节: 科学符号 | 1.4 科学说明

章节大纲

Introduction
::导言

Very large and very small quantities and measures are often used to provide information for magazines, textbooks, television, newspapers, and the internet.
::杂志、教科书、电视、报纸和互联网经常使用大量和非常小的数量和措施提供信息。

Some examples are:
::举例如下:

The distance between the sun and Neptune is 4,500,000,000 km.
::太阳和海王星之间的距离是4500万公里

The diameter of an electron is approximately 0.000 000 000 000 22 inches.
::电子的直径约为10,000,000,000,000,22英寸。

Scientific notation is a convenient way to represent such numbers. How can you write the numbers above using scientific notation?
::科学标记是代表此类数字的方便方式。您如何使用科学标记写上上述数字 ?

Operations with Scientific Notation
::具有科学标记的操作

To represent a number in scientific notation means to express the number as a product of two factors : a number between 1 and 10 (including 1) and a power of 10. A positive real number $\begin{aligned} x \end{aligned}$ is said to be written in scientific notation if it is expressed as

\begin{aligned} x = a \times 10^{n} \end{aligned}

where

\begin{aligned} 1 \leq a < 10 and n \in I (n i s a n i n t e g e r) . \end{aligned}

In other words, a number in scientific notation is a single, nonzero digit followed by a decimal point and other digits, all multiplied by a power of 10.
::在科学标记中代表数字,表示数字是两个因素的产物:一个数字在1到10之间(包括1个),10的功率是10。如果以x=ax10n表示,则实际正数x在科学标记中写为x=ax10n,其中1a<10和nI(n为整数)。换句话说,科学标记中的数字是单数,非零位数,然后是小数点和其他数字,所有数字都乘以10的功率。

When working with numbers written in scientific notation, you can use the rules below. These rules are verified by example in Examples 2 and 3, below.
::当使用以科学符号书写的数字时,您可以使用以下的规则。以下例例2和例3对这些规则进行了验证。

Scientific Notation Rules
::科学说明规则

For

\begin{aligned} m, n \in I \end{aligned}

\begin{aligned} (A \times 10^{n}) + (B \times 10^{n}) = (A + B) \times 10^{n} \end{aligned}

\begin{aligned} (A \times 10^{n}) - (B \times 10^{n}) = (A - B) \times 10^{n} \end{aligned}

\begin{aligned} (A \times 10^{m}) \times (B \times 10^{n}) = (A \times B) \times (10^{m + n}) \end{aligned}

\begin{aligned} (A \times 10^{m}) \div (B \times 10^{n}) = (A \div B) \times (10^{m - n}) \end{aligned}

Writing and Operating with Scientific Notation
::以科学标志书写和操作

Watch the following video for an overview on converting between decimal notation and scientific notation, and on numbers in scientific notation:
::观看以下视频,以概述小数点数和科学符号之间的转换以及科学符号中的数字:

Examples
::实例

Example 1
::例1

Write the following numbers using scientific notation:
::使用科学符号写下下列数字:

a) 2679000
:a) 2679000

Solution:
::解决方案 :

\begin{aligned} 2679000 & = 2.679 \times 1000000 \\ 2.679 \times 1000000 & = 2.679 \times 10^{6} \end{aligned}

The exponent, $\begin{aligned} n = 6 \end{aligned}$ , represents the position of the decimal point that is 6 places to the right of the standard position of the decimal point.
::引号 n=6 表示小数点的位置,即小数点标准位置右侧的6位。

b) 0.00005728
:b) 0.0005728

Solution:
::解决方案 :

\begin{aligned} 0.00005728 & = 5.728 \times 0.00001 \\ 5.728 \times 0.00001 & = 5.728 \times \frac{1}{100000} \\ 5.728 \times \frac{1}{100000} & = 5.728 \times \frac{1}{10^{5}} \\ 5.728 \times \frac{1}{100000} & = 5.728 \times 10^{- 5} \end{aligned}

