将 5.5858585858... 表示为有理数

有理数的形式是 p/q，其中 p 和 q 是整数，q ≠ 0。由于数字的基本结构，p/q 形式，大多数人发现很难区分分数和有理数。当一个有理数被除法时，输出是十进制形式，可以是结束也可以是重复的。

3、-3、4、-4、5 等是有理数的一些示例，因为它们可以用分数形式表示为 3/1、4/1 和 5/1。

A rational number is a sort of real number that has the form p/q where q≠0. When a rational number is split, the result is a decimal number, which can be either a terminating or a recurring decimal.

十进制数到有理数的转换

第1步：获得重复小数并把它等于x

第 2 步：通过从重复数字顶部删除条并列出重复数字至少两次，以十进制形式写出数字。对于示例，写 x = $0.\bar9$因为 x = 0.999…。和 x = $0.\overline{15}$当 x = 0.151515……

第 3 步：确定有条的位数。

第四步：如果重复小数有1位重复，则乘以10，如果有两位重复，则乘以100，三位重复则乘以1000，依此类推。

步骤 5：将步骤 4 中获得的数字减去步骤 2 中获得的数字。

第 6 步：将等式两边除以 x 系数。

第 7 步：在最后以最简单的形式写出有理数。

将 5.5858585858... 表示为有理数

解决方案：

Given: 5.5858585858 or $5.\overline{585}$

lets assume x = 5.5858585858… ⇢ (1)

And there are two digits after decimal which are repeating,

So, multiply equation (1) both sides by 100,

So $100 x = 558.\overline{5858}$        ⇢ (2)

Now subtract equation (1) from equation (2)

$100x - x = 558.\overline{5858} - 5.\overline{5858}$

         99x = 553

             x = 553/99

                = 553/99

5.5858585858 can be expressed 553/99 as rational number

类似问题

问题1：将小数改写为有理数。 0.666666666……？

解决方案：

Given: 0.66666..  or $0.\bar{6}$

Let’s assume x = 0.66666… ⇢ (1)

And there are one digits after decimal which are repeating, so we will multiply equation 1 both sides by 10.

So 10x = $6.\bar{6}$                   ⇢ (2)

Now subtract equation (1) from equation (2)

$10x - x = 6.\bar{6} - 0.\bar{6}$

       9x = 6

         x = 6/9     

           = 2/3                 

0.666666…  can be expressed 2/3 as rational number

问题2：将小数改写为有理数。 0.69696969……？

解决方案：

Given: 0.696969.. or $0.\overline{69}$

Let’s assume x = 0.696969… ⇢ (1)

And there are two digits after decimal which are repeating, so multiply equation (1) both sides by 100.

So $100x = 69.\overline{69}$   ⇢ (2)

Now subtract equation (1) from equation (2)

$100x - x = 69.\overline{69} - 0.\overline{69}$

       99x = 69

           x = 69/99

             = 23/33                    

0.69696969… can be expressed 23/33 as rational number

问题 3：将 1.3737237... 表示为一个有理数，形式为 p/q，其中 p 和 q 没有公因数。

解决方案：

Given : 1.373737… or $1.\overline{37}$

lets assume x = 1.373737….   eq. 1

And there are two digits after decimal which are repeating

so we will multiply equation 1 both sides by 100

so $100 x = 137.\overline{37}$                           eq. 2

now subtract equation 1 from equation 2

$100x - x = 137.\overline{37}- 1.\overline{37}$

       99x = 136

           x = 136/99                    

1.373737…. can be expressed 126/99 in form of p/q as rational number 

问题 4：将 10.827827827... 表示为一个有理数，形式为 p/q，其中 p 和 q 没有公因数。

解决方案：

Given: 10.827827827… or $10.\overline{827}$

Let’s assume x = 10.827827827…        ⇢ 1

And there are three digits after decimal which are repeating

So multiply equation 1 both sides by 1000

So $1000 x = 10827.\overline{827}$           ⇢ (2)

Now subtract equation (1) from equation (2)

$1000x - x = 10827.\overline{827}- 10.\overline{827}$

       999x = 10817

             x = 10817/999

10.927927927 can be expressed 10817/999 in form of p/q as rational number

问题5：将小数改写为有理数。 0.79797979……？

解决方案：

Given: 0.797979.. or $0.\overline{79}$

Let’s assume x = 0.797979… ⇢ (1)

And there are two digits after decimal which are repeating, so multiply equation (1) both sides by 100.

So $100x = 79.\overline{79}$   ⇢ (2)

Now subtract equation (1) from equation (2)

$100x - x = 79.\overline{79} - 0.\overline{79}$

       99x = 79

           x = 79/99

              = 79/33                    

0.79797979… can be expressed 79/33 as rational number

问题6：将小数改写为有理数。 0.555555……？

解决方案：

Given: 0.555555..  or $0.\bar5$

Let’s assume x = 0.555555… ⇢ (1)

And there are one digits after decimal which are repeating, so we will multiply equation 1 both sides by 10.

So $10x = 5.\bar5$                   ⇢ (2)

Now subtract equation (1) from equation (2)

$10x - x = 5.\bar{5} - 0.\bar{5}$

       9x = 5

         x = 5/9    

            = 5/9               

0.555555…  can be expressed 5/9 as rational number

问题 7：将 6.684684684... 表示为一个有理数，形式为 p/q，其中 p 和 q 没有公因数。

解决方案：

Given: 6.684684684 or 6.684bar

Step 1: Write the number in decimal form by removing bar from the top of repeating digits and listing repeating digits at least twice

Lets assume x = 6.684684684… ⇢ (1)

Step 2: There are three digits after decimal which are repeating, So, multiply equation (1) both sides by 1000,

So 1000 x = 6684.684684       ⇢ (2)

Now subtract equation (1) from equation (2)

1000x – x   =  6684. 684684 – 6.684684684

         999x = 6678

Divide both sides of the equation by the x coefficient.

         999x/999 = 6678/999

                      x = 6678/999

                         = 2226/ 333

                         = 742/111

6.684684684 can be expressed 742/111 as rational number