Course: 1.2 定序实际数字

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排序真实数字
Mariah can spin a basketball on her finger for a quarter of an hour, Chita can spin one for 10 minutes, and Dakara can spin one for 650 seconds. Who can spin the basketball on her finger the longest?
::玛利亚的手指上可以旋转一个篮球四分之一小时,奇塔的手指上可以旋转一个10分钟,达卡的手指上可以旋转一个650秒。谁能把篮球打得最长?

We need to put all these different ways of measuring time in the same unit.
::我们需要将所有这些不同的时间计量方法放在同一个单位。

Putting Real Numbers in Order
::按顺序排列真实数字

Real numbers can be listed in order even if they are different types of real numbers. The easiest way to do this is to convert all into decimals.
::即使真实数字是不同类型的真实数字,也可以按顺序列出真实数字。最容易的方法就是将所有数字转换为小数。

Let's do the following problems.
::让我们解决以下的问题。

Plot $\begin{aligned} 1.25, \frac{7}{2} \end{aligned}$ , and $\begin{aligned} 2 \sqrt{6} \end{aligned}$ on a number line.
::计 1.2572 和26 在数字线上。

One way to compare numbers is to use a number line. To plot these numbers, convert them all to decimals. $\begin{aligned} 1.25, \frac{7}{2} = 3.5 \end{aligned}$ , and $\begin{aligned} 2 \sqrt{6} \approx 4.899 \end{aligned}$ (The symbol $\begin{aligned} \approx \end{aligned}$ means approximately .) Draw your number line and plot the points. Recall that 0 is called the origin.
::比较数字的一种方法是使用数字行。要绘制这些数字,请将其转换为小数数。 1.25, 72=3.5, 和 264. 899 (符号意指约。) 绘制数字线并绘制点。请注意, 0 被称为来源。

Depending on your scale, you can have hash marks at half-values or only even values. The placement of each number on the number line is an approximate representation of each number.
::视您的规模而定,您可以拥有半值或甚至值的散列标记。数字行中每个数字的排列是每个数字的大致表示。

List $\begin{aligned} \frac{3}{4}, 1.23, \sqrt{2}, \frac{2}{3}, 1 \end{aligned}$ and $\begin{aligned} \frac{8}{7} \end{aligned}$ in order from least to greatest.
::名单341.23、2、33、1和87,从最少到最多顺序。

First, write each number as a decimal.
::首先,将每个数字以小数点记为小数点。

$\begin{aligned} \frac{3}{4} = 0.75, 1.23, \sqrt{2} \approx 1.4142, \frac{2}{3} = 0.6 \bar{6}, 1, \frac{8}{7} = 1. \bar{142857} \end{aligned}$ . Now, write the decimals, in order, starting with the smallest and ending with the largest: $\begin{aligned} 0.6667, 0.75, 1, 1.1428, 1.23, 1.4142 \end{aligned}$
::34=0.75,1.23,21.4142,23=0.66,1,87=1.142857。现在,按序写小数小数,从最小开始,到最大:0.6667,0.75,1.1428,1.23,1.4142。

Finally, exchange the decimals with the original numbers: $\begin{aligned} \frac{2}{3}, \frac{3}{4}, 1, \frac{8}{7}, 1.23, \sqrt{2} \end{aligned}$
::最后,将小数点与原数(23,34,1,87,1.23)交换。

Replace the blank between $\begin{aligned} - \frac{5}{3} \end{aligned}$ ______ $\begin{aligned} - \frac{π}{2} \end{aligned}$ with <, > or =.
::将 - 53 2 之间的空白替换为 < 、 > 或 = 。

Write both numbers in decimals. $\begin{aligned} - \frac{5}{3} = - 1.6 \bar{6}, - \frac{π}{2} \approx - 1.57079 \end{aligned}$ . This means that $\begin{aligned} - \frac{π}{2} \end{aligned}$ is the larger number, so $\begin{aligned} - \frac{5}{3} < - \frac{π}{2} \end{aligned}$ .
::以小数写两个数字。 -531.66,21.57079。这意味着%2是最大的数字,所以~532。

Examples
::实例

Example 1
::例1

Earlier, you were asked who can spin the basketball on her finger the longest.
::之前有人问你谁能用她手指上最长的篮球

To compare these different measurements of time, let's put them all in minutes. A quarter of an hour is a quarter of 60 minutes or $\begin{aligned} 60 \div 4 = 15 \end{aligned}$ . And there are 60 second in a minute, so 650 seconds is $\begin{aligned} 650 \div 60 = 10.8333333... \end{aligned}$ minutes. From this, we know that a quarter of an hour is the most and therefore Mariah can spin the basketball on her finger the longest.
::为了比较这些不同的时间测量方法,让我们把它们都放在分钟内。四分之一小时是四分之一60分钟或604=15。一分钟有60秒, 所以650秒是650 60=10.8333333... 分钟。从这个角度来说, 我们知道最多一个小时是四分之一小时, 因此Mariah可以把篮球打在她的手指上, 时间最长一点。