The exponent, $\begin{aligned} n = - 5 \end{aligned}$ , represents the position of the decimal point that is 5 places to the left of the standard position of the decimal point.
::引号 n5 表示小数点的位置,即小数点标准位置左边的5位。

One advantage of scientific notation is that calculations with large or small numbers can be done by applying the laws of .
::科学标记的一个优点是,可以通过适用......法则进行大数或小数的计算。

Example 2
::例2

Complete the following table:
::填写下表:

Expression in Scientific Notation	Expression in Standard Form	Result in Standard Form	Result in Scientific Notation
$\begin{aligned} 1.3 \times 10^{5} + 2.5 \times 10^{5} \end{aligned}$	$\begin{aligned} 130, 000 + 250, 000 \end{aligned}$	$\begin{aligned} 380, 000 \end{aligned}$	$\begin{aligned} 3.8 \times 10^{5} \end{aligned}$
$\begin{aligned} 3.7 \times 10^{- 2} + 5.1 \times 10^{- 2} \end{aligned}$	$\begin{aligned} 0.037 + 0.051 \end{aligned}$	$\begin{aligned} 0.088 \end{aligned}$	$\begin{aligned} 8.8 \times 10^{- 2} \end{aligned}$
$\begin{aligned} 4.6 \times 10^{4} - 2.2 \times 10^{4} \end{aligned}$	$\begin{aligned} 46, 000 - 22, 000 \end{aligned}$	$\begin{aligned} 24, 000 \end{aligned}$	$\begin{aligned} 2.4 \times 10^{4} \end{aligned}$
$\begin{aligned} 7.9 \times 10^{- 2} - 5.4 \times 10^{- 2} \end{aligned}$	$\begin{aligned} 0.079 - 0.054 \end{aligned}$	$\begin{aligned} 0.025 \end{aligned}$	$\begin{aligned} 2.5 \times 10^{- 2} \end{aligned}$

Note that the numbers in the last column have the same power of 10 as those in the 1st column.
::请注意,最后一栏的数值与第一栏的数值具有10的相同功率。

Example 3
::例3

Complete the following table:
::填写下表:

Expression in Scientific Notation	Expression in Standard Form	Result in Standard Form	Result in Scientific Notation
$\begin{aligned} (3.6 \times 10^{2}) \times (1.4 \times 10^{3}) \end{aligned}$	$\begin{aligned} 360 \times 1, 400 \end{aligned}$	$\begin{aligned} 504, 000 \end{aligned}$	$\begin{aligned} 5.04 \times 10^{5} \end{aligned}$
$\begin{aligned} (2.5 \times 10^{3}) \times (1.1 \times 10^{- 6}) \end{aligned}$	$\begin{aligned} 2, 500 \times 0.0000011 \end{aligned}$	$\begin{aligned} 0.00275 \end{aligned}$	$\begin{aligned} 2.75 \times 10^{- 3} \end{aligned}$
$\begin{aligned} (4.4 \times 10^{4}) \div (2.2 \times 10^{2}) \end{aligned}$	$\begin{aligned} 44, 000 \div 220 \end{aligned}$	$\begin{aligned} 200 \end{aligned}$	$\begin{aligned} 2.0 \times 10^{2} \end{aligned}$
$\begin{aligned} (6.8 \times 10^{- 4}) \div (3.2 \times 10^{- 2}) \end{aligned}$	$\begin{aligned} 0.00068 \div 0.032 \end{aligned}$	$\begin{aligned} 0.02125 \end{aligned}$	$\begin{aligned} 2.125 \times 10^{- 2} \end{aligned}$

Note that for multiplication, the power of 10 is the result of adding the exponents of the powers in the 1st column. For division, the power of 10 is the result of subtracting the exponents of the powers in the 1st column.
::请注意,对于乘法,10的功率是第1栏中增加权力指数的结果。对于分法,10的功率是减去第1栏中权力指数的结果。

