10.11 代表带有十进制的理性数字-interactive

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• 选择节以十进制代表理性数字

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  以十进制代表理性数字

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  Mathematicians
  ::数学家

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  You may have read  the previous lessons in which you  have looked at how math is used across nine different professional fields. However, one field which you haven’t looked at yet should be the most obvious of all. In this chapter, you will learn about mathematicians and how the field has changed throughout history. Y ou will focus on pure mathematicians and applied mathematicians. A pure mathematician focuses more on advancing mathematics, looking for patterns and new relationships that extend our ability to interpret the universe. An applied mathematician uses math as a tool to solve real-world problems. Y ou will use , , radicals, and as a lens through which to observe how mathematicians have directly contributed to the growth of humanity.
  ::您可能已经读过您以前研究过数学如何在九个不同专业领域应用的教训。 然而,一个您尚未研究的领域应该是最显而易见的。 在本章中,您将学习数学家和整个历史过程的变化。 您将关注纯数学家和应用数学家。 一个纯数学家将更多关注促进数学,寻找模式和新关系,以扩展我们解释宇宙的能力。 一个应用数学家将数学用作解决现实世界问题的工具。 您将使用, 激进, 并作为一个观察数学家如何直接促进人类成长的透镜。

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  Modern mathematicians look at math concepts that are extremely complicated and have been built upon for thousands of years. To understand modern mathematics, you will be examining the mathematical advancements that brought us to where we are today.
  ::现代数学家研究极其复杂的数学概念,这些概念已经发展了数千年。 为了理解现代数学,你将研究使我们来到今天的数学进步。

   

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  Types of Numbers
  ::数字类型类型

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  In  Africa, in approximately 20,000 BCE, you have the first evidence of counting. What is referred to as the Ishango bone has carved notches in a pattern shown below: 
  ::在非洲,在大约20,000个生物浓度和浓度指数中,你有第一个计算证据。

  

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  Around the year 3,400 BCE came the first number system from the Sumerians of Mesopotamia, modern day Iraq. This number system set the stage for our current number system. Real numbers are the set of numbers which can be found on a number line . A set means a collection of distinct objects or in this case, numbers. The set of real numbers can be broken down into smaller subsets which each have their own subsets. A subset is a group of objects or numbers taken from another set.
  ::大约在3400年左右, BCE是现代伊拉克美索不达米亚苏美尔人的第一个数字系统。 这个数字系统为我们目前的数字系统铺设了舞台。 实际数字是数字线上可以找到的一组数字。 一组是不同对象的集合, 或在此情况下, 数字。 一组是真实数字, 可以细分为较小的子集, 每个子集都有自己的子集。 子集是一组来自另一组的物体或数字 。

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  Real numbers can be broken down into two subsets: rational numbers and irrational numbers. If you look at the real numbers and only take the numbers which can be written as a fraction, this subset would be the rational numbers are the subset of real numbers which can be written as a fraction. Irrational numbers are the subset of real numbers which cannot be written as a fraction. Y ou saw an example of these earlier in the chapter. If you looked at the rational numbers and only took the numbers without a fractional part, this subset would be the integers . If you look at the integers and only took the positive integers and 0, this subset would be the whole numbers . If you  looked at the whole numbers and only took the non- zero numbers, this subset would be called the natural numbers . The natural numbers are similar to the number system used by the ancient Sumerians.
  ::真实数字可以分为两个子集: 理性数字和非理性数字。 如果您查看真实数字, 并且只将数字作为分数来写, 这个子集将是理性数字是真实数字的子集, 可以作为一个分数来写。 误差数字是无法作为一个分数来写的真实数字的子集。 您可以在章节中先看到这些子集。 如果您查看理性数字, 并且只将数字除去一个分数, 这个子集将是整数。 如果您查看整数, 并且只采用正整数和正整数 0, 这个子集将是整数。 如果您查看整个数字, 并且只采用非零数字, 这个子集将被称为自然数字。 自然数字与古苏门亚人使用的数字系统相似 。

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  Use the interactive below for a visual explanation of how the set of real numbers can be broken down.
  ::使用下面的交互数据来直观解释 如何细分真实数字组 。

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  CK-12 PLIX Interactive: Number System
  ::CK-12 PLIX 互动:数字系统

   

   

   

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  Proving Rationality
  ::证明合理性

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  Earlier, you learned that a rational number is any number which can be written as a fraction. To prove that a number is rational you can write it as a fraction. This fraction expresses that value as a ratio of p to q or  $\begin{array}{r}\frac{p}{q}\end{array}$  where p and q are both integers. The number  $\begin{array}{r}\frac{1}{3}\end{array}$  is a rational number because it can be written as a ratio of one integer to another. The integer 1 represents p, and the integer 3 represents q.
  ::早些时候,您学会了一个理性数字是可以作为一个分数写入的任何数字。为了证明一个数字是合理的,您可以把它写成一个分数。这个分数表示该数值是p-q或pq之比,而p和q两者都是整数。数字13是一个合理数字,因为它可以写成一个整数与另一个整数之比。整数1代表 p,整数3代表 q。

