4.14有理数的性质

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  有理数的性质

  

  [Figure 1]

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  Eva just learned about rational numbers in math class. She learned that rational numbers are numbers that be written as a ratio of two integers. She understands that numbers like  $\begin{array}{r}\frac{1}{3}\end{array}$  and  $\begin{array}{r}\frac{9}{5}\end{array}$ are rational numbers. Eva's friend Mike says that all integers are rational numbers too, but Eva isn't sure if that is true. After all, integers don't look like fractions. How could Mike help to convince Eva that integers are rational numbers too?
  ::Eva刚刚在数学课中学会了理性数字。 她知道理性数字是两个整数之比。 她理解13和9 5这样的数字是理性数字。 Eva的朋友Mike说,所有整数都是理性数字, 但Eva并不确定这是否属实。 毕竟,整数看起来不像分数。 Mike怎么能帮助Eva相信整数也是理性数字呢?

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  In this concept, you will learn how to identify rational numbers.
  ::在这个概念中,你将学会如何确定合理数字。

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  Identifying Rational Numbers
  ::识别有理数字

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  A  ratio  is a comparison between two numbers. Ratios can be written in words, as fractions, or using a colon.
  ::A 比率是两个数字的比较。 比率可以用文字、 分数或使用冒号来写。

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  Here is an example.
  ::举一个例子。

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  Suppose your class has 12 boys and 13 girls. You could say that the ratio of boys to girls is:
  ::假设你的班级有12个男孩和13个女孩。你可以说,男孩与女孩的比例是:

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  • 12 to 13
    ::12至13 12至13
  • 12:13
  • $\begin{array}{r}\frac{12}{13}\end{array}$

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  All three forms are equivalent ways of expressing the same ratio.
  ::所有三种表格都是表示相同比例的同等方式。

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  The  rational numbers  are the set of numbers that can be written as a ratio of two integers. In other words, any number that can be written as a fraction where both the  numerator  and the  denominator  are integers is a  rational number . Any number that cannot be written as a fraction in this way is not a rational number and is considered an  irrational number .
  ::理性数字是可以写成两个整数之比的一组数字。换句话说,当分子和分母都是整数时,任何可以写成一个分数的数字都是合理的数字。不能以这种方式写成一个分数的任何数字都不是合理的数字,而是不合理的数字。

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  Let's look at an example.
  ::让我们举个例子。

  $\begin{array}{r}\frac{\text{-}2}{3}\end{array}$

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  This number is already written as a fraction so it is definitely a rational number.
  ::此数字已作为一个小数写入, 因此它绝对是一个合理的数字 。

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  The answer is that  $\begin{array}{r}\frac{\text{-}2}{3}\end{array}$  is a rational number because it is the ratio of -2 to 3.
  ::答案是 -2 3是一个合理的数字,因为它是-2比3之比。

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  Let's look at another example.
  ::让我们再举一个例子。

  10

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  This number is not already written as a fraction; however, you could rewrite it as a fraction. Remember that any number divided by 1 is just itself. So you have
  ::此数字尚未作为一个分数写成; 但是, 您可以把它重新写成一个分数。 记住, 任何除以 1 的数都是它本身 。 因此, 您可以将它重新写成一个分数 。

  $\begin{array}{r}10=\frac{10}{1}\end{array}$

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  The answer is that because 10 can be written as the ratio of 10 to 1, it is a rational number. In fact, all integers are rational numbers because they can all be written as a ratio of themselves to 1.
  ::答案是,因为10可以被写成10比1的比例,所以这是一个合理的数字。事实上,所有整数都是合理的数字,因为它们都可以被写成自己与1的比例。

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  Numbers written in decimal form can also be rational numbers as long as they are repeating or terminating decimals.
  ::以小数格式书写的数字也可以是合理数字,只要它们重复或终止小数。

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  Here is an example.
  ::举一个例子。

  0.687

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  This number goes out to the thousandths place. You can read this number as 687 thousandths. Reading it in this way can help you to rewrite this number in fraction form:
  ::此数字输出到千个地方。 您可以读取这个数字为 687 千个。 这样阅读它可以帮助您以分数形式重写此数字 :

  $\begin{array}{r}\frac{687}{1000}\end{array}$

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  Because 0.687 can be written as the ratio of 687 to 1000, it is a rational number.
  ::由于0.687可以写成687与1000之比,因此这是一个合理的数字。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about Eva, who just learned about rational numbers.
  ::早些时候,你得到一个伊娃的问题, 她刚刚学会了理性数字。