Example 4
::例4

Calculate each of the following:
::计算以下各点:

a) $\begin{aligned} 4.6 \times 10^{4} + 5.3 \times 10^{5} \end{aligned}$
:a) 4.6×104+5.3×105

Solution:
::解决方案 :

Before the rule

\begin{aligned} (A \times 10^{n}) + (B \times 10^{n}) = (A + B) \times 10^{n} \end{aligned}

can be used, one of the numbers must be rewritten so that the powers of 10 are the same.
::在规则 (Ax10n) +(Bx10n) =(A+B) x10n = (A+B) = (A+B) x10n 之前,必须改写其中的一个数字,以使10 的功率相同。

Rewrite $\begin{aligned} 4.6 \times 10^{4} \end{aligned}$
::重写 4.6x104

$\begin{aligned} 4.6 \times 10^{4} = (0.46 \times 10^{1}) \times 10^{4} \end{aligned}$ The power $\begin{aligned} 10^{1} \end{aligned}$ indicates the number of places to the right that the decimal point must be moved to return 0.46 to the original number of 4.6.
::4.6x104=(0.46x101)×104 电源101表示右侧必须移动小数点将0.46返回原4.6的位数。

$\begin{aligned} (0.46 \times 10^{1}) \times 10^{4} = 0.46 \times 10^{5} \end{aligned}$ Add the exponents of the power.
:0.46x101) ×104=0.46x105 加上电源的推动者。

Rewrite the question and substitute $\begin{aligned} 4.6 \times 10^{4} \end{aligned}$ with $\begin{aligned} 0.46 \times 10^{5} \end{aligned}$ .
::重写问题,用0.46x105取代4.6x104。

$\begin{aligned} 0.46 \times 10^{5} + 5.3 \times 10^{5} \end{aligned}$

Apply the rule $\begin{aligned} (A \times 10^{n}) + (B \times 10^{n}) = (A + B) \times 10^{n} . \end{aligned}$
::应用规则 (Ax10n)+(Bx10n)=(A+B)x10n。

\begin{aligned} (0.46 \times 10^{5}) + (5.3 \times 10^{5}) = (0.46 + 5.3) \times 10^{5} \\ (0.46 + 5.3) \times 10^{5} = 5.76 \times 10^{5} \\ 4.6 \times 10^{4} + 5.3 \times 10^{5} = 5.76 \times 10^{5} \end{aligned}

b) $\begin{aligned} 4.7 \times 10^{- 3} - 2.4 \times 10^{- 4} \end{aligned}$
:b) 4.7×10-3-3-2.4×10-4

Solution:
::解决方案 :

Before the rule

\begin{aligned} (A \times 10^{n}) - (B \times 10^{n}) = (A - B) \times 10^{n} \end{aligned}

can be used, one of the numbers must be rewritten so that the powers of 10 are the same.
::在规则(Ax10n)-(Bx10n)=(A-B)x10n =(A-B)x10n 之前,必须改写其中的一个数字,以使10的权力相同。

Rewrite $\begin{aligned} 4.7 \times 10^{- 3} \end{aligned}$
::重写 4. 7x10- 3

$\begin{aligned} 4.7 \times 10^{- 3} = (47 \times 10^{- 1}) \times 10^{- 3} \end{aligned}$ The power $\begin{aligned} 10^{- 1} \end{aligned}$ indicates the number of places to the left that the decimal point must be moved to return 47 to the original number of 4.7.
::4.7x10-3=(47x10-1-1)x10-3 功率 10-1表示左侧的位数,小数点必须移动到小数点后将47返回原来的4.7。

$\begin{aligned} (47 \times 10^{- 1}) \times 10^{- 3} = 47 \times 10^{- 4} \end{aligned}$ Add the exponents of the power.
:47x10-1)×10-3=47x10-4,加上权力的推动者。

