10.12 将重复的十进制转换成小数-interactive

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  将重复的十进制转换成小数

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  Egypt
  ::埃及

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  date back to 3,100 BCE in ancient Egypt. Egyptians were the first to develop a number system that included non- whole numbers . The Egyptian number system used fractions to represent values between whole numbers. The symbol that they used to signify a fraction was the following:
  ::埃及人首先开发了一个包含非完整数字的数系统。 埃及数字系统使用分数表示整个数字之间的数值。 他们用来表示一个分数的符号如下:

  

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  This symbol was placed above a number to represent the reciprocal of that number. For example, if it were placed above the number 8, it would represent  $\begin{align*}\frac{1}{8}.\end{align*}$   Egyptians did not need a symbol for $\begin{align*}\frac{2}{8} \end{align*}$  because they could use the symbol for  $\begin{align*}\frac{1}{4}.\end{align*}$   Nor did they need a symbol for  $\begin{align*}\frac{3}{8}\end{align*}$   because $\begin{align*}\frac{3}{8} = \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\end{align*}$  .
  ::这个符号被置于数字之上,以表示数字的对等性。例如,如果它位于数字8之上,它代表18个。 埃及人不需要28个符号,因为他们可以使用14个符号,也不需要38个符号,因为38=18+18+18。

  

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  By exclusively using fractions, ancient Egyptians avoided the problem of non-terminating decimals. In this section, you will continue your investigation of non-terminating decimals. Egyptians invented fractions because they allowed them to represent any fractional quantity while preserving accuracy. In our modern number system, our calculations are only as accurate as the place value to which we round . T ake a look at how you can write a repeating decimal as a fraction to avoid the need to round.
  ::古埃及人完全使用分数来避免不终止小数的问题。 在本节中, 您将继续调查非终止小数。 埃及人发明分数, 因为他们允许他们代表任何分数数量, 同时保存准确性 。 在我们现代的数字系统中, 我们的计算方法只能和我们圆在一起的位置值一样准确 。 请看看您如何将重复的小数数作为分数写出来, 以避免圆圈的必要性 。

   

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  What Does it Mean?
  ::这意味着什么?

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  If you have eaten a third of a pizza, it means that you ate  $\begin{align*}0.\overline{3}\end{align*}$  pizzas. How is it possible to eat a never-ending decimal of pizza? Does that mean that you ate an infinite amount of pizza? Unfortunately, you only ate a finite amount of pizza, but let’s take a closer look at what it means when a decimal repeats. The decimal  $\begin{align*}0.\overline{3}\end{align*}$   means  $\begin{align*}0.33333…\end{align*}$  or $\begin{align*}\frac{3}{10} + \frac{3}{100} + \frac{3}{1,000} + … \end{align*}$  The amount is increasing forever, but each place value is one-ten th as small as the last. The reason we have difficulty writing  $\begin{align*}\frac {1}{3}\end{align*}$  is that  our base ten system of numbers makes it difficult to divide 10 into thirds. This value can be more easily expressed as a fraction.
  ::如果你吃了三分之一的比萨饼,这意味着你吃了0.3个比萨饼。怎么可能吃不完的比萨小数点?这是否意味着你吃了无限量的比萨饼?不幸的是,你只吃了有限量的比萨饼,但让我们仔细看看小数点的重复意味着什么。小数点0。小数点3意味着0.333333...或310+3100+31,000+...这个数量一直在持续增加,但每个地方的价值都比最后的少十分之一。我们难以写13个是因为我们的10个基本数字系统很难将10个数字分成3/3。这个数值可以更方便地表达成一个分数。

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  Use the interactive below to further explore this idea by examining where repeating decimals fall on a number line .
  ::利用以下互动方式进一步探讨这一想法,审查重复小数点在数字线上的位置。

   

   

   

   

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  Discussion  Question
  ::讨论问题

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  Mathematically, $\begin{align*}0.\overline{9}=1.0.\end{align*}$  How could you prove this?  Join the discussion in the !
  ::从数学角度讲,你怎么能证明这一点?

   

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  Repeating Decimals
  ::重复十进数

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  To write a repeating decimal as a fraction, you will need to use an equation.
  ::要将重复的十进制写为分数, 您需要使用方程式 。

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  1. Begin by writing $\begin{align*}x\end{align*}$  = the repeating number.
     ::开始写入 x = 重复编号 。
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  2. Multiply both sides of the equation by a power of 10 which will move the decimal to the right of the repeating number.
     ::乘以方程的两边乘以10的功率,将小数点移到重复数字右侧。
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  3. Subtract the equation from  step 1 from the equation in  step 2.
     ::将方程从第1步从第2步的方程中减去第1步的方程。
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  4. Solve the resulting equation.
     ::解决由此产生的等式。

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  Example
  ::示例示例示例示例

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  Write  $\begin{align*}0.\overline{7}\end{align*}$  as a fraction.
  ::写0。 7作为一个分数。

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  1. Begin by writing $\begin{align*}x\end{align*}$  = the repeating number.
  ::1. 开始为写入 x = 重复编号。

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    $\begin{align*}x = 0.\overline{7}\end{align*}$
  ::x=0. 7

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  2. Multiply both sides of the equation by a power of 10 which will move the decimal to the right of the repeating number.
  ::2. 将方程的两边乘以10的功率,将小数点向重复数字右移至小数点。

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  The repeating number is seven. To move the decimal to the right of the 7, you need to multiply by 10. This gives you the following: $\begin{align*}10x = 7.\overline{7}\end{align*}$ .
  ::重复编号为 7 。 要将小数点移到 7 点右侧, 您需要乘以 10 。 这给了您以下的 : 10x= 7 。

