理性数字和不合理数字的属性

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Properties of Rational versus Irrational Numbers 
::合理和不合理数字的属性

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Not all square roots are irrational, but any square root that can’t be reduced to a form with no radical signs in it is irrational. For example, $\begin{array}{r}\sqrt{49}\end{array}$ is rational because it equals 7, but $\begin{array}{r}\sqrt{50}\end{array}$ can’t be reduced farther than $\begin{array}{r}5\sqrt{2}\end{array}$ . That factor of $\begin{array}{r}\sqrt{2}\end{array}$ is irrational, making the whole expression irrational.
::并非所有的平方根都是非理性的,但任何不能被缩小为没有激进迹象的形式的平方根都是非理性的。 比如,49是理性的,因为它等于7,但50不能比52更进一步。 2是非理性的,使整个表达变得不合理。

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Identifying Rational and Irrational Numbers 
::确定合理和不合理数字

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Identify which of the following are and which are irrational numbers.
::确定以下哪些是非理性数字,哪些是非理性数字。

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a) 23.7
:a) 23.7

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23.7 can be written as $\begin{array}{r}23\frac{7}{10}\end{array}$ , so it is rational.
::23.7可以写为23710,因此是合理的。

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b) 2.8956
::b) 2.8956

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2.8956 can be written as $\begin{array}{r}2\frac{8956}{10000}\end{array}$ , so it is rational.
::2.8956可以写为289561000,所以是合理的。

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c) $\begin{array}{r}\pi \end{array}$
::c) ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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$\begin{array}{r}\pi =3.141592654\dots \end{array}$ We know from the definition of $\begin{array}{r}\pi \end{array}$ that the decimals do not terminate or repeat, so $\begin{array}{r}\pi \end{array}$ is irrational.
::3141592654... 我们从 的定义中知道 小数不会终止或重复, 所以是不合理的。

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d) $\begin{array}{r}\sqrt{6}\end{array}$
::d) 6 6

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  $\begin{array}{r}\sqrt{6}=\sqrt{2}\times \sqrt{3}\end{array}$ . We can’t reduce it to a form without radicals in it, so it is irrational.
::6=2 ×3. 我们无法将它降为没有激进的形态, 因此它是非理性的。

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Repeating Decimals
::重复十进数

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Any number whose decimal representation has a finite number of digits is rational, since each decimal place can be expressed as a fraction. For example, $\begin{array}{r}3.\overline{27}=3.272727272727\dots \end{array}$ This decimal goes on forever, but it’s not random; it repeats in a predictable pattern. Repeating decimals are always rational; this one can actually be expressed as $\begin{array}{r}\frac{36}{11}\end{array}$ .
::任何小数表示的十进制数字都有一定数量的数字是合理的, 因为每个小数位可以以小数表示。 例如, 3. 27 3. 32 727 272. 27... 这个小数会永远持续下去, 但这不是随机的; 它会以可预测的模式重复。 重复小数总是合理的; 这个小数位实际上可以表示为 3611 。

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Expressing Decimals as Fractions 
::以分数表示十进数

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Express the following decimals as fractions.
::以下列小数分数表示。

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a.) 0.439
::a) 0.439

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0.439 can be expressed as $\begin{array}{r}\frac{4}{10}+\frac{3}{100}+\frac{9}{1000}\end{array}$ , or just $\begin{array}{r}\frac{439}{1000}\end{array}$ . Also, any decimal that repeats is rational, and can be expressed as a fraction.
::0.439 可以表示为410+3100+91000, 或只有4391000。此外, 重复的任何小数点是合理的, 可以表示为分数 。

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b.) $\begin{array}{r}0.25\overline{38}\end{array}$
:b) 0.2538'

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$\begin{array}{r}0.25\overline{38}\end{array}$ can be expressed as $\begin{array}{r}\frac{25}{100}+\frac{38}{9900}\end{array}$ , which is equivalent to $\begin{array}{r}\frac{2513}{9900}\end{array}$ .
::2538 以25100+389900表示,相当于25139900。

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Classify Real Numbers
::分类 Real 数字

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We can now see how real numbers fall into one of several categories.
::我们现在可以看到实际数字如何属于几个类别之一。