Rewrite the question and substitute $\begin{aligned} 4.7 \times 10^{- 3} \end{aligned}$ with $\begin{aligned} 47 \times 10^{- 4} \end{aligned}$ .
::重写问题,以47x10-4取代4.7x10-3。

$\begin{aligned} 47 \times 10^{- 4} - 2.4 \times 10^{- 4} \end{aligned}$

Apply the rule $\begin{aligned} (A \times 10^{n}) - (B \times 10^{n}) = (A - B) \times 10^{n} . \end{aligned}$
::应用规则 (Ax10n)-(Bx10n)=(A-B)x10n。

\begin{aligned} (47 \times 10^{- 4}) - (2.4 \times 10^{- 4}) & = (47 - 2.4) \times 10^{- 4} \\ (47 \times 10^{- 4}) - (2.4 \times 10^{- 4}) & = 44.6 \times 10^{- 4} \end{aligned}

The answer must be written in scientific notation.
::答案必须写在科学符号中。

\begin{aligned} 44.6 \times 10^{- 4} = (4.46 \times 10^{1}) \times 10^{- 4} & Apply the law of exponents \\ - add the  exponents of the power. \\ 4.46 \times 10 \times 10^{- 4} = 4.46 \times 10^{- 3} \\ 4.7 \times 10^{- 3} - 2.4 \times 10^{- 4} = 4.46 \times 10^{- 3} \end{aligned}

::44.6×10-4=(4.46×10×101)×10-4pp 前列者法 - 增加权力的前列者4. 46×10×10-4=4.46×10-4=4.46×10-4=4.46×10-34.7×10-3-3-2-2.4×10-4=4.46×10-3

c) $\begin{aligned} (7.3 \times 10^{5}) \times (6.8 \times 10^{4}) \end{aligned}$
:c) (7.3×105)×(6.8×104)

Solution:
::解决方案 :

\begin{aligned} 7.3 \times 10^{5} \times 6.8 \times 10^{4} & Apply the rule \\ (A \times 10^{m}) \times (B \times 10^{n}) = (A \times B) \times (10^{m + n}) . \\ = (7.3 \times 6.8) \times (10^{5 + 4}) \\ = (49.64) \times (10^{9}) \\ = 49.64 \times 10^{9} & Write the answer in scientific notation. \\ = (4.964 \times 10^{1}) \times 10^{9} & Apply the law of exponents \\ - add the exponents of the power. \\ 49.64 \times 10^{9} = 4.964 \times 10^{10} \\ (7.3 \times 10^{5}) \times (6.8 \times 10^{4}) = 4.964 \times 10^{10} \end{aligned}

::7.3x6xxxx.6.8x10104 应用规则 (Ax10m) ×(Bx10n) = (AxB) ×(10m+n) = (7.3x6.8) x(105+4) = (49.64) x(109) = 49.64xxxx10) ×109 在科学符号中写上答案 = (4. 964x101) ×109 pply acentents 的法则 - 添加功率的引号. 49. 64x109= 4. 964x1010 (7.3x105) x(6.8x104) = 4.964x1010

d) $\begin{aligned} (4.8 \times 10^{9}) \div (5.79 \times 10^{7}) \end{aligned}$
:d) (4.8×109)(5.79×107)

Solution:
::解决方案 :

\begin{aligned} (4.8 \times 10^{9}) \div (5.79 \times 10^{7}) & Apply the rule \\ (A \times 10^{m}) \div (B \times 10^{n}) = (A \div B) \times (10^{m - n}) . \\ = (4.8 \div 5.79) \times 10^{9 - 7} & Apply the law of exponents \\ - subtract the exponents of the power. \\ = (0.829) \times 10^{2} & Write the answer in scientific notation. \\ = (8.29 \times 10^{- 1}) \times 10^{2} & Apply the law of exponents \\ - add the exponents of the power. \\ = 8.29 \times 10^{1} \end{aligned}