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  Discussion Questions
  ::讨论问题 讨论问题

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  1. How can we prove that the number 7 is a rational number? Can that be extended to all whole numbers?
     ::我们如何能够证明7号数字是一个合理的数字?这能否扩大到所有数字?
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  2. How can we prove that the number $\begin{array}{r}12\frac{1}{2}\end{array}$  is a rational number? Can that be extended to all mixed numbers?
     ::我们如何能够证明1212数字是一个合理的数字?这能否扩大到所有混合数字?
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  3. How can we prove that the number 0.3 is a rational number? Can that be extended to all decimals values?
     ::我们怎样才能证明0.3数字是一个合理的数字?这能否扩大到所有小数点值?
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  4. How can we prove that 0 is a rational number?
     ::我们如何证明0是一个合理的数字?

    

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  Fractions and Decimals
  ::小数和十进数

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  Now that you know that any decimal can be written as a fraction, you  might assume that any fraction can be written as a decimal, but remember that not all decimals are finite! To write a fraction as a decimal divide the numerator by the denominator .
  ::既然您知道任何小数点可以作为一个分数写入, 您就可以假设任何分数可以以小数点写成, 但记住并非所有小数点都是有限的 。 要将小数点写成小数点, 分子除以分母 。

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  Example
  ::示例示例示例示例

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  Express the fraction  $\begin{array}{r}\frac{4}{9}\end{array}$   as a decimal .
  ::将分数49作为小数表示。

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  To find the decimal value of  $\begin{array}{r}\frac{4}{9}\end{array}$ you need to divide 4 by 9. However, when you do this you may notice a pattern:
  ::要找到49的十进制值,您需要将4除以9。 但是,当您这样做时,您可能会注意到一个模式:

  $$\begin{array}{rl}& \mathrm{0.44...}\\ 9)& \overline{4.00}\\ -& \underset{\_}{36}\\ & 40\\ -& \underset{\_}{36}\\ & 4\end{array}$$

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  The 4 repeats over and over. This type of decimal is called a non- terminating decimal which means a decimal that contains an infinite number of digits. A terminating decimal is a decimal that contains a finite number of digits. Non-terminating decimals can be broken down into two subsets: repeating decimals and non-repeating decimals. A repeating decimal is a non-terminating decimal with a group of digits that endlessly repeat. A non-repeating decimal is a non-terminating decimal without a group of digits that endlessly repeat. The repetition may not always be easily visible, but all rational non-terminating decimals are repeating decimals.
  ::重复 4 重复 4 重复 4 重复 4 重复 4 重复 4 。 重复 4 重复 4 重复 4 。 重复 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

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  Another example of a repeating decimal is $\begin{array}{r}\frac{1}{7}\end{array}$ . When you divide 1 by 7, you get the decimal 0.142857142857… The pattern involves more numbers than the example  $\begin{array}{r}\frac{4}{9}\end{array}$  but still repeats. To write a repeating decimal place a line, called a vinculum, over the numbers that repeat. Since the digits 142857 repeat in the decimal value of the fraction $\begin{array}{r}\frac{1}{7}\end{array}$ . Y ou  can say that $\begin{array}{r}\frac{1}{7}=0.\overline{142857}\end{array}$ . Similarly, you can  say that $\begin{array}{r}\frac{4}{9}=0.\bar{4}\end{array}$ .
  ::重复小数点的另一个例子是17。当将小数点除以1除以7时,您可以得到小数点为0.142857142857...这个模式涉及的数字比例49多,但仍重复。要写一个重复的小数点,在重复的数点上加上一条小数点,叫作文库卢。由于数字142857重复小数点为第17点的小数点值,你可以说17=0.142857。同样,你可以说49=0.4。

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  Use the interactive below to explore non-terminating numbers further.
  ::使用以下互动方式进一步探讨非终止数字。

   

  

  INTERACTIVE

  Representing Rational Numbers with Decimals

  

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  • Move the red point to change the value of the numerator.
    ::移动红色点以更改分子值。
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  • Move the blue point to change the value of the denominator.
    ::移动蓝色点以改变分母值。

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  Discussion Question
  ::讨论问题

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  Do you think that any rational number can be written as a fraction? How can we know what the corresponding fraction is for a repeating number?
  ::您是否认为任何合理数字可以作为一个分数写下来? 我们怎么知道重复数字的相应分数是多少?

  Summary

  播放段落 A non-terminating decimal contains an infinite number of digits. ::不终止的十进制小数含有无限数字。 播放段落 A t erminating decimal is a decimal that contains a finite number of digits.  ::终止小数小数是包含一定数字的小数小数。 播放段落 A repeating decimal is a non-terminating decimal with a group of digits that endlessly repeat in a pattern. ::重复的十进制是一个不终止的十进制小数,其中一组数字在一个模式中无休止地重复。 播放段落 A non-repeating decimal is a non-terminating decimal without a group of digits that endlessly repeat. ::不重复的十进制是非终止的十进制,没有一组无休止重复的数字。