Course: 2.10 理性数字和不合理数字的属性 | The Way To Learn

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

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, $\begin{aligned} \sqrt{49} \end{aligned}$ is rational because it equals 7, but $\begin{aligned} \sqrt{50} \end{aligned}$ can’t be reduced farther than $\begin{aligned} 5 \sqrt{2} \end{aligned}$ . That factor of $\begin{aligned} \sqrt{2} \end{aligned}$ 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。

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

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。

a) 23.7
:a) 23.7

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

b) 2.8956
::b) 2.8956

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

c) $\begin{aligned} π \end{aligned}$
::c) ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} π = 3.141592654 \dots \end{aligned}$ We know from the definition of $\begin{aligned} π \end{aligned}$ that the decimals do not terminate or repeat, so $\begin{aligned} π \end{aligned}$ 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。

d) $\begin{aligned} \sqrt{6} \end{aligned}$
::d) 6 6

$\begin{aligned} \sqrt{6} = \sqrt{2} \times \sqrt{3} \end{aligned}$ . 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。

Repeating Decimals
::重复十进数

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, $\begin{aligned} 3. \bar{27} = 3.272727272727 \dots \end{aligned}$ 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 $\begin{aligned} \frac{36}{11} \end{aligned}$ .
::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 。

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

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

a.) 0.439
::a) 0.439

0.439 can be expressed as $\begin{aligned} \frac{4}{10} + \frac{3}{100} + \frac{9}{1000} \end{aligned}$ , or just $\begin{aligned} \frac{439}{1000} \end{aligned}$ . 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。

b.) $\begin{aligned} 0.25 \bar{38} \end{aligned}$
:b) 0.2538'

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

Classify Real Numbers
::分类 Real 数字

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。

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)。

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。

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

a) 0
::a) 0 0

Integer
::整数

b) -1
::b) -1

Integer
::整数

c) $\begin{aligned} \frac{π}{3} \end{aligned}$
:c) %3

Irrational (Although it's written as a fraction, $\begin{aligned} π \end{aligned}$ is irrational, so any fraction with $\begin{aligned} π \end{aligned}$ 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。)

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

Irrational
::无理

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

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

Example
::示例示例示例示例

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。

Example 1
::例1

$\begin{aligned} \frac{100}{99} \frac{\sqrt{3}}{3} - \sqrt{.075} \frac{2 π}{3} \end{aligned}$

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

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

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

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

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

$\begin{aligned} - \sqrt{.075}, \frac{\sqrt{3}}{3}, \frac{100}{99}, \frac{2 π}{3} \end{aligned}$

Review
::回顾

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。

$\begin{aligned} \sqrt{0.25} \end{aligned}$

$\begin{aligned} \sqrt{1.35} \end{aligned}$

$\begin{aligned} \sqrt{20} \end{aligned}$

$\begin{aligned} \sqrt{25} \end{aligned}$

$\begin{aligned} \sqrt{100} \end{aligned}$

$\begin{aligned} \sqrt{π^{2}} \end{aligned}$

$\begin{aligned} \sqrt{2 \cdot 18} \end{aligned}$

Write 0.6278 as a fraction.
::将0.6278作为一个分数写下来。

Place the following numbers in numerical order, from lowest to highest. $\begin{aligned} \frac{\sqrt{6}}{2} \frac{61}{50} \sqrt{1.5} \frac{16}{13} \end{aligned}$
::按数字顺序排列下列数字,从最低到最高。 6261501.51613

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。

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。

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