Section: 科学记数法中的数字运算 | 2.18科学记数法中的数字运算

Section outline

Diana has always loved the ocean, so this summer she is going to a camp at the Ocean Institute! On her first day she learns that the Atlantic Ocean has an area of $\begin{aligned} 4.108 \times 10^{7} \end{aligned}$ square miles while the Indian Ocean has an area of $\begin{aligned} 2.835 \times 10^{7} \end{aligned}$ square miles. Diana can tell that the Atlantic Ocean is bigger than the Indian Ocean, but she's not sure by how much. How can Diana figure out how much bigger the area of the Atlantic Ocean is compared to the area of the Indian Ocean?
::戴安娜一直热爱海洋,所以今年夏天,她要去海洋研究所的一个营地。在她的第一天,她了解到大西洋的面积是4.108×10 7平方英里,而印度洋的面积是2.835×10 7平方英里。戴安娜可以发现大西洋比印度洋大,但她不知道有多少。戴安娜怎么才能知道大西洋的面积比印度洋大多少?

In this concept, you will learn how to add, subtract, multiply, and divide numbers in scientific notation.
::在此概念中,您将学会如何在科学符号中添加、减、乘、除数字。

Operations with Numbers Written in Scientific Notation
::在科学标注中写入数字的操作

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. Scientific notation makes it easier to read, write, and calculate with very large and very small numbers.
::当一个数字以科学符号写成时,它作为至少1个但少于10个的数字和10个数字的产物而写成,10个数字的功率是10个。科学符号使得以非常大和非常小的数字读、写和计算更容易。

You can add, subtract, multiply, and divide numbers that are written in scientific notation while still keeping the numbers in scientific notation.
::您可以添加、减、乘和除以以科学符号书写的数字,同时将数字保留在科学符号中。

Here are the steps for adding or subtracting two numbers written in scientific notation.
::以下是在科学符号中增加或减去两个数字的步骤。

Rewrite the number with the smaller exponent so that it has the same exponent as the number with the larger exponent by moving the decimal point of its decimal number.
::用较小的引言重写数字,以便通过移动小数点的小数点,使数字与数字的引言相同。

Add/subtract the decimal numbers. The power of 10 will not change.
::添加/减小数数字。10的功率不会改变。

Convert your result to scientific notation if necessary.
::必要时将结果转换为科学符号。

Here is an example.
::举一个例子。

Add $\begin{aligned} (5.7 \times 10^{4}) + (4.87 \times 10^{5}) \end{aligned}$ .
::加(5.7×10 4)+(4.87×10 5)

First, notice that the are 4 and 5. You will need to rewrite $\begin{aligned} 5.7 \times 10^{4} \end{aligned}$ so that it has an exponent of 5. Because you need to increase the exponent by 1, you will move the decimal point one space to the left.
::首先,请注意是4和5, 您需要重写 5. 7 × 10 4, 这样它就有5 的指数, 因为需要增加 1, 您需要将小数点 1 的空格移到左边。

$\begin{aligned} 5.7 \times 10^{4} \end{aligned}$ becomes $\begin{aligned} 0.57 \times 10^{5} \end{aligned}$
::5.7 × 10 4 变成 0.57 × 10 5

Now, rewrite the problem.
::现在,重写问题。

$\begin{aligned} (0.57 \times 10^{5}) + (4.87 \times 10^{5}) \end{aligned}$

Next, add the decimal numbers using what you have learned about decimal addition. The power of 10 stays the same.
::下一步,使用您所学到的关于小数添加的内容添加小数数。10的功率保持不变。

$\begin{aligned} (0.57 + 4.87) \times 10^{5} \\ 5.44 \times 10^{5} \end{aligned}$

Last, check to make sure that your answer is in scientific notation. 5.44 is at least 1 but less than 10, so your answer is in scientific notation.
::最后,检查以确保您的答案在科学符号中。 5. 44 至少有 1 个, 少于 10 个, 所以您的答案在科学符号中。

The answer is $\begin{aligned} (5.7 \times 10^{4}) + (4.87 \times 10^{5}) = 5.44 \times 10^{5} \end{aligned}$ .
::答案是(5.7×10 4)+(4.87×10 5)=5.44×10 5。

Here are the steps for multiplying or dividing two numbers in scientific notation.
::这是科学符号中两个数字的乘法或除法步骤。

Multiply/divide the decimal numbers.
::乘/divide小数数字。

Multiply/divide the powers of 10 by adding/subtracting their exponents.
::乘/减10国的权力,增加/减其指数。

Convert your answer to scientific notation if necessary.
::必要时将答案转换为科学符号。

Here is an example of multiplying two decimal numbers.
::下面是两位小数乘以两个小数的例子。

Multiply $\begin{aligned} (3.4 \times 10^{- 2}) (6.2 \times 10^{6}) \end{aligned}$ .
::乘数(3.4 × 10 - 2) (6.2 × 10 6) 。