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  3. Subtract the equation from  step  1 from the equation in  step 2.
  ::3. 将第1步的方程从第2步的方程中减去第1步的方程。

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  $\begin{align*}10&x = 7.\overline{7}\\ - \:\:\:\:&\underline{x = 0.\overline{7}}\\ 9&x = 7\end{align*}$
  ::10x7 7x0 7x9x7

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  4. Solve the resulting equation. 
  ::4. 解决由此产生的等式。

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    $\begin{align*}9x &= 7\\ x &= \frac{7}{9}\end{align*}$
  ::9x=7x=79

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  Use the interactive below to explore this further.
  ::利用以下互动方式进一步探讨这一问题。

   

   

   

   

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  Discussion Question
  ::讨论问题

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  $\begin{align*}0.\overline{7}\end{align*}$  can be written as $\begin{align*}\frac{7}{9}.\end{align*}$   $\begin{align*}0.\overline{56}\end{align*}$  can be written as $\begin{align*}\frac{56}{99}.\end{align*}$   $\begin{align*}0.\overline{115}\end{align*}$  can be written as $\begin{align*}\frac{115}{999}.\end{align*}$  Do you think you notice a pattern? Will it work for all repeating numbers?
  ::0 7 可以写为 79. 0. 56 可以写为 5699. 0. 115 可以写为 1159999.

   

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  Mixed Repeating Decimals
  ::混合重复十进数

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  A mixed repeating decimal is a decimal that does not repeat until after the tenths place. The value  $\begin{align*}\tfrac{7}{12} = 0.58\overline{3}\end{align*}$  is an example of this. To write a mixed repeating decimal, you will use the same steps as before.
  ::混合重复的十进制是十进制后才会重复的十进制。值 712=0.58 3 就是一个例子。要写混合重复的十进制,您将使用与之前相同的步骤。

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  1. Begin by writing $\begin{align*}x\end{align*}$  = the repeating number.
     ::开始写入 x = 重复编号 。
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  2. Multiply both sides of the equation by a power of 10 which will move the decimal to the right of the repeating number.
     ::乘以方程的两边乘以10的功率,将小数点移到重复数字右侧。
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  3. Subtract the equation from  step  1 from the equation in  step 2.
     ::将方程从第1步从第2步的方程中减去第1步的方程。
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  4. Solve the resulting equation.
     ::解决由此产生的等式。

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  Example
  ::示例示例示例示例

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  Write  $\begin{align*}0.8\overline{6}\end{align*}$  as a fraction.
  ::写0.8分之六

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  1. Begin by writing $\begin{align*}x\end{align*}$  = the repeating number.
  ::1. 开始为写入 x = 重复编号。

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    $\begin{align*}x = 0.8\overline{6}\end{align*}$
  ::x=0.8 6

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  2. Multiply both sides of the equation by a power of 10 which will move the decimal to the right of the repeating number.
  ::2. 将方程的两边乘以10的功率,将小数点向重复数字右移至小数点。

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  The repeating number is six. To move the decimal to the right of the 6, you need to multiply by 100 which gives you the following:
  ::重复编号为 6。 要将小数点移到 6 点右侧, 您需要乘以 100 , 这样您就可以获得以下条件 :

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    $\begin{align*}100x = 86.\overline{6}\end{align*}$
  ::100=86 6个

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  3. Subtract the equation from step 1 from the equation in step 2.
  ::3. 将第1步的方程从第2步的方程中减去第1步的方程。

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  $\begin{align*}100&x = 86.\overline{6}\\ - \:\:\:\:\:&\underline{x = 0.8\overline{6}}\\ 99&x = 85.8\end{align*}$
  ::100x86 6x0.8 6x99x85.8

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  4. Solve the resulting equation.
  ::4. 解决由此产生的等式。

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  $\begin{align*}99x &= 85.8 \\ x &= \frac{85.8}{99}\end{align*}$
  ::99x=85.8x=85.899

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  Since your answer has a decimal in the fraction, you must multiply the numerator and denominator by a power of ten which will produce an equivalent fraction with no decimals. Multiplying the numerator and denominator by 10 gives you  your answer:
  ::由于您的回答在分数中有一个小数点,您必须把分子和分母乘以10的功率,从而产生一个相等的分数,而没有小数点。乘以分子和分母乘以10,您就可以得到答案:

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  $\begin{align*}x = \frac{858}{990}\end{align*}$
  ::x=858990

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  We can reduce this fraction to  $\begin{align*}\frac{429}{495} = \frac{143}{165} = \frac{13}{15}\end{align*}$ .
  ::我们可以将这一分数减少到429495=143165=1315。

   

   

   

   

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  Discussion Question
  ::讨论问题

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  The value π represents the ratio between circumference and diameter. This ratio is a ratio between two numbers, but π doesn’t repeat. Why do you think that is?
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  Summary

  播放段落 To write a repeating decimal as a fraction: 播放段落 Start by creating an equation where  x= "the repeating number". ::以创建 X = “ 重复数字” 的方程式开始 。 播放段落 Multiply both sides of the equation by a power of 10 which will move the decimal to the right of the repeating number. ::乘以方程的两边乘以10的功率,将小数点移到重复数字右侧。 播放段落 Subtract the equation from the first step from the equation in the second step. ::将方程从第一步从第二步的方程中减去。 播放段落 Solve the resulting equation. ::解决由此产生的等式。 ::将重复的十进制成一个分数 : 以创建 X = “ 重复数字” 的方程式开始 。 将方程式的两边乘以 10 的功率乘以 10 , 该功率将把十进制小数乘以重复数字的右侧 。 将方程式从第一个步骤从第二步的方程式中减去。 解决由此产生的方程式 。