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If a real number can be expressed as a rational number , it falls into one of two categories. If the denominator of its simplest form is one, then it is an integer . If not, it is a fraction (this term also includes decimals, since they can be written as fractions.)
::如果一个实际数字可以以合理数字表示,则它属于两类中的一种。如果其最简单形式的分母为一种,则它是一个整数。如果不是,它是一个分数(这个术语也包括小数,因为它们可以作为分数来写)。

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If the number cannot be expressed as the ratio of two integers (i.e. as a fraction), it is irrational .
::如果数字不能以两个整数(即分数)之比表示,则不合理。

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Classify the following real numbers.
::将以下真实数字分类 。

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a) 0
::a) 0 0

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Integer
::整数

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b) -1
::b) -1

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Integer
::整数

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c) $\begin{array}{r}\frac{\pi }{3}\end{array}$
:c) %3

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Irrational (Although it's written as a fraction,  $\begin{array}{r}\pi \end{array}$  is irrational, so any fraction with  $\begin{array}{r}\pi \end{array}$  in it is also irrational.) 
::讽刺(尽管它是一个小数, 是非理性的, 所以任何小数, 也是非理性的。)

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d) $\begin{array}{r}\frac{\sqrt{2}}{3}\end{array}$
:d) 23

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Irrational
::无理

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e) $\begin{array}{r}\frac{\sqrt{36}}{9}\end{array}$
:e) 369

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Rational (It simplifies to $\begin{array}{r}\frac{6}{9}\end{array}$ , or $\begin{array}{r}\frac{2}{3}\end{array}$ .)
::合理(简化为69或23)。

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Example
::示例示例示例示例

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Place the following numbers in numerical order , from lowest to highest.
::按数字顺序排列以下数字,从最低到最高。

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Example 1
::例1

$\begin{array}{r}\frac{100}{99}\frac{\sqrt{3}}{3}-\sqrt{.075}\frac{2\pi }{3}\end{array}$

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Since $\begin{array}{r}-\sqrt{.075}\end{array}$ is the only negative number, it is the smallest.
::- 075是唯一负数,是最小数。

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Since $\begin{array}{r}100>99\end{array}$ , $\begin{array}{r}\frac{100}{99}>1\end{array}$ .
::自100>99,10099>1以来。

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Since the $\begin{array}{r}\sqrt{3}<s\end{array}$ , then $\begin{array}{r}\frac{\sqrt{3}}{3}<1\end{array}$ .
::从3 < 3 < 开始,然后是33 < 1。

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Since $\begin{array}{r}\pi >3\end{array}$ , then $\begin{array}{r}\frac{\pi }{3}>1\Rightarrow \frac{2\pi }{3}>2\end{array}$
::自% 3 起, 然后% 3 > 1 @% 2% 3 > 2

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This means that the ordering is:
::这意味着命令是:

$\begin{array}{r}-\sqrt{.075},\frac{\sqrt{3}}{3},\frac{100}{99},\frac{2\pi }{3}\end{array}$

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Review 
::回顾

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For questions 1-7, classify the following numbers as an integer, a rational number or an irrational number.
::对于问题1-7,将下列数字分类为整数、合理数字或不合理数字。

1. $\begin{array}{r}\sqrt{0.25}\end{array}$
2. $\begin{array}{r}\sqrt{1.35}\end{array}$
3. $\begin{array}{r}\sqrt{20}\end{array}$
4. $\begin{array}{r}\sqrt{25}\end{array}$
5. $\begin{array}{r}\sqrt{100}\end{array}$
6. $\begin{array}{r}\sqrt{{\pi }^2}\end{array}$
7. $\begin{array}{r}\sqrt{2\cdot 18}\end{array}$
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8. Write 0.6278 as a fraction.
   ::将0.6278作为一个分数写下来。
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9. Place the following numbers in numerical order, from lowest to highest. $\begin{array}{r}\frac{\sqrt{6}}{2}\frac{61}{50}\sqrt{1.5}\frac{16}{13}\end{array}$
   ::按数字顺序排列下列数字,从最低到最高。 6261501.51613
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10. Use the marked points on the number line and identify each proper fraction.
    ::使用数字行上的标记点并确定每个适当的分数。

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Review (Answers)
::回顾(答复)

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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 。