1.2 实际数字

章节大纲

• 选择节常规

  折叠 展开

  常规

  全部折叠 展开全部

  • 选择活动新闻通告

    

    新闻通告 讨论区
• 选择节实际数字

  折叠 展开

  实际数字

  播放段落

  Introduction
  ::导言

  播放段落

  Suppose you and three friends were playing a game in which you each drew a number from a hat, and the person with the highest number won. Let's say you drew the number $\begin{align*} \frac{3}{2}\end{align*}$ , while your friends drew the numbers $\begin{align*}\sqrt{3}\end{align*}$ , 1.7, and $\begin{align*} \frac{\pi}{3}\end{align*}$ , respectively. Could you figure out who won the game? You'll learn how to classify, order , and graph real numbers so you can compare values such as these.
  ::假设你和三个朋友在玩一个游戏, 你们每人从帽子上抽出一个数字, 最高数的人赢了。 假设你画了32号数字, 而你的朋友分别画了3, 1. 7和 3号数字。 你能找出谁赢了游戏吗? 你会学会如何分类、 排序和图表真实数字, 这样你就可以比较这些数值 。

  播放段落

  Real Numbers
  ::实际数字

  播放段落

  Real numbers have always played an important role in mathematics, and their classification provides greater insight into their use. Real numbers are all the numbers on the number line, and there are infinitely many of them. Their types and categories are important because they can give you more information about the problem you are looking at. There are also imaginary numbers, a topic to be  discussed later in this chapter . In fact, certain types of numbers can direct you to further formulas or definitions in mathematics.
  ::真实数字在数学中一直起着重要作用,它们的分类能更深入地了解它们的用途。 真实数字是数字线上的所有数字, 并且有无限多的数字。 它们的类型和类别很重要, 因为它们能为您提供更多关于您所看到的问题的更多信息。 还有虚构数字, 本章后面要讨论的话题。 事实上, 某些类型的数字可以引导您在数学中找到更多的公式或定义 。

  播放段落

  When you began to learn numbers, or when you teach numbers to  a child, you count beginning with the number 1. For instance, 1, 2, 3, 4, ... . These are classified as counting or natural numbers.
  ::当开始学习数字时,或者当向儿童教授数字时,从数字1开始算起。例如,1,2,3,4,.。这些被归类为计数或自然数字。

  播放段落

  Not long after understanding how to count, you learn the idea of "none," when you are counting an item for which there are none to count. The number zero is introduced to identify quantities of no objects. This number is added to the counting numbers to create the set of whole numbers: 0, 1, 2, 3, ... .
  ::在了解如何计算之后不久, 当你在计数一个无可计数的项目时, 你就会知道“ 无” 的概念。 数字为零, 用于识别无天体的数量 。 此数字被添加到计数数中, 以创建全数集 : 0, 1, 2, 3,... 。

  播放段落

  Then we discover numbers that are not positive when we attempt to take a larger number away from a smaller number. A debt is created that can be represented by negative numbers. The whole numbers along with their opposites (negatives) make up the set of integers : ... -3, -2, -1, 0, 1, 2, 3, ... .
  ::然后我们发现当我们试图从较小数字中取出更多数字时,数字并不正数。 债务产生时, 负数可以代表债务。 整数和反数( 负数) 构成整数组 : ... - 3, - 2, - 1, 0, 1, 2, 2, 3, ...