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  Mike is trying to convince Eva that integers are rational numbers too. Eva understands that a rational number is a number that can be written in fraction form as a ratio of two integers.
  ::Mike试图让Eva相信整数也是合理数字。 Eva理解,一个合理数字是一个可以以分数形式以两个整数的比例表示的数字。

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  If Mike wanted to convince Eva that integers are rational numbers too, he could give a specific example. For example, the integer -20.
  ::如果Mike想让Eva相信整数也是合理的数字,他可以举一个具体的例子。例如整数 -20。

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  -20 does not look like the ratio of two integers. However, it can be rewritten in fraction form.
  ::-20 看上去不像两个整数的比率。 但是, 它可以以分数形式重写 。

  $\begin{array}{r}\text{-}20=\frac{\text{-}20}{1}\end{array}$

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  Because any integer could be similarly written as a ratio of itself to 1, all integers are rational numbers. In fact, integers make up a subset of the set of rational numbers.
  ::因为任何整数都可以类似地写成自己与1的比例,所以所有的整数都是理性数字。事实上,整数是一组理性数字的一个子集。

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  Example 2
  ::例2

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  Show that the following number is rational by writing it as a ratio in fraction form.
  ::以分数形式写成比例以显示以下数字是合理的 。

  .85

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  You read this number as eighty-five hundredths. You can convert it to a fraction:
  ::您将这个数字读为八千五百分之一。 您可以将其转换成一个分数 :

  $\begin{array}{r}\frac{85}{100}\end{array}$

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  The answer is that .85 is a rational number because it can be written as the ratio  $\begin{array}{r}\frac{85}{100}\end{array}$ .
  ::答案是.85是一个合理的数字,因为它可以写成85 100的比率。

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  Example 3
  ::例3

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  Determine whether or not the following number is rational. If it is rational, show how it can be written as a ratio in fraction form.
  ::确定以下数字是否合理。 如果合理, 请显示如何以分数形式将其写成比例 。

  -4

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  This number is not already written as a fraction; however, it can be rewritten as a fraction.
  ::此数字尚未作为一个分数写成; 但是, 它可以作为一个分数重新写成 。

  $\begin{array}{r}\text{-}4=\frac{\text{-}4}{1}\end{array}$

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  The answer is that because -4 can be written as the ratio of -4 to 1, it is a rational number.
  ::答案是,因为 -4 可以写为 -4比1的比例,这是一个合理的数字。

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  Example 4
  ::例4

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  Determine whether or not the following number is rational. If it is rational, show how it can be written as a ratio in fraction form.
  ::确定以下数字是否合理。 如果合理, 请显示如何以分数形式将其写成比例 。

  $\begin{array}{r}0.33\bar{3}\end{array}$

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  This is a  repeating decimal  and all repeating decimals are rational. This means you can rewrite this number in fraction form as the ratio of two integers.
  ::这是重复的十进制, 所有重复的十进制数字都是合理的。 这意味着您可以重写此数字, 以分数为两个整数的比例 。

  $\begin{array}{r}0.33\bar{3}=\frac{1}{3}\end{array}$

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  The answer is that because  $\begin{array}{r}0.33\bar{3}\end{array}$  can be written as the ratio of 1 to 3, it is a rational number.
  ::答案是,因为0.333 3可以写成1比3的比例,所以这是一个合理的数字。

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  Example 5
  ::例5

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  Determine whether or not the following number is rational. If it is rational, show how it can be written as a ratio in fraction form.
  ::确定以下数字是否合理。 如果合理, 请显示如何以分数形式将其写成比例 。

  $\begin{array}{r}\pi \approx 3.14159\dots \end{array}$

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  $\begin{array}{r}\pi \end{array}$ , pronounced like “pie,” is an example of a number that is not rational. It is an  irrational number . Its decimals continue on forever, but never repeat. It cannot be rewritten in fraction form.
  ::, 像“ pie ” 一样, 是一个数字不合理的例子。 这是一个不合理的数字。 它的十进制会永远持续下去, 但永远不再重复。 它不能以小数形式重写 。

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  Review
  ::回顾

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  Rewrite each number as the ratio of two integers to prove that each number is rational.
  ::将每个数字重写为两个整数的比率,以证明每个数字是合理的。

  1. −11
  2. $\begin{array}{r}3\frac{1}{6}\end{array}$
  3. 9
  4. 0.08
  5. -0.34
  6. 0.678
  7. $\begin{array}{r}\frac{4}{5}\end{array}$
  8. -19
  9. 25
  10. 0.17
  11. 0.2347
  12. -17
  13. 347
  14. 87
  15. -97