The decimal part of a number begins with a certain bit, and the number in which one or several numbers appear repeatedly in turn is called cyclic decimal. Cyclic decimals have cyclic nodes (cyclic points).

English name: cyclic decimal

When two numbers are divided, if you can't get the integer quotient, there will be two situations: one is to get a finite decimal. One is to get infinite decimals.

The infinitesimal decimal of the previous number or section is called cyclic decimal, such as 2. 1666...* (mixed cyclic decimal), 35.232323 ... (cyclic decimal), 20.333333 ... (cyclic decimal), etc. , in which the numbers that appear repeatedly in turn are called cyclic decimals. The abbreviation of cyclic decimal is to omit all the digits after the first cyclic segment, and add a dot above the first two digits and the last two digits of the first cyclic segment. [1] For example:

2.966666 ... abbreviated as 2. 96(6 has a dot on it; It says "2.96, six cycles")

35.232323 ... abbreviated as 35.23(2 and 3 have a dot respectively; It says "35.23, 23 cycles")

36.568568 ... abbreviated as 36.568 (there is a dot on 5 and 8 respectively; It says "36.568, five to eight cycles")

Cyclic decimals can be converted into fractions by summing geometric series (attached link: geometric series). For example, the chemical method in the picture.

Therefore, in the classification of numbers, cyclic decimals belong to rational numbers.

For example, edit this paragraph

In the problem of circulating decimals, the most famous one is whether 0.999… is equal to 1 [2]. The algebraic method is as follows:

Prove:

Suppose X=0.999. ...

∵

10X = 9.999 ... emergency service number ...

that is

9x = 9

∴

x = 1

The above reasoning process is relatively strict, not the so-called 0.3= 1/3 and 0.9.

Lichangbai 1947 Comments: There is something wrong with this certificate. Because I didn't notice the infinite complexity. In fact, the above proof has two results, one is:

x= 1

That is, the results that have been obtained above. But if you go from

10x=9.99 ...

Let's go, divide both sides by 10 at the same time, and we get.

x=0.999 ....

Only one of these two results should be correct. Obviously, the result of x=0.999 ... is more credible than the result of x= 1. It is not appropriate to infer infinity without careful investigation.

I have proved that 1 is not equal to 0.999. ...

It is very easy to prove 0.9 ... ≠ 1 by using logic.

Please compare the following two formulas:

1 = 1- 1/ 10(n→∞)( 1)

1 = 1- 1/ 10+ 1/ 10(n→∞)(2)

These two formulas are obviously not exactly the same, but there are differences. Therefore, only one should be correct, but not both. If you are a little more careful, you will find that the right side of (1. 1) is one less110 than the right side of (1.2). So the formula (1.2) is definitely correct, but the formula (1. 1) is not valid.

But the right side of (1. 1) is 0.9. ...

And it is considered that110 = 0 will lead to equality of any number.

If you think

110 = 0 (directly inferring 0.9…= 1) (3)

And think it is strictly equal, then because of the "strict equality" can be infinitely recursive, that is, get:

2× 1/ 10=0, (4)

3× 1/ 10=0, (5)

…

Infinitely increase, there will always be a moment to get:

10× 1/ 10=0。 (6)

But the right side of (1.2.4) is equal to 1 instead of 0, which is an obvious fact.

Push it down in the same way, and any two numbers can be equal. This is obviously absurd.

The calculated results also prove this point. But calculus is needed. Therefore, it is omitted. You can check the related articles of Li Changbai's Mathematical Network.

Strictly speaking, the above methods are flawed, and the real methods are as follows:

According to the definition of cyclic decimal:

Such as 1/3,

When divided by three, the remaining digits are 1. If this continues, according to induction, there will be countless 3' s after this decimal, and all of them are 3' s, so 1/3 = 0.3 3 cycles.

Then we look at the period of 0.9 9.

We use 11to calculate, but the difference is that we don't want to divide1at once, we just abdicate.

The first step is to get more than 0.9 0. 1, which is no problem and does not violate any operating rules.

By this calculation, it can be concluded that11divided by 0.9 can be expressed as 9 cycles.

That is, the 0.9 9 cycle is equal to 1.