First, multiply the decimal numbers using what you have learned about decimal multiplication.
::首先,使用关于十进制乘法的知识乘以十进制数字。

$\begin{aligned} 3.4 \times 6.2 = 21.08 \end{aligned}$

Next, multiply the powers of 10 by adding their exponents.
::接下来,乘以10的功率增加他们的推手

$\begin{aligned} 10^{- 2} \times 10^{6} = 10^{- 2 + 6} = 10^{4} \end{aligned}$

Now, combine the results.
::现在,合并结果。

$\begin{aligned} \begin{array}{rcl} (3.4 \times 10^{- 2}) (6.2 \times 10^{6}) & = & (3.4 \times 6.2) \times (10^{- 2} \times 10^{6}) \\ = & 21.08 \times 10^{4} \end{array} \end{aligned}$

Finally, convert your answer to scientific notation. You will need to rewrite $\begin{aligned} 21.08 \times 10^{4} \end{aligned}$ so that the decimal number is at least 1 but less than 10. Move its decimal point one space to the left. Then, to keep the overall value the same, increase the exponent on the 10 by 1.
::最后,您要将答案转换为科学符号。您需要重写21. 08 × 10 4, 这样小数点数至少为 1, 但小数点小于 10 。将其小数点 1 的空格向左移动。然后, 要保持总值不变, 请将总值增加十点的指数 1 。

$\begin{aligned} 21.08 \times 10^{4} \end{aligned}$ becomes $\begin{aligned} 2.108 \times 10^{5} \end{aligned}$
::21.08 × 10 4 变成 2.108 × 10 5

The answer is $\begin{aligned} (3.4 \times 10^{- 2}) (6.2 \times 10^{6}) = 2.108 \times 10^{5} \end{aligned}$ .
::答案是(3.4 × 10 - 2) (6.2 × 10 6) = 2.108 × 10 5。

Here is an example of dividing two decimal numbers.
::下面是两个小数数字的除法示例。

Divide $\begin{aligned} (8.4 \times 10^{5}) \div (1.4 \times 10^{- 2}) \end{aligned}$ .
::差数(8.4×10 5 ) (1.4×10 - 2) 。

First, divide the decimal numbers using what you have learned about .
::首先,将小数数除以您所学到的。

$\begin{aligned} 8.4 \div 1.4 = 6 \end{aligned}$

Next, divide the powers of 10 by subtracting their exponents. Remember that subtracting a negative number is the same as adding the positive version of the number.
::下一步, 将 10 的功率除以 10 。请记住 , 减去负数等于添加正数。

$\begin{aligned} 10^{5} \div 10^{- 2} = 10^{5 - (- 2)} = 10^{5 + 2} = 10^{7} \end{aligned}$

Now, combine the results.
::现在,合并结果。

$\begin{aligned} \begin{array}{rcl} (8.4 \times 10^{5}) \div (1.4 \times 10^{- 2}) & = & (8.4 \div 1.4) \times (10^{5} \div 10^{- 2}) \\ = & 6 \times 10^{7} \end{array} \end{aligned}$

Last, check to make sure that your answer is in scientific notation. 6 is at least 1 but less than 10, so your answer is in scientific notation.
::最后,检查以确保您的答案在科学符号中。 6 表示至少 1, 但小于 10, 所以您的答案在科学符号中。

The answer is $\begin{aligned} (8.4 \times 10^{5}) \div (1.4 \times 10^{- 2}) = 6 \times 10^{7} \end{aligned}$ .
::答案是(8.4 × 10 5) (1.4 × 10 - 2) = 6 × 10 7。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Diana at the Ocean Institute.
::早些时候,你在海洋研究所遇到关于戴安娜的问题

Diana learned that the Atlantic Ocean has an area of $\begin{aligned} 4.108 \times 10^{7} \end{aligned}$ square miles while the Indian Ocean has an area of $\begin{aligned} 2.835 \times 10^{7} \end{aligned}$ square miles. She wants to know how much bigger the area of the Atlantic Ocean is compared to the area of the Indian Ocean.
::戴安娜了解到大西洋的面积为4.108×10 7平方英里,而印度洋的面积为2.835×10 7平方英里。她想知道大西洋的面积与印度洋的面积相比有多大。

In order to figure this out, Diana will need to subtract the two areas.
::为了弄清这一点,戴安娜需要减去这两个区域。

$\begin{aligned} (4.108 \times 10^{7}) - (2.835 \times 10^{7}) \end{aligned}$

First, she should notice that the exponents are both 7. Because the exponents are the same, she does not need to rewrite either number.
::首先,她应注意到,出处均为7,因为出处相同,她不需要重写两个号码。