  播放段落

  Next we explore numbers  that are part of a whole. These numbers, which may come in the form of a fraction or decimal, represent portions of the whole, or a combination of wholes and portions of the  whole. These are called rational numbers because each one can be represented as a ratio. Rational numbers include all of the integers as well. Here are some examples of integers  represented as fractions:  $\begin{align*}\frac{12}{1},-\frac{150}{3},\frac{1}{-1}, \ \mathrm{and} \ \frac{-74}{2}\end{align*}$ . Other examples of rational numbers include  $\begin{align*}\frac{1}{4}, \frac{5}{4}, \ \mathrm{and} \ \frac{3}{7}\end{align*}$ . 
  ::接下来我们探索整体中的一部分数字。 这些数字可以是分数或小数, 可以是整体的一部分, 也可以是整体中整体和部分的组合。 这些称为理性数字, 因为每个数字都可以以比例表示。 理性数字也包括所有整数。 下面是以分数表示的整数的例子: 121, -1503, 1-1, 和-742。 其他合理数字的例子包括14, 54 和 37。

  播放段落

  Finally, there are some very unique numbers in that they cannot be represented as a ratio, and are thus called irrational numbers. They are very important to mathematics, so they are not disregarded. For example,  $\begin{align*}\pi \end{align*}$  is essential to calculating the dimensions and the area of circles, spheres, and cylinders. There are also square (and higher-degree) roots that cannot be written as decimals, but are essential to s olving problems, such as  calculating the sides of a triangle. Some examples of irrational roots are  $\begin{align*}\sqrt{2}\end{align*}$  and  $\begin{align*}\sqrt[3]{7}\end{align*}$ .
  ::最后,有一些非常独特的数字,因为它们不能被作为一个比率来表示,因此被称为非理性数字。它们对于数学非常重要,因此它们不被忽略。例如, ____对于计算圆圈、球体和圆柱体的尺寸和面积至关重要。 还有平方(和高度)根不能写成小数,但对于解决问题至关重要,例如计算三角形的两边。一些非理性根的例子有2个和73个。

  

  播放段落

  Types of Real Numbers
  ::实际数字类型

  Type of Number	Definition	Examples
  Real Numbers	Numbers that correspond to numbers on the number line	6, -.9785,  $\begin{align*}\pi \end{align*}$ ,  $\begin{align*}-\frac{17}{432}\end{align*}$
  Irrational Numbers	Numbers that cannot be written as a fraction or ratio of integers	$\begin{align*}\frac{\pi }{6}, \sqrt{2}, \sqrt[9]{67}, -13\sqrt{14}\end{align*}$
  Rational Numbers	播放段落 Numbers that can be written as a fraction or ratio, where  $\begin{align*}a\end{align*}$  and $\begin{align*}b\end{align*}$  are integers and  $\begin{align*}\left\{{\frac{a}{b}} \ \mathrm{such \ that} \ {b\neq 0}\right\}\end{align*}$ ::可以以分数或比率写成的数字,其中a和b为整数,{ab为b0}	$\begin{align*}\frac{17}{19}\end{align*}$ , 6, -19,  $\begin{align*}5.\overline{5}\end{align*}$
  Integers	Positive and negative whole numbers and zero	-2, 14, -399, 0
  Whole Numbers	Numbers that are not negative and have no fractional or decimal part	0, 1, 2, 3, ...
  Counting or Natural Numbers	Numbers that are used to count: whole numbers and do not include zero or any negatives	1, 2, 3, ....

  播放段落

  Classifying Real Numbers
  ::分类真实数字

  播放段落

  See the video below for examples of  how the set of real numbers is broken down into the subsets of rational and irrational numbers. It further explains how rational numbers are partitioned into integers, whole numbers, and counting or natural numbers. It also demonstrates how to compare real numbers using >, <, ≥, ≤, and = symbols. 
  ::有关真实数字组如何细分为理性和非理性数字子集的示例,请见下文视频。它进一步解释了合理数字组如何分割为整数、整数、计数或自然数字。它还展示了如何使用 > 、 < 、 _ 、 、 和 = 符号比较实际数字。

   

  播放段落

  Graphing and Ordering Real Numbers  
  ::图形和排列实际数字

  播放段落

  Every non-integer real number can be positioned between two integers. Many times you will need to organize real numbers to determine the least value, greatest value, or both. This is usually done on a number line.
  ::每个非整数真实数字都可以在两个整数之间定位。 您需要多次组织真实数字来决定最小值、 最大值或两者。 这通常是在数字行上完成的 。

