2.17科学记数法中数字的比较

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  科学记数法中数字的比较

  

  [Figure 1]

   

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  Michelle’s brother is taking chemistry this year and is currently learning about electrons and protons. He has learned that electrons are particles with a negative charge while protons are particles with a  positive  charge. He has also learned that the mass of an electron is about  $\begin{array}{r}9.1094\times {10}^{-31}\end{array}$  kilograms while the mass of a proton is about  $\begin{array}{r}1.6726\times {10}^{-27}\end{array}$  kilograms. At first he thinks that electrons have a greater mass than protons because 9.1094 is greater than 1.6726, but now he’s not sure. How can Michelle use her knowledge of scientific notation to help her brother to compare the mass of an electron with the mass of a proton?
  ::米歇尔的弟弟今年正在接受化学,目前正在学习电子和质子。他已经知道电子是负电荷粒子,质子是正电荷粒子。他还知道电子的质量大约为9.1094×10-31公斤,质子的质量大约为1.6726×10-27公斤。 起初,他认为电子的质量大于质子,因为9.1094大于1.6726,但现在他不确定。 米歇尔如何利用科学识别知识帮助她的兄弟将电子的质量与质子的质量进行比较?

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  In this concept, you will learn how to compare and  order  numbers in scientific notation.
  ::在这个概念中,你会学会如何比较和排序科学符号中的编号。

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  Comparing Numbers Written in Scientific Notation
  ::比较在科学符号中写入的编号

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  Recall that 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 and 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.
  ::回顾当一个数字在科学符号中写下时,它是一个至少有1个但少于10个的数的产物,10个的功率是10个的,大数(数字大于1个)是10个的正数。小数(数字在0到1个之间)是10个的负数。10个的具体功率是10个表示数字有多大或多小。

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  You can compare two numbers written in scientific notation by looking at their powers of 10. The number with the greater power of 10 will be the greater number. If two numbers have the same power of 10, then compare the decimal numbers to determine the greater number.
  ::您可以通过查看它们10的功率来比较两个以科学符号写成的数字。 10的功率越大,数字就越大。 如果两个数字的功率与10相同,那么比较小数数,以确定更多的数字。

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  Here is an example.
  ::举一个例子。

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  Compare  $\begin{array}{r}8.43\times {10}^6\end{array}$  and  $\begin{array}{r}2.38\times {10}^8\end{array}$ .
  ::比较 8.43 × 10 6 和 2.38 × 10 8 。

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  First, notice the   are 6 and 8. Because the exponents are different, you know that the number with the greater exponent is the greater number. 8 is greater than 6.
  ::首先,注意是6和8, 因为出价不同, 你知道出价越高的数越多。 8 大于 6。

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  $\begin{array}{r}2.38\times {10}^8\end{array}$  is the greater number.
  ::2.38 × 10 8 数目较多。

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  The answer is  $\begin{array}{r}8.45\times {10}^6<2.38\times {10}^8\end{array}$ .
  ::答案是8.45 × 10 6 < 2.38 × 10 8 。

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  Here is another example.
  ::下面是另一个例子。

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  Compare  $\begin{array}{r}3.2\times {10}^{-10}\end{array}$  and  $\begin{array}{r}1.2\times {10}^{-9}\end{array}$ .
  ::比较3.2×10-10和1.2×10-9。

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  First, notice the exponents are -10 and -9. Even though the exponents are negative, because they are different you still know that the number with the greater exponent is the greater number. -9 is greater than -10.
  ::首先,注意指数是 -10和 -9. 尽管指数是负的, 因为它们是不同的, 你仍然知道, 指数越高, 数字就越大。 - 9 大于 - 10 。

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  $\begin{array}{r}1.2\times {10}^{-9}\end{array}$  is the greater number.
  ::1.2 × 10- 9是数量较多的。

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  The answer is  $\begin{array}{r}3.2\times {10}^{-10}<1.2\times {10}^{-9}\end{array}$ .
  ::答案是3.2 × 10 - 10 < 1.2 × 10 - 9 。

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  Here is one more example.
  ::这里还有一个例子。

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  Compare  $\begin{array}{r}5.65\times {10}^5\end{array}$  and  $\begin{array}{r}5.56\times {10}^5\end{array}$ .
  ::比较5.65×10 5和5.56×10 5。

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  First, notice the exponents are 5 and 5. Because the exponents are the same, you will have to compare the decimal numbers to determine the greater number. 5.65 is greater than 5.56.
  ::首先,注意指数是5和5,因为指数是相同的,你必须比较小数数,以确定数字。5.65大于5.56。

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  $\begin{array}{r}5.65\times {10}^5\end{array}$  is the greater number.
  ::5.65×10 5是人数较多的。