Example 2
::例2

List $\begin{aligned} - \frac{1}{4}, \frac{3}{2}, - \sqrt{3}, \frac{3}{5} \end{aligned}$ , and 2 in order from greatest to least.
::清单14、32、-3、35和2,从最大到最低顺序。

Write all the real numbers as decimals. $\begin{aligned} - \frac{1}{4} = - 0.25, \frac{3}{2} = 1.5, - \sqrt{3} \approx - 1.732, \frac{3}{5} = 0.6, 2 \end{aligned}$ In order, the numbers are: $\begin{aligned} 2, \frac{3}{2}, \frac{3}{5}, - \frac{1}{4}, - \sqrt{3} \end{aligned}$
::将所有实际数字都写为小数。 - 140. 25, 32=1.5, - 31.732, 35=0. 6, 2 按顺序排列, 数字为: 2, 32, 35, - 14, - 3。

Example 3
::例3

Compare $\begin{aligned} \sqrt{7} \end{aligned}$ and 2.5 by using <, >, or =.
::使用 < 、 > 或= 来比较7和2.5。

$\begin{aligned} \sqrt{7} \approx 2.646 \end{aligned}$ . Therefore, it is larger than 2.5. Comparing the two numbers, we have $\begin{aligned} \sqrt{7} > 2.5 \end{aligned}$ .
::72.646. 因此,它大于2.5。与这两个数字相比,我们有7>2.5。

Review
::回顾

Plot the following numbers on a number line. Use an appropriate scale.
::在数字行中绘制以下数字。使用适当的比例尺。

$\begin{aligned} - 1, 0.3, \sqrt{2} \end{aligned}$

$\begin{aligned} - \frac{1}{4}, - 2 \frac{1}{2}, 3.15 \end{aligned}$

$\begin{aligned} 1.4, \frac{5}{6}, \sqrt{9} \end{aligned}$

$\begin{aligned} - \sqrt{6}, \frac{4}{3}, π \end{aligned}$

Order the following sets of numbers from least to greatest.
::排序以下数组,从最小数到最大数组。

$\begin{aligned} - 4, - \frac{9}{2}, - \frac{1}{3}, - \frac{1}{4}, - π \end{aligned}$

$\begin{aligned} 0, - \frac{1}{2}, \frac{4}{5}, \frac{1}{6}, - \sqrt{\frac{1}{3}} \end{aligned}$

Order the following sets of numbers from greatest to least.
::顺序排列以下从最大到最小的一组数字。

$\begin{aligned} 3.68, 4 \frac{1}{2}, 5, 3 \frac{11}{12}, \sqrt{10} \end{aligned}$

$\begin{aligned} - 2, - \frac{6}{5}, - \frac{11}{4}, - \sqrt{5}, - \sqrt{3} \end{aligned}$

Compare each pair of numbers using <, >, and =.
::用 <, > 和 = 来比较每一对数字。

$\begin{aligned} - \frac{1}{4} \underline{} - \frac{3}{8} \end{aligned}$

$\begin{aligned} \sqrt{8} \underline{} 2.9 \end{aligned}$

$\begin{aligned} - 2 \frac{8}{9} \underline{} - 2.75 \end{aligned}$

$\begin{aligned} \frac{10}{15} \underline{} \frac{8}{12} \end{aligned}$

$\begin{aligned} - \sqrt{50} \underline{} - 5 \sqrt{2} \end{aligned}$

$\begin{aligned} 1 \frac{5}{6} \underline{} 1.95 \end{aligned}$

Calculator Challenge Locate the button $\begin{aligned} e \end{aligned}$ on your scientific calculator. $\begin{aligned} e \end{aligned}$ is called the natural number and will be used in the Exponential and Logarithmic Functions chapter.

Press the $\begin{aligned} e \end{aligned}$ button. What is $\begin{aligned} e \end{aligned}$ equivalent to?
::按下 e 按钮。 e 等量什么 ?

What type of real number do you think $\begin{aligned} e \end{aligned}$ is?
::你认为真实数字是什么类型?

Which number is larger? $\begin{aligned} e \end{aligned}$ or $\begin{aligned} π \end{aligned}$ ?
::哪个数字更大?

Which number is larger? $\begin{aligned} e \end{aligned}$ or $\begin{aligned} \sqrt{7} \end{aligned}$
::哪个数字更大? e 或 7

::计算器挑战计算器定位您科学计算器上的按钮 e 。 e 被称为自然数字, 并将用于指向和对数函数章节。按下 e 按钮。 e 键是什么等量 ? 您认为实际数字是哪一类? 哪个数字更大? e 或 ? 哪个数字更大? e 或 ? 哪个数字更大? e 或 7

Review (Answers)
::回顾(答复)

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。

Section outline

General

排序真实数字

Putting Real Numbers in Order ::按顺序排列真实数字

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Review ::回顾

Review (Answers) ::回顾(答复)

Putting Real Numbers in Order
::按顺序排列真实数字

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Review
::回顾

Review (Answers)
::回顾(答复)