point is in both expressions.
  ::查看第一个示例并注意两个表达式中的小数点位置。

  

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  So the exponent on the ten acts to move the decimal point over to the right. An exponent of 4 moves it 4 places and an exponent of 3 would move it 3 places.
  ::所以十点上的推手将小数点移到右边。四点的推手将它移到四点,三点的推手将它移到三点。

  

  

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  This makes sense because each time you multiply by 10, you move the decimal point one place to the right. 1.07 times 10 is 10.7, then 10.7 times 10 again is 107.0, and so on.
  ::这是有道理的,因为每次乘以10, 就会将小数点一移到右边。 1.07乘以10乘以10是10.7, 然后再乘以10.7乘以10乘以10, 也就是107.0, 等等。

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  Similarly, if you look at the later examples in the table, you can see that a negative exponent on the 10 means the decimal point moves that many places to the left. This is because multiplying by $\begin{array}{r}{10}^{\text{-}1}\end{array}$ is the same as multiplying by $\begin{array}{r}\frac{1}{10},\end{array}$  which is like . So instead of moving the decimal point one place to the right for every multiple of 10, we move it one place to the left for every multiple of $\begin{array}{r}\frac{1}{10}.\end{array}$
  ::同样,如果看看表格中后面的例子,你可以看到,10点上的负指数意味着小数点向左移动许多位数。这是因为乘以10-1与乘以110相同,这就像。所以,不把小数点向右移到10点,而是将小数点向右移到110点,我们把小数点向左移到110点的每个位数向左移一个位数。

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  That’s how to convert numbers from scientific notation to standard form. When we’re converting numbers to scientific notation, however, we have to apply the whole process backwards. First we move the decimal point until it’s immediately after the first nonzero digit; then we count how many places we moved it. If we moved the decimal  to the left, the exponent on the 10 is positive ; if we moved it to the right,  the exponent is negative.
  ::这就是如何将数字从科学符号转换为标准格式。 但是,当我们将数字转换为科学符号时,我们必须将整个过程向后应用。 首先,我们将小数点移到第一位非零位数之后;然后我们算一下我们移到左边的多少地方。 如果我们将小数点移到左边,10点上的引号是正数;如果我们将其移到右边,引号是负数。

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  F or example, to write 0.000032 in scientific notation, we’d first move the decimal five places to the right to get 3.2; then, since we moved it right, the exponent on the 10 should be negative five, so 0.000032  in scientific notation is $\begin{array}{r}3.2\times {10}^{\text{-}5}.\end{array}$
  ::例如,为了在科学符号中写下0.00032, 我们首先将小数点后5位移到右侧, 以获得3.2; 然后,既然我们移动正确, 10位的指数应该是负5, 所以科学符号中的0.00032是3.2x10-5。

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  You can double-check whether you’ve got the right direction by comparing the number in scientific notation with the number in standard form, and thinking “Does this represent a big number or a small number?” A positive exponent on the 10 represents a number bigger than 10 and a negative exponent represents a number smaller than 10, and you can easily tell if the number in standard form is bigger or smaller than 10 just by looking at it.
  ::您可以通过比较科学符号中的数字与标准格式的数字,并思考“这是代表一个大数字还是一个小数字?” 来重复检查你是否得到了正确的方向。 10上的一个正表征代表一个大于10的数字,而负表征代表一个小于10的数字,你可以很容易地通过查看它来判断标准格式中的数字是大于还是小于10。

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

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

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  a) 63
  :a) 63

  $$\begin{array}{r}63=6.3\times 10=6.3\times {10}^1\end{array}$$

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  b) 9,654
  :b) 9 654

  $$\begin{array}{r}9,654=9.654\times 1,000=9.654\times {10}^3\end{array}$$

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  c) 653,937,000
  :c) 653 937 000美元

  $$\begin{array}{r}653,937,000=6.53937000\times 100,000,000=6.53937\times {10}^8\end{array}$$

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  d) 0.003
  ::d) 0.0003

  $$\begin{array}{r}0.003=3\times \frac{1}{1000}=3\times {10}^{\text{-}3}\end{array}$$

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  e) 0.000056
  :e)0.00056

  $$\begin{array}{r}0.000056=5.6\times \frac{1}{100,000}=5.6\times {10}^{\text{-}5}\end{array}$$

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  f) 0.00005007
  ::f) 0.0005007

  $$\begin{array}{r}0.00005007=5.007\times \frac{1}{100,000}=5.007\times {10}^{\text{-}5}\end{array}$$

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  Evaluating Expressions
  ::评价表达式

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  The key to evaluating expressions involving scientific notation is to group the powers of 10 together and deal with them separately.
  ::评价涉及科学标记的表达方式的关键是将10项权力组合在一起,分别处理。