  播放段落

  Examples
  ::实例

  播放段落

  Example 1
  ::例1

  播放段落

  Classify the following numbers :
  ::分类以下数字:

  播放段落

  a) 0
  ::a) 0 0

  播放段落

  Solution:
  ::解决方案 :

  播放段落

  Zero is a whole number, an integer, a rational number , and a real number.
  ::零是一个整数,一个整数,一个合理数,一个真实数。

   

  播放段落

  b) –1
  :b)-1

  播放段落

  Solution:
  ::解决方案 :

  播放段落

  –1 is an integer, a rational number, and a real number.
  ::-1是一个整数,一个合理的数字和一个真实的数字。

   

  播放段落

  c) $\begin{align*}\frac{\pi}{3}\end{align*}$
  :c) %3

  播放段落

  Solution:
  ::解决方案 :

  播放段落

  Even though  $\begin{align*}\frac{\pi}{3}\end{align*}$ is a fraction, $\begin{align*}\pi\end{align*}$  is not an integer, so  $\begin{align*}\frac{\pi}{3}\end{align*}$  is an irrational number and a real number.
  ::3虽然是一个分数,但不是整数,所以3是一个非理性的数字和真实的数字。

   

  播放段落

  d) $\begin{align*}\frac{\sqrt{36}}{9}\end{align*}$
  :d) 369

  播放段落

  Solution:
  ::解决方案 :

  $$\begin{align*}\frac{\sqrt{36}}{9} = \frac{6}{9} = \frac{2}{3}\end{align*}$$

  播放段落

  This is a rational number and a real number.
  ::这是一个合理的数字和一个真实的数字。

  播放段落

  Example 2
  ::例2

  播放段落

  Plot the following rational numbers on a number line:
  ::在数字行上绘制下列合理数字 :

  播放段落

  a) $\begin{align*}\frac{2}{3}\end{align*}$
  :a) 23

  播放段落

  Solution:
  ::解决方案 :

  播放段落

  Upon dividing 2 by 3, the result is a repeating decimal,  $\begin{align*}0.\overline{6}\end{align*}$ . So  $\begin{align*}\frac{2}{3} = 0.\overline{6}\end{align*}$ , which is between 0 and 1.
  ::2除以3后,结果为重复小数点0.6。所以23=0.6,介于0和1之间。

  

  播放段落

  b) $\begin{align*}-\frac{3}{7}\end{align*}$
  :b)-37

  播放段落

  Solution:
  ::解决方案 :

  播放段落

  Upon dividing -3 by 7, the result is a repeating decimal,  $\begin{align*}-0.\overline{428571}\end{align*}$ . So   $\begin{align*}-\frac{3}{7}=-0.\overline{428571}\end{align*}$ , which is between –1 and 0.

  
  ::将 - 3 除以 7 后, 结果是重复小数小数, - 0. 428571 。 所以 - 37\ 0. 428571 , 介于 - 1 和 0 之间 。

  播放段落

  c) $\begin{align*}\frac{57}{16}\end{align*}$
  :c) 5716

  播放段落

  Solution:
  ::解决方案 :

  播放段落

  Upon dividing 57 by 16, the result is a finite decimal,  $\begin{align*}3.5625\end{align*}$ . So    $\begin{align*}\frac{57}{16} = 3.5625\end{align*}$ ,  which is between 3 and 4.
  ::在将57除以16时,结果为小数小数点为3.5625。5716=3.5625,在3到4之间。

  播放段落

  Example 3
  ::例3

  播放段落

  Compare $\begin{align*} \frac{\pi}{15}\end{align*}$ and $\begin{align*}\tfrac{\sqrt{3}}{\sqrt{75}}.\end{align*}$
  ::比较15和375。

  播放段落

  Solution:
  ::解决方案 :

