The method of dividing the cycle into fractions can be derived by moving the cycle segment, or can be calculated by the summation formula of infinite recursive proportional sequence. Below we use the method of conjecture verification to deduce.
(A) the pure circular decimal into a fraction
As we all know, a finite fraction can be divided into fractions whose mother is 10, 100, 1000 ... So, what kind of denominator can a pure cyclic decimal be reduced to? Let's start with a simple one-digit recurring decimal. For example, @ ①, @ ② ... When they are divided into component numbers, what can their denominators be written as?
Think about it: Could it be 10? No Because110 = 0.1< @ ①, 3/10 = 0.3 > @ ②; Could it be 8? No Because 1/8 = 0. 125 > @ ①, 3/8 = 0.375 > @ ②; So, what could it be? Because110 < @ ① <1/8,3/10 < @ ② < 3/8, the mother may be 9. Let's test our conjecture:1/9 =1÷ 9 = 0.11... = @ ①; 3/9= 1/3= 1÷3=0.333……= @②。
The calculation results show that our guess is correct. Then, can a pure cyclic decimal with all cyclic segments being one digit be written as a fraction with denominator of 9? Let's make @ ③ and @ ④ into components according to our own guess, and then verify them.
@ ③ = 4/9 Verification: 4/9 = 4÷ 9 = 0.444 ...
@ ④ = 6/9 = 2/3 Verification: 2/3 = 2 ÷ 3 = 0.666 ...
After the above conjecture and verification, we can draw the following conclusions: when a cyclic segment is the pure cyclic fractional component number of a number, the number composed of cyclic segments is the numerator and 9 is the denominator; Then, if you can cut it, cut it again.
How to turn a pure cyclic decimal with a cyclic segment of two digits into a number? For example, @ ⑤, @⑤ ... How much can their denominator be written?
Think about it: Could it be 100? No Because12/100 = 0.12 < @ ⑤,13/100 = 0.13 < @ ⑤. Could it be 98? No Because 12/98 ≈ 0. 1224 > @ ⑤,13/98 ≈ 0.1327 > @⑤; How much could it be? Because12/100 ÷ @ ⑤12/98,13/100 ÷ @ ⑤13/98. Whether it is correct or not remains to be verified.
12/99= 12÷99=0. 12 12 12……=@⑤;
13/99= 13÷99=0. 13 13 13……=@⑥。
The verification results show that our guess is correct. Then, can all the pure cyclic decimals with two digits in the cyclic segment be written as fractions with denominator of 99? Let's use the method of conjecture to divide @ ⑦ and @ ⑧ into components and check them.
@ ⑦ = 15/99 = 5/33, checking calculation: 5/33 = 5 ÷ 33 = 0.1515. ...
@ ⑧ =18/99 = 2/1,checking calculation: 2/11= 2 ÷1= 0 ..
After this conjecture and verification, we can draw the following conclusions: when the cyclic segment is a pure cyclic fractional component number with two digits, the number composed of a cyclic chain is used as the numerator and 99 as the denominator; Then, if you can cut it, cut it again.
Can we infer that the cyclic segment is a method of three-digit pure cyclic decimal?
Because when the cycle segment is a single-digit pure cycle fractional part, the denominator is 9, and when the cycle segment is a two-digit pure cycle fractional part, the denominator is 99, so when the cycle segment is a three-digit pure cycle fractional part, we guess that the denominator is 999, and the numerator is also a number composed of cycle segments. Let's check again. If this conjecture is correct, then we can push it backwards.
Attached drawings (drawings)
The experiment proved that our guess was completely correct. Accordingly, when the cyclic segment is a pure cyclic decimal component with four digits, 9999 will be used as the denominator. Practice has proved that it is also correct. Therefore, the method of decimal places of components in a pure cycle is:
Use numbers, such as 9, 99, 999 ..... as the denominator, and the number of 9 is the same as the number of digits in the loop segment; Use a number compose of ring nodes as a molecule; Provide the number of points that can be reduced finally.
Second, turn mixed cycle decimals into fractions.
We studied how to turn a pure cyclic decimal into a fraction by the method of conjecture verification, and then studied how to turn a mixed cyclic decimal into a fraction by this method.
Or start with a simple number, such as:
Attached drawings (drawings)
..... What are the characteristics of numerator and denominator respectively when there is only one mixed cyclic decimal component in this cyclic segment?
Think of it this way: mixed cyclic decimals have cyclic parts and acyclic parts. Can it be rewritten as the sum of a pure cyclic decimal and a finite decimal, and then converted into a component number? Let's have a try.
Attached drawings (drawings)
Observing the above process, can you see the characteristics of mixed cycle decimal with only one digit in the cycle segment? It is easy to see that their denominators are all numbers consisting of a 9 and several 0s. Careful observation shows that the number of zeros is exactly the same as the number of acyclic parts. What are the characteristics of its molecules? It is not difficult to see that their molecules are smaller than those formed by the acyclic part and the first link point. How much smaller is it? Let's calculate:
( 1)2 1- 19=2 (2)543-489=54 (3)696-627=69
After careful observation, it is not difficult to see that the molecule is only a number composed of acyclic parts, which is smaller than the number composed of acyclic parts and the first link point. Is this law universal? Let's use the above rules.
Attached drawings (drawings)
Component number, and verify its correctness.
Attached drawings (drawings)
Verification: 352/1125 = 352 ÷1125 = 0.312888. ...
The result of verification is completely correct. Then, the cyclic part is a fraction formed by a mixed cyclic decimal of two digits. Do numerator and denominator have the same law? Molecules are numbers composed of one acyclic part less than the number composed of the acyclic part and the first cyclic part of decimals; The denominator is a number consisting of 9 and 0. The number of 0 is the same as the number of acyclic parts, and the number of 9 is the same as the number of cyclic parts. Let's guess.
Attached drawings (drawings)
Divide it into several parts and then verify it.
Attached drawings (drawings)
Practice has proved that our guess is correct. Then, you can mix cyclic decimals, whether the cyclic segment is three digits or four digits ..., and so on? Let's put
Attached drawings (drawings)
After the part number, verify it again.
Attached drawings (drawings)
The result of verification is also correct, indicating that our guess may be correct. This method is indeed correct. Of course, when we use the method of guessing and verifying, we may not always guess correctly. If it is not correct, it needs to be modified according to the specific situation, and then verified until it is correct.
The method of conjecture verification is an important way for human beings to explore the unknown. Many scientific laws are discovered by guessing, and then constantly verified, guessed and verified. Conjecture verification is also an important mathematical thinking method. We should not only explain specific knowledge to students, but also let them learn to use this way of thinking from an early age.
Notes about words not stored in fonts:
@ ① Original word 0. 1, plus 1.
@ ② Original word 0.3, plus 3.
@ ③ The original word is 0.4, plus 4.
@ ④ The original word is 0.6, plus 6.
@ ⑤ Original word 0. 12, add 12.
@ ⑥ Original word 0. 13, plus 13.
@ ⑦ Original word 0. 15, plus 15.
@ 8 Original word 0. 18, add 18.