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  a)  $\begin{array}{r}(3.2\times {10}^6)\cdot (8.7\times {10}^{11})\end{array}$
  :a) (3.2×106) (8.7×1011)

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  $\begin{array}{r}(3.2\times {10}^6)(8.7\times {10}^{11})=\underset{27.84}{\underset{⏟}{3.2\times 8.7}}\times \underset{{10}^{17}}{\underset{⏟}{{10}^6\times {10}^{11}}}=27.84\times {10}^{17}.\end{array}$ But $\begin{array}{r}27.84\times {10}^{17}\end{array}$ isn’t in proper scientific notation, because it has more than one digit before the decimal point. We need to move the decimal point one more place to the left and add 1 to the exponent, which gives us $\begin{array}{r}2.784\times {10}^{18}.\end{array}$
  :3.2x106)(8.7x1011) =3.2x8.727.84xx106x10111017=27.84x1017。但27.84x1017不是适当的科学标记,因为它在小数点前有一个以上的位数。我们需要将小数点向左移动一个多处位置,并在前列增加一个位置,这给了我们2.784x1018。

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  b)  $\begin{array}{r}(5.2\times {10}^{\text{-}4})\cdot (3.8\times {10}^{\text{-}19})\end{array}$
  :b) (5.2×10-4)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

  $$\begin{array}{rl}(5.2\times {10}^{\text{-}4})(3.8\times {10}^{\text{-}19})& =\underset{19.76}{\underset{⏟}{5.2\times 3.8}}\times \underset{{10}^{\text{-}23}}{\underset{⏟}{{10}^{\text{-}4}\times {10}^{\text{-}19}}}\\ & =19.76\times {10}^{\text{-}23}\\ & =1.976\times {10}^{\text{-}22}\end{array}$$

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  c)  $\begin{array}{r}(1.7\times {10}^6)\cdot (2.7\times {10}^{\text{-}11})\end{array}$
  :c) (1.7×106)___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

  $$\begin{array}{r}(1.7\times {10}^6)(2.7\times {10}^{\text{-}11})=\underset{4.59}{\underset{⏟}{1.7\times 2.7}}\times \underset{{10}^{\text{-}5}}{\underset{⏟}{{10}^6\times {10}^{\text{-}11}}}=4.59\times {10}^{\text{-}5}\end{array}$$

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  When we use scientific notation in the real world, we often round off our calculations. Since we’re often dealing with very big or very small numbers, it can be easier to round off so that we don’t have to keep track of as many digits—and scientific notation helps us with that by saving us from writing out all the extra zeros. For example, if we round off 4,227,457,903 to 4,200,000,000, we can then write it in scientific notation as simply $\begin{array}{r}4.2\times {10}^9.\end{array}$
  ::当我们在现实世界中使用科学符号时,我们常常绕过自己的计算。 由于我们经常处理非常大或非常小的数字,因此我们比较容易四舍五入,这样我们就不必跟踪那么多数字,而科学符号通过节省我们写出所有额外零数来帮助我们做到这一点。 比如,如果我们四舍五入4,227,457,903到4,200,000,000,000,000,0,000,0,000,0,000,0,000,000,0,000,0,000,000,0,000,000,0,000,000,0,000,000,0,000,000,0,000,000,2,2,2,200,000,000,000,000,000。

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  When rounding , we often talk of significant figures or significant digits . Significant figures include
  ::在四舍五入时,我们经常谈论重要数字或重要数字。

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  • all nonzero digits
    ::全部非十位数
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  • all zeros that come before a nonzero digit and after either a decimal point or another nonzero digit
    ::在非零位数之前以及在小数点或另一个非零位数之后出现的所有零

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  For example, the number 4000 has one significant digit; the zeros don’t count because there’s no nonzero digit after them. But the number 4000.5 has five significawnt digits: the 4, the 5, and all the zeros in between. And the number 0.003 has three significant digits: the 3 and the two zeros that come between the 3 and the decimal point.
  ::比如,4000数字有一个重要数字;零数字没有计算,因为后面没有非零数字。 但是,4000.5数字有五个符号数字:4、5和中间的所有零。 而003数字有三个重要数字:3和小数点之间的3和2个零。

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  Rounding to the Correct Amount of Significant Figures 
  ::四舍四入得出重要数字正确数额

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  Evaluate the following expressions. Round to 3 significant figures and write your answer in scientific notation.
  ::评估以下表达式。 轮到 3 个重要数字, 并在科学符号中写下您的答复 。

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  a) $\begin{array}{r}(3.2\times {10}^6)÷(8.7\times {10}^{11})\end{array}$
  :a) (3.2×106) (8.7×1011)

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  b) $\begin{array}{r}(5.2\times {10}^{-4})÷(3.8\times {10}^{-19})\end{array}$
  :b) (5.2×10-4)(3.8×10-19)