Now, she can subtract the decimal numbers using what she has learned about decimal subtraction. The power of 10 stays the same.
::现在,她可以使用她所学到的关于小数减法的知识来减去小数数字。 10的功率保持不变。

$\begin{aligned} (4.108 - 2.835) \times 10^{7} \\ 1.273 \times 10^{7} \end{aligned}$

The answer is that the Atlantic Ocean is $\begin{aligned} 1.273 \times 10^{7} \end{aligned}$ more square miles than the Indian Ocean.
::答案是,大西洋比印度洋多1 273×10 7平方英里。

Example 2
::例2

At its closest, the planet Neptune is 4,300,000,000 kilometers away from Earth. A group of astronauts from Earth want to make it to Neptune in 20,000 days. If they travel the same number of kilometers each day, how many kilometers will they travel each day? Convert both numbers to scientific notation before solving.
::离地球最近的海王星海王星距离地球4300,000,000公里。一组来自地球的宇航员想在20,000天内到达海王星。如果他们每天旅行的公里数相同,那么他们每天要旅行多少公里?在解决之前,将这两个数字转换成科学符号。

First, convert both numbers to scientific notation.
::首先,将两个数字转换为科学符号。

$\begin{aligned} \begin{array}{rcl} 4, 300, 000, 000 & = & 4.3 \times 10^{9} \\ 20, 000 & = & 2 \times 10^{4} \end{array} \end{aligned}$

Next, notice that the astronauts want to travel the same distance each day. You will need to divide the total distance by the number of days to find the distance they will need to travel each day.
::接下来,请注意宇航员希望每天旅行的距离相同。您需要将总距离除以天数, 才能找到他们每天旅行的距离。

$\begin{aligned} (4.3 \times 10^{9}) \div (2 \times 10^{4}) \end{aligned}$

Now, divide the decimal numbers using what you have learned about decimal division.
::现在,将小数数除以您所学的小数分数。

$\begin{aligned} 4.3 \div 2 = 2.15 \end{aligned}$

Next, divide the powers of 10 by subtracting their exponents.
::其次,将10的权责分割为减去其权责。

$\begin{aligned} 10^{9} \div 10^{4} = 10^{9 - 4} = 10^{5} \end{aligned}$

Now, combine the results.
::现在,合并结果。

$\begin{aligned} \begin{array}{rcl} (4.3 \times 10^{9}) \div (2 \times 10^{4}) & = & (4.3 \div 2) \times (10^{9} \div 10^{4}) \\ = & 2.15 \times 10^{5} \end{array} \end{aligned}$

The answer is that the astronauts will need to travel $\begin{aligned} 2.15 \times 10^{5} \end{aligned}$ or 215,000 kilometers per day. They will be traveling for almost 55 years!
::答案是宇航员每天需要旅行2.15×10 5或215 000公里。他们将旅行近55年!

Example 3
::例3

Add $\begin{aligned} (3.4 \times 10^{3}) + (5.6 \times 10^{4}) \end{aligned}$ .
::加(3.4×10 3)+(5.6×10 4)

First, notice that the exponents are 3 and 4. You will need to rewrite $\begin{aligned} 3.4 \times 10^{3} \end{aligned}$ so that it has an exponent of 4. Because you need to increase the exponent by 1, you will move the decimal point one space to the left.
::首先,请注意前列数为 3 和 4 。您需要重写3.4 x 10 3, 这样它就有4 的前列数。因为需要将前列数增加 1, 您需要将小数点 1 的空格移到左边。

$\begin{aligned} 3.4 \times 10^{3} \end{aligned}$ becomes $\begin{aligned} 0.34 \times 10^{4} \end{aligned}$
::3.4 × 10 3 改为 0.34 × 10 4

Now, rewrite the problem.
::现在,重写问题。

$\begin{aligned} (0.34 \times 10^{4}) + (5.6 \times 10^{4}) \end{aligned}$

Next, add the decimal numbers using what you have learned about decimal addition. The power of 10 stays the same.
::下一步,使用您所学到的关于小数添加的内容添加小数数。10的功率保持不变。

$\begin{aligned} (0.34 + 5.6) \times 10^{4} \\ 5.94 \times 10^{4} \end{aligned}$

Last, check to make sure that your answer is in scientific notation. 5.94 is at least 1 but less than 10, so your answer is in scientific notation.
::最后,检查以确保您的答案在科学符号中。 5. 94 至少有 1 个, 少于 10 个, 所以您的答案在科学符号中。

The answer is $\begin{aligned} (3.4 \times 10^{3}) + (5.6 \times 10^{4}) = 5.94 \times 10^{4} \end{aligned}$ .
::答案是(3.4×10 3)+(5.6×10 4)=5.94×10 4。