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  The answer is  $\begin{array}{r}5.65\times {10}^5>5.56\times {10}^5\end{array}$ .
  ::答案是5.65×10 5 > 5.56×10 5。

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  Sometimes you will want to compare two numbers, but only one of them is in scientific notation. In this case, convert the number that is not in scientific notation into scientific notation first. Then, compare the two numbers.
  ::有时你会想要比较两个数字,但其中只有一个是科学符号。在这种情况下,首先将不是科学符号的数字转换为科学符号。然后比较这两个数字。

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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 Michelle’s brother, who is learning about electrons and protons.
  ::更早之前, 米歇尔的哥哥正在学习电子和质子,

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  He wants to know how the mass of an electron compares to the mass of a proton. He knows that the mass of an electron is about  $\begin{array}{r}9.1094\times {10}^{-31}\end{array}$  kilograms while the mass of a proton is about  $\begin{array}{r}1.6726\times {10}^{-27}\end{array}$  kilograms.
  ::他想知道电子质量如何与质子质量相比。他知道电子质量约为9.1094×10-31公斤,质子质量约为1.6726×10-27公斤。

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  First, Michelle should notice that the numbers are in scientific notation and the exponents are -31 and -27. The exponents are different, so the number with the greater exponent is the greater number. -27 is greater than -31.
  ::首先,米歇尔应该注意到,数字在科学符号中,指数是 -31和 -27。指数是不同的,指数是更大的数字。 -27是大于 -31。

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  $\begin{array}{r}1.6726\times {10}^{-27}\end{array}$  is the greater number.
  ::1.6726 × 10 - 27是数量较多的。

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  The answer is  $\begin{array}{r}9.1094\times {10}^{-31}<1.6726\times {10}^{-27}\end{array}$ . The mass of an electron is smaller than the mass of a proton. In fact, it is almost 2000 times smaller! 
  ::答案是 9. 1094 × 10 - 31 < 1. 6726 × 10 - 27. 电子质量小于质子质量。 事实上,电子质量几乎是质子质量的2000倍!

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  Example 2
  ::例2

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  Compare  $\begin{array}{r}3.4\times {10}^5\end{array}$  and  $\begin{array}{r}34,000,000\end{array}$ .
  ::比较3.4×10 5和34,000,000。

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  First, notice that the second number is not in scientific notation. Rewrite 34,000,000 in scientific notation. Find the first non- zero  digit and put a  decimal point  to its right. Here, the 3 is the first non-zero digit.
  ::首先, 请注意第二个数字不是科学符号。 在科学符号中重写 34 000 000 。 查找第一个非零位数, 右侧放一个小数点。 这里, 3 是第一个非零位数 。

  $\begin{array}{r}3.4\end{array}$

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  Next, count how many spaces you needed to move the decimal point to get from 34,000,000 to 3.4.
  ::接下来, 计数您需要多少空格才能移动小数点, 从34 000 000 移动到 3. 4 。

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  34,000,000 going to 3.4: Move the decimal point 7 spaces to the left.
  ::3.4亿至3.4:将小数点7空格向左移动。

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  Now, put everything together. Your number in scientific notation is 3.4 multiplied by 10 to the power of 7.
  ::现在,把一切都集中起来。你在科学符号中的数字是3.4乘以10乘以7乘以7的功率。

  $\begin{array}{r}3.4\times {10}^7\end{array}$

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  Next, compare the numbers.
  ::接下来,比较数字。

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  $\begin{array}{r}3.4\times {10}^5\end{array}$  and  $\begin{array}{r}3.4\times {10}^7\end{array}$
  ::3.4 × 10 5和3.4 × 10 7

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  Notice the exponents are 5 and 7. Because the exponents are different, you know that the number with the greater exponent is the greater number. 7 is greater than 5.
  ::注意出处是5和7, 因为出处是不同的, 你知道出处较大的数字是更大的数字。 7大于5。

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  $\begin{array}{r}3.4\times {10}^7\end{array}$  is the greater number.
  ::3.4 × 10 7 数目较多。

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  The answer is  $\begin{array}{r}3.4\times {10}^5<3.4\times {10}^7\end{array}$  which means  $\begin{array}{r}3.4\times {10}^5<34,000,000\end{array}$ .
  ::答案是3.4 × 10 5 < 3.4 × 10 7,这意味着3.4 × 10 5 < 34,000,000。

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  Example 3
  ::例3

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  Compare  $\begin{array}{r}4.5\times {10}^7\end{array}$  and  $\begin{array}{r}4.5\times {10}^9\end{array}$ .
  ::比较4.5×10 7和4.5×10 9。