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  Solution
  ::解决方案

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  It’s easier if we convert to fractions and THEN separate out the powers of 10.
  ::如果我们转换成分数, 并分离出10分的权力,

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  a)  $\begin{array}{r}(3.2\times {10}^6)÷(8.7\times {10}^{11})\end{array}$

  $$\begin{array}{rlrl}(3.2\times {10}^6)÷(8.7\times {10}^{11})& =\frac{3.2\times {10}^6}{8.7\times {10}^{11}}& & \text{separate out the powers of 10}\\ & =\frac{3.2}{8.7}\times \frac{{10}^6}{{10}^{11}}& & \text{evaluate each fraction (round to 3 s.f.)}\\ & =0.368\times {10}^{(6-11)}\\ & =0.368\times {10}^{\text{-}5}& & \text{remember only 1 number before the decimal!}\\ & =3.68\times {10}^{\text{-}6}\end{array}$$

  
  ::a) (3.2×106) (8.7×1011) (3.2×1011) (8.7×1011) (8.7×1011) =3.2×1068.7×1011) =3.2×68.7×1011 = 3.28.7×1061011 的功率除以 10= 3. 28.7×1061011 的功率。) =0. 368×10(6-11) =0. 368×10-5 仅包含小数前的 1 个数 。= 3.68×10-6

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  b)  $\begin{array}{r}(5.2\times {10}^{\text{-}4})÷(3.8\times {10}^{\text{-}19})\end{array}$

  $$\begin{array}{rlrl}(5.2\times {10}^{\text{-}4})÷(3.8\times {10}^{\text{-}19})& =\frac{5.2\times {10}^{\text{-}4}}{3.8\times {10}^{\text{-}19}}& & \text{separate the powers of 10}\\ & =\frac{5.2}{3.8}\times \frac{{10}^{\text{-}4}}{{10}^{\text{-}19}}& & \text{evaluate each fraction (round to 3 s.f.)}\\ & =1.37\times {10}^{((\text{-}4)-(\text{-}19))}\\ & =1.37\times {10}^{15}\end{array}$$

  
  ::b) (5.2×10-4)(3.8×10-19)(5.2×10-4)(3.8×10-19)=5.2×10-43.8×10-19)=5.2×10-43.8×10-19 将10=5.23.8×10-410-410-19评价每一部分的功率分开(四舍五入至3 s.f)=1.37×10(4)-(19)=1.37×1015)

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  Example
  ::示例示例示例示例

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  Example 1
  ::例1

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  Evaluate the following expression. Round to 3 significant figures and write your answer in scientific notation.
  ::评估以下表达式。 回合到 3 个重要数字, 并在科学符号中写下您的答复 。

  $\begin{array}{r}(1.7\times {10}^6)÷(2.7\times {10}^{\text{-}11})\end{array}$

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  $$\begin{array}{rlrl}(1.7\times {10}^6)÷(2.7\times {10}^{\text{-}11})& =\frac{1.7\times {10}^6}{2.7\times {10}^{\text{-}11}}& & \text{next we separate the powers of 10}\\ & =\frac{1.7}{2.7}\times \frac{{10}^6}{{10}^{\text{-}11}}& & \text{evaluate each fraction (round to 3 s.f.)}\\ & =0.630\times {10}^{(6-(\text{-}11))}\\ & =0.630\times {10}^{17}\\ & =6.30\times {10}^{16}\end{array}$$

  
  :1.7×106) (2.7×10-11)=1.7×1062.7×10-11 下一步我们将10=1.72.7×10610-11的功率分离出来 10=1.72.7×10610-11评价每个碎片(圆到 3 s.f.)=0.630×10(6-(11))=0.630×1017=6.30×1016)

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  Note that we have to leave in the final zero to indicate that the result has been rounded.
  ::请注意,我们必须在最后零点离开,以表明结果四舍五入。

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  Review
  ::回顾

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  Write the numerical value of the following.
  ::写入下列数值。

  1. $\begin{array}{r}3.102\times {10}^2\end{array}$
  2. $\begin{array}{r}7.4\times {10}^4\end{array}$
  3. $\begin{array}{r}1.75\times {10}^{\text{-}3}\end{array}$
  4. $\begin{array}{r}2.9\times {10}^{\text{-}5}\end{array}$
  5. $\begin{array}{r}9.99\times {10}^{\text{-}9}\end{array}$

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

  6. 120,000
  7. 1,765,244
  8. 12
  9. 0.00281
  10. 0.000000027

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  How many significant digits are in each of the following?
  ::以下每个数字中有多少位数?

  11. 38553000
  12. 2754000.23
  13. 0.0000222
  14. 0.0002000079

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  Round each of the following to two significant digits.
  ::以下各回合至两位重要数字。

  15. 3.0132
  16. 82.9913

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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 。