  播放段落

  Step 1: S implify the 2nd fraction in order to better compare the two numbers.
  ::第1步:简化第2部分,以便更好地比较这两个数字。

  播放段落

  First, the denominator can be simplified by rewriting 75 into its factors, 3 and 25:
  ::首先,可将75项要素改写为3和25,从而简化分母:

  $$\begin{align*}\sqrt{75} = \sqrt{3 \cdot 25}.\end{align*}$$

  播放段落

  Next, take  the square root of 25, because the square root of 25 is a whole number:

  $$\begin{align*}\sqrt{3 \cdot 25}=\sqrt{3}\cdot \sqrt{25}=5\sqrt{3}.\end{align*}$$
  ::接下来,以25的平方根为平方根, 因为25的平方根是一个整数: 325=325=53。

  播放段落

  Thus, the 2nd fraction simplifies to:
  ::因此,第2个分数简化为:

  $$\begin{align*}\frac{\sqrt{3}}{\sqrt{75}}=\frac{\sqrt{3}}{5\sqrt{3}}=\frac{1}{5}.\end{align*}$$

  播放段落

  Step 2:  Rewrite $\begin{align*} \frac{\pi}{15}\end{align*}$ to compare it to $\begin{align*}\frac{1}{5}:\end{align*}$
  ::步骤2:重写 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

  $$\begin{align*} \frac{\pi}{15}=\frac{\pi}{3\times 5}=\frac{\pi}{3}\times \frac{1}{5}.\end{align*}$$

  播放段落

  Since $\begin{align*}\pi >3\end{align*}$ ,   $\begin{align*}\frac{\pi}{3}>1\end{align*}$ . Then,  $\begin{align*}\frac{\pi}{3}\times \frac{1}{5}> \frac{1}{5}.\end{align*}$   Therefore , $\begin{align*}\frac{\pi}{15}> \frac{\sqrt{3}}{\sqrt{75}}.\end{align*}$
  ::3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, , 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, , , 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15,

   

  播放段落

  Example 4
  ::例4

  播放段落

  For the numbers  $\begin{align*} \tfrac{\sqrt{12}}{2}, 1.5\cdot \sqrt{3}, \tfrac{3}{2}, \tfrac{2\sqrt{5}}{\sqrt{20}}\end{align*}$ :
  ::122,1.53,32,2520:

  播放段落

  a) Classify each number:
  :a) 将每一数字分类:

  播放段落

  Solution:
  ::解决方案 :

  播放段落

  S implify the numbers in order to classify them:
  ::简化数字以便进行分类:

  播放段落

  $\begin{align*} \frac{\sqrt{12}}{2}=\frac{\sqrt{4\times 3}}{2}=\frac{2\sqrt{3}}{2}=\sqrt{3}.\end{align*}$ This is an irrational number. An irrational number is a type of real number.
  ::122=4x32=232=3。这是一个非理性数字。非理性数字是一种实际数字。

  播放段落

  $\begin{align*}1.5\cdot \sqrt{3}. \end{align*}$ This number cannot be simplified, but since it is a multiple of an irrational number, it is also irrational. In other words, we cannot get rid of the irrational part, so we cannot write it as a rational number. It is also a real number.
  ::1.5+3. 这个数字无法简化, 但它是一个不合理数字的倍数, 也是不合理的。 换句话说, 我们不能摆脱不合理部分, 所以我们不能把它写成一个合理的数字。 它也是一个真实的数字 。

  播放段落

  $\begin{align*} \frac{3}{2}. \end{align*}$ Since this number is in the form of a proper fraction, it is also a rational number and a real number.
  ::32. 由于这个数字以适当分数的形式出现,它也是一个合理数字和实际数字。