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  First, notice the exponents are 7 and 9. The number with the greater exponent is the greater number. 9 is greater than 7.
  ::首先,通知出处为7和9。 出处越大,数字越多。 9比7多。

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  $\begin{array}{r}4.5\times {10}^9\end{array}$  is the greater number.
  ::4.5×10 9是数量较多的。

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  The answer is  $\begin{array}{r}4.5\times {10}^7<4.5\times {10}^9\end{array}$ .
  ::答案是4.5 × 10 7 < 4.5 × 10 9 。

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  Example 4
  ::例4

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  Compare  $\begin{array}{r}5.6\times {10}^{-3}\end{array}$  and  $\begin{array}{r}7.8\times {10}^{-5}\end{array}$ .
  ::比较5.6×10-3和7.8×10-5。

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  First, notice the exponents are -3 and -5. The number with the greater exponent is the greater number. -3 is greater than -5.
  ::首先,注意出处是 -3 和 -5. 出处越大,数字就越大。 - 3 大于 - 5。

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  $\begin{array}{r}5.6\times {10}^{-3}\end{array}$  is the greater number.
  ::5.6 x 10 - 3是数量较多的。

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  The answer is  $\begin{array}{r}5.6\times {10}^{-3}>7.8\times {10}^{-5}\end{array}$ .
  ::答案是5.6×10-3 > 7.8×10-5。

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  Example 5
  ::例5

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  Compare  $\begin{array}{r}8.9\times {10}^2\end{array}$  and  $\begin{array}{r}9.8\times {10}^2\end{array}$ .
  ::比较8.9×10.2和9.8×10.2。

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  First, notice the exponents are 2 and 2. Because the exponents are the same, you will have to compare the decimal numbers to determine the greater number. 9.8 is greater than 8.9.
  ::首先,注意指数是2和2, 因为指数是相同的, 您必须比较小数数, 才能确定更大的数字 。 9. 8 大于 8. 9 。

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  $\begin{array}{r}9.8\times {10}^2\end{array}$  is the greater number.
  ::9.8 × 10 2 数目较多。

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  The answer is  $\begin{array}{r}8.9\times {10}^2<9.8\times {10}^2\end{array}$ .
  ::答案是8.9 × 10 2 < 9.8 × 10 2 。

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  Review
  ::回顾

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  Compare the following. Write  $\begin{array}{r}<\end{array}$ ,  $\begin{array}{r}>\end{array}$ , or  $\begin{array}{r}=\end{array}$  for each blank.
  ::比较以下。为每个空白撰写 < 、 > 或 = 。

  1. $\begin{array}{r}2.1\times {10}^6\underset{\_}{}8.9\times {10}^5\end{array}$  
  2. $\begin{array}{r}0.00000212\underset{\_}{}2.12\times {10}^{-5}\end{array}$  
  3. $\begin{array}{r}4.26\times {10}^{10}\underset{\_}{}4,260,000,000\end{array}$  
  4. $\begin{array}{r}7.2\times {10}^{-3}\underset{\_}{}1.25\times {10}^{-1}\end{array}$  
  5. $\begin{array}{r}2.1\times {10}^5\underset{\_}{}3.1\times {10}^4\end{array}$  
  6. $\begin{array}{r}8.1\times {10}^6\underset{\_}{}8.9\times {10}^5\end{array}$  
  7. $\begin{array}{r}7.2\times {10}^{-4}\underset{\_}{}8.9\times {10}^{-5}\end{array}$  
  8. $\begin{array}{r}3.3\times {10}^2\underset{\_}{}3.1\times {10}^3\end{array}$  
  9. $\begin{array}{r}1.21\times {10}^4\underset{\_}{}1.89\times {10}^2\end{array}$  
  10. $\begin{array}{r}5.5\times {10}^6\underset{\_}{}5.51\times {10}^6\end{array}$  
  11. $\begin{array}{r}1.61\times {10}^3\underset{\_}{}1.89\times {10}^2\end{array}$  

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  Order the following from greatest to least. Where necessary, convert to scientific notation.
  ::从最大到最小的顺序。 必要时, 转换为科学符号 。

  12. $\begin{array}{r}9.2\times {10}^{10},6.4\times {10}^{15},2.1\times {10}^{20},1.7\times {10}^{15}\end{array}$  
  13. $\begin{array}{r}5.63\times {10}^{-5},4.16\times {10}^{-3},3.42\times {10}^{-6},8.71\times {10}^{-3}\end{array}$  
  14. $\begin{array}{r}2.21\times {10}^{-5},0.0000202,2.02\times {10}^{-4},0.0000222\end{array}$  
  15. $\begin{array}{r}86,102,000,8.61\times {10}^8,86,120,000,8.61\times {10}^7\end{array}$  

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  Review (Answers)
  ::回顾(答复)