Example 4
::例4

Multiply $\begin{aligned} (1.2 \times 10^{4}) (3.4 \times 10^{4}) \end{aligned}$ .
::乘以 (1.2 × 10 4) (3.4 × 10 4) 。

First, multiply the decimal numbers using what you have learned about decimal multiplication.
::首先,使用关于十进制乘法的知识乘以十进制数字。

$\begin{aligned} 1.2 \times 3.4 = 4.08 \end{aligned}$

Next, multiply the powers of 10 by adding their exponents.
::接下来,乘以10的功率增加他们的推手

$\begin{aligned} 10^{4} \times 10^{4} = 10^{4 + 4} = 10^{8} \end{aligned}$

Now, combine the results.
::现在,合并结果。

$\begin{aligned} \begin{array}{rcl} (1.2 \times 10^{4}) (3.4 \times 10^{4}) & = & (1.2 \times 3.4) \times (10^{4} \times 10^{4}) \\ = & 4.08 \times 10^{8} \end{array} \end{aligned}$

Last, check to make sure that your answer is in scientific notation. 4.08 is at least 1 but less than 10, so your answer is in scientific notation.
::最后,检查以确保您的答案在科学符号中。 4. 08是至少 1, 但小于 10, 所以您的答案在科学符号中。

The answer is $\begin{aligned} (1.2 \times 10^{4}) (3.4 \times 10^{4}) = 4.08 \times 10^{8} \end{aligned}$ .
::答案是(1.2×10 4) (3.4×10 4) = 4.08× 10 8。

Example 5
::例5

Subtract $\begin{aligned} (5.6 \times 10^{4}) - (3.2 \times 10^{4}) \end{aligned}$ .
::减(5.6×10 4)-(3.2×10 4)

First, notice that the exponents are both 4. Because the exponents are the same, you do not need to rewrite either number.
::首先,请注意前列词是4, 因为前列词是相同的, 您不需要重写任何一个数字。

Now, subtract the decimal numbers using what you have learned about decimal subtraction. The power of 10 stays the same.
::现在,使用小数减法所学的减去小数数减去小数数。 10的功率保持不变。

$\begin{aligned} (5.6 - 3.2) \times 10^{4} \\ 2.4 \times 10^{4} \end{aligned}$

Last, check to make sure that your answer is in scientific notation. 2.4 is at least 1 but less than 10, so your answer is in scientific notation.
::最后,检查以确保您的答案在科学符号中。 2. 4 至少为 1, 但小于 10, 所以您的答案在科学符号中。

The answer is $\begin{aligned} (5.6 \times 10^{4}) - (3.2 \times 10^{4}) = 2.4 \times 10^{4} \end{aligned}$ .
::答案是(5.6×10 4)-(3.2×10 4)=2.4×10 4。

Review
::回顾

Add, subtract, multiply or divide.
::添加、减、乘或除。

$\begin{aligned} (3.4 \times 10^{3}) + (5.4 \times 10^{3}) \end{aligned}$

$\begin{aligned} (5.4 \times 10^{4}) - (1.3 \times 10^{4}) \end{aligned}$

$\begin{aligned} (6.7 \times 10^{5}) + (5.4 \times 10^{5}) \end{aligned}$

$\begin{aligned} (13.4 \times 10^{3}) - (5.4 \times 10^{3}) \end{aligned}$

$\begin{aligned} (6.4 \times 10^{3}) (5.1 \times 10^{3}) \end{aligned}$

$\begin{aligned} (12.5 \times 10^{5}) \div (2 \times 10^{3}) \end{aligned}$

$\begin{aligned} (5.4 \times 10^{5}) + (4.4 \times 10^{5}) \end{aligned}$

$\begin{aligned} (12.2 \times 10^{3}) - (10.1 \times 10^{3}) \end{aligned}$

$\begin{aligned} (5.6 \times 10^{3}) + (4.5 \times 10^{3}) \end{aligned}$

$\begin{aligned} (3.3 \times 10^{4}) (1.2 \times 10^{2}) \end{aligned}$

$\begin{aligned} (24.6 \times 10^{5}) \div (6 \times 10^{3}) \end{aligned}$

$\begin{aligned} (266 \times 10^{- 6}) + (8.6 \times 10^{- 6}) \end{aligned}$

$\begin{aligned} (7.14 \times 10^{4}) - (5.5 \times 10^{4}) \end{aligned}$

$\begin{aligned} (2.56 \times 10^{- 3}) \times (3.8 \times 10^{6}) \end{aligned}$

$\begin{aligned} (4.97 \times 10^{8}) \div (7 \times 10^{5}) \end{aligned}$

Review (Answers)
::回顾(答复)

Section outline

Operations with Numbers Written in Scientific Notation ::在科学标注中写入数字的操作

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Operations with Numbers Written in Scientific Notation
::在科学标注中写入数字的操作

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)