4.8x109) (5.79x107) 应用规则(Ax10m) (Bx10n) =(AB) ×(10m-n) =(4.85.79) x109-7 ppplication the expenters of the power of the power.=(0.829)×102Write the answord in science notation.=(8.29x10-1)×102pplication the expenters =(8.29x101)

Example 5
::例5

a) Express the following product in scientific notation: $\begin{aligned} (4 \times 10^{12}) (9.2 \times 10^{7}) \end{aligned}$ .
:a) 以科学符号表示下列产品4x1012)(9.2x107)。

Solution :
::解决方案 :

Apply the rule:
::适用规则:

\begin{aligned} (A \times 10^{m}) \times (B \times 10^{n}) = (A \times B) \times (10^{m + n}) . \end{aligned}

Ax10m)x(Bx10n)=(AxB)x(10m+n)

\begin{aligned} (4 \times 10^{12}) \times (9.2 \times 10^{7}) = (4 \times 9.2) \times (10^{12 + 7}) \\ (4 \times 9.2) \times (10^{12 + 7}) = 36.8 \times 10^{19} \end{aligned}

Express the answer in scientific notation:
::以科学符号表示回答:

\begin{aligned} 36.8 \times 10^{19} = (3.68 \times 10^{1}) \times 10^{19} \\ (3.68 \times 10^{1}) \times 10^{19} = 3.68 \times 10^{20} \\ (4 \times 10^{12}) (9.2 \times 10^{7}) = 3.68 \times 10^{20} \end{aligned}

b) Express the following quotient in scientific notation: $\begin{aligned} \frac{6400000}{0.008} \end{aligned}$ .
:b) 在科学编号中以下列商数表示:640000.008。

Solution:
::解决方案 :

Begin by expressing the numerator and the denominator in scientific notation:
::首先是表达分子和科学符号中的分母:

\begin{aligned} \frac{6.4 \times 10^{6}}{8.0 \times 10^{- 3}} \end{aligned}

Apply the rule $\begin{aligned} (A \times 10^{m}) \div (B \times 10^{n}) = (A \div B) \times (10^{m + n}) . \end{aligned}$
::应用规则 (Ax10m) {(Bx10n) = (A{B}) x(10m+n) 。

\begin{aligned} (6.4 \times 10^{6}) \div (8.0 \times 10^{- 3}) & = (6.4 \div 8.0) \times (10^{6 - (- 3)}) \\ = (0.8) \times (10^{9}) \\ = 0.8 \times 10^{9} \\ = (8.0 \times 10^{- 1}) \times 10^{9} \\ = 8.0 \times 10^{- 1} \times 10^{9} \\ = 8.0 \times 10^{8} \\ \frac{6400000}{0.008} = 8.0 \times 10^{8} \end{aligned}

c) If

\begin{aligned} a & = 0.000415 \\ b & = 521 \\ c & = 71640 \end{aligned}

c) 如 a=0.000415b=521c=71640

calculate the value for $\begin{aligned} \frac{a b}{c} \end{aligned}$ . Express the answer in scientific notation.
::计算 abc 的值。在科学符号中表示答案。

Solution:
::解决方案 :

\begin{aligned} 0.000415 & = 4.15 \times 10^{- 4} \\ 521 & = 5.21 \times 10^{2} \\ 71640 & = 7.1640 \times 10^{4} \end{aligned}

Use the values in scientific notation to determine the value for $\begin{aligned} \frac{a b}{c} \end{aligned}$ :
::使用科学符号中的值确定 abc 的值:

\begin{aligned} \frac{a b}{c} = \frac{(4.15 \times 10^{- 4}) (5.21 \times 10^{2})}{7.1640 \times 10^{4}} \end{aligned}