  播放段落

  $\begin{align*} \frac{2\sqrt{5}}{\sqrt{20}}=\frac{2\sqrt{5}}{\sqrt{4\times 5}}=\frac{2\sqrt{5}}{2\sqrt{5}}=1.\end{align*}$ This number can be simplified to an integer. All positive integers can be expressed as natural numbers, whole numbers, and rational numbers. Integers are a special kind of real number.
  ::2520=254x5=2525=1. 此数字可以简化为整数。 所有正数整数都可以以自然数、整数和合理数表示。 整数是一种特殊的真数类型 。

   

  播放段落

  b) Order the four numbers:
  :b) 命令四个数字:

  播放段落

  Solution:
  ::解决方案 :

  播放段落

  The four numbers are ordered as follows: $\begin{align*}1<\frac{3}{2}<\sqrt{3} <1.5\cdot \sqrt{3}\end{align*}$ .
  ::这四个号码的顺序如下:1<32<3<3<1.53。

  播放段落

  $\begin{align*}1<\frac{3}{2}\end{align*}$ since the numerator is larger than the denominator and $\begin{align*}\frac{3}{2}=1.5\end{align*}$ .
  ::1<32,因为分子大于分母和32=1.5。

  播放段落

  $\begin{align*}\frac{3}{2}<\sqrt{3}\end{align*}$ since we can see on our calculators that $\begin{align*}\sqrt{3}\approx 1.7\end{align*}$ .
  ::32<3,因为我们可以看到我们的计算器 3+1.7。

  播放段落

  $\begin{align*}\sqrt{3} < 1.5\cdot \sqrt{3}\end{align*}$ since multiplying by 1.5 makes any positive number larger.
  ::自乘以1.5后, 3<1.5/13 使正数增加。

  播放段落

  Review
  ::回顾

  播放段落

  Classify the numbers below as natural, whole, integer, irrational, rational, and/or real. Include all the categories that apply to the number.
  ::将以下数字分类为自然、整体、整数、不合理、合理和/或真实。包括适用于数字的所有类别。

  1. $\begin{align*}\mathrm{-}19\end{align*}$  
  2. $\begin{align*}\sqrt{0.25}\end{align*}$
  3. $\begin{align*}\tfrac{0}{4}\end{align*}$  
  4. $\begin{align*}\sqrt{1.35}\end{align*}$
  5. $\begin{align*}\mathrm{-}\tfrac{1}{9}\end{align*}$  
  6. $\begin{align*}\tfrac{1}{5}\end{align*}$  
  7. $\begin{align*}4\sqrt{2}\end{align*}$
  8. $\begin{align*}5\end{align*}$  
  9. $\begin{align*}\mathrm{-}2\pi\end{align*}$  
  播放段落
  10. $\begin{align*}\sqrt{100}\end{align*}$
      
      Order the following numbers from least to greatest:
      ::100 将以下数字从最小到最大顺序排列如下:
  11. 
      $$\begin{align*}\frac{\sqrt{6}}{2} && \frac{61}{50} && \sqrt{1.5} && \frac{16}{13}\end{align*}$$
  12. 
      $$\begin{align*}\mathrm{-}7\frac{{1}}{2} && \mathrm{-}3 && 5.5 && 7\frac{1}{2} && \mathrm{-}1.5\end{align*}$$
  13.  
      $$\begin{align*}\tfrac{9}{10} && \mathrm{-}2.2 && \sqrt{4} && \mathrm{-}0.3\end{align*}$$
  14. 
      $$\begin{align*}\sqrt{3} && 1.85 && \tfrac{26}{10} && 1 && \tfrac{3}{4}\end{align*}$$  
  播放段落
  15. Find the value of each marked point:
      
      

      

       
      ::查找每个标记点的值 :

   

  播放段落

  Review (Answers)
  ::回顾(答复)

  播放段落

  Please see the Appendix.
  ::请参看附录。

  播放段落

  Resource: Classifying Numbers
  ::资源: 分类数字

  播放段落

  Watch the following video for additional review on the classification of numbers.
  ::观看以下录像,以便进一步审查数字分类。