::abc=( 4.15x10- 4)( 521x102) 7. 160x104

In the numerator, apply the rule $\begin{aligned} (A \times 10^{m}) \times (B \times 10^{n}) = (A \times B) \times (10^{m + n}) . \end{aligned}$
::在分子中,应用规则 (Ax10m) x(Bx10n) = (AxB) x(10m+n)。

\begin{aligned} \frac{(4.15 \times 10^{- 4}) (5.21 \times 10^{2})}{7.1640 \times 10^{4}} = \frac{(4.15 \times 5.21) \times (10^{- 4} \times 10^{2})}{7.1640 \times 10^{4}} \\ \frac{(4.15 \times 5.21) \times (10^{- 4} \times 10^{2})}{7.1640 \times 10^{4}} = \frac{21.6215 \times 10^{- 2}}{7.1640 \times 10^{4}} \\ Apply the rule (A \times 10^{m}) \div (B \times 10^{n}) = (A \div B) \times (10^{m - n}) . \\ \frac{21.6215 \times 10^{- 2}}{7.1640 \times 10^{4}} = (21.6215 \div 7.1640) \times (10^{- 2 - 4}) \\ (21.6215 \div 7.1640) \times (10^{- 2 - 4}) = 3.018 \times 10^{- 6} \end{aligned}

415×10-4)(5.21×102)7.1640×104=(415×5.5)xxxxxxx(10-4-4xxxx)7.1640×104(4.15×5.21)x(10-4xxxx)7.1640xx(10-10)x(Ax10m){(Bx10n)=(AB)x(10-9mn)21.6215-27.27-1640×104=(21.62-15)7.1640x(10-2-4)(21.62.15_7.1640)x(10-10-2-4)=0.1810-6

Review
::回顾

Express each of the following in scientific notation:
::在科学符号中表达以下各点:

42000
0.00087
150.64
56789
0.00947

Express each of the following in standard form:
::以标准格式表述以下各点:

$\begin{aligned} 4.26 \times 10^{5} \end{aligned}$
$\begin{aligned} 8 \times 10^{4} \end{aligned}$
$\begin{aligned} 5.967 \times 10^{10} \end{aligned}$
$\begin{aligned} 1.482 \times 10^{- 6} \end{aligned}$
$\begin{aligned} 7.64 \times 10^{- 3} \end{aligned}$

Perform the indicated operations and express the answer in scientific notation:
::执行指定的操作,并在科学符号中表示答案:

$\begin{aligned} 8.9 \times 10^{4} + 4.3 \times 10^{5} \end{aligned}$
$\begin{aligned} 8.7 \times 10^{- 4} - 6.5 \times 10^{- 5} \end{aligned}$
$\begin{aligned} (5.3 \times 10^{6}) \times (7.9 \times 10^{5}) \end{aligned}$
$\begin{aligned} (3.9 \times 10^{8}) \div (2.8 \times 10^{6}) \end{aligned}$

For the given values, perform the indicated operations for $\begin{aligned} \frac{a b}{c} \end{aligned}$ , and express the answer in scientific notation and standard form:
::对于给定值,为 abc 执行指定的操作,并以科学符号和标准格式表示答案:

15.

\begin{aligned} a & = 76.1 \\ b & = 818000000 \\ c & = 0.000016 \end{aligned}

::15.a=76.1b=81800000c=0.000016

16.

\begin{aligned} a & = 9.13 \times 10^{9} \\ b & = 5.45 \times 10^{- 23} \\ c & = 1.62 \end{aligned}

::a=9.13×109b=5.45×10-23c=1.62

Review (Answers)
::回顾(答复)

Please s ee the Appendix.
::请参看附录。

章节大纲

Introduction ::导言

Operations with Scientific Notation ::具有科学标记的操作

Scientific Notation Rules ::科学说明规则

Writing and Operating with Scientific Notation ::以科学标志书写和操作

Examples ::实例

Review ::回顾

Review (Answers) ::回顾(答复)

Introduction
::导言

Operations with Scientific Notation
::具有科学标记的操作

Scientific Notation Rules
::科学说明规则

Writing and Operating with Scientific Notation
::以科学标志书写和操作

Examples
::实例

Review
::回顾

Review (Answers)
::回顾(答复)