Certificate of completion

There is no limit (nothing to do with cyclic decimal) and cyclic decimal algorithm!

Only decimal division and cyclic decimal definition are used!

Edit this paragraph.

Special attention is paid to:

The definition of irrational number is infinite acyclic decimal, so it can be judged that infinite acyclic decimal is irrational number (because definition is also judgment).

repeating decimal

The pure cyclic decimal is rewritten as a component quantity, and the molecule is a number composed of numbers in a cyclic segment; All the numbers in the denominator are 9, and the number of 9 is the same as that in the loop segment.

For example:

0. 1= 1/9 0. 1234= 1234/9999

Mixed cycle: the mixed cycle decimal is rewritten as a component number, and the numerator is the number connected by the first cycle node, MINUS the difference of the number composed of the number of the acyclic part; The first few digits of the denominator are 9 and the last few digits are 0. The number of 9 is the same as the number of cyclic parts, and the number of 0 is the same as the number of acyclic parts.

For example: 0.1234 = (1234-1)/9990 0.558898 = (558898-55)/99900.

This concept is wrong.

The number of decimal places in a finite decimal is limited.

The number of decimal places in a circular decimal is infinite.

Therefore, the statement of finite cyclic decimal is wrong in itself. I hope that the authorized editor can change the definition of this entry.

Uncertainty; uncertain

Infinite cyclic decimal is a form of infinity, and there is no definite value, that is to say, it is impossible to get the exact value of cyclic decimal, so it is impossible to perform four operations on infinity. Take 0.333333 ... for example, it can be written as a set of 0.3+0.03+0.0003+ ... +3* 10 (n is an infinite natural number), or 0.3+0.003+0.0003+ ... or 0.3+0.03+0. These expressions can describe the infinite cycle of decimal 0.333333 ... When n is a fixed value, it is obvious that the values described by these three expressions are different, but when n is infinite, therefore, as an infinite quantity, the value of the infinite cycle decimal is uncertain, plus 0.333333 ...+0.33333 ... = 0.66666 .. It cannot be applied. Infer 0.000...3+0.000...3=0.000...6, which seems to prove that whenever 0.333333 ...+0.333333 ... = 0.666666 ..., but don't forget that 0.33333 ... is an infinite quantity. You must prove that the addition before 0.000...3+0.000 ... 3 equals 0.000...6 can be established, but it is impossible to prove every link, because there are infinite links to prove. If you stop proving deduction at any time, the conclusion will be invalid. Even if you can continue to deduce, you will never reach the end of derivation and proof. This proof can only be infinitely close and can never be completed.

Although infinity has no fixed value, some infinity has a limit value. For example, the limit value of infinite loop is decimal 0.99999 ... equal to 1, and the mathematical expression is: when x tends to 1, limX= 1. By using the concept of limit, infinite quantity can be transformed into a fixed value, and the purpose of participating in operation can be realized. Infinity itself cannot perform four operations, and only the limit value of infinity can apply the arithmetic of addition, subtraction, multiplication and division.

Not all infinite quantities have limit values. If the infinite cycle decimal number is 0.33434, there is no limit value. ...

relative dimension

In mathematics, the proof of the relationship between two numbers is generally expressed by subtraction and by the directed distance between two points on the coordinate axis in the geometric sense. For the convenience of understanding, the discussion range is set in the real number range, using a and b, when a-b >; When there is a & gtb；; When a-b

Maybe you can easily use this proof: 0.999...-0.666 ...= 0.333 ... >; 0, so 0.999...& gt0.666...； ...; Pupils can easily understand "0.343434 ...-0.1212 = 0.222222 ... >; 0, so 0.343434 ...>0.121212 ... "; But this understanding is incorrect. Mathematics is a rigorous subject. Only a correct theoretical basis can lead to a correct result, but a correct result does not necessarily lead to a correct theoretical basis. This is like knowing that coal can burn, but you don't necessarily know why coal can burn, under what conditions coal can burn and under what conditions it can't burn; It's like knowing that your ears can hear sound, but you don't necessarily know why your ears can hear sound, under what conditions you can hear sound and under what conditions you can't hear sound.

Addition and subtraction prove that the size of two numbers only applies to finite numbers with definite values, and the comparison between infinite numbers and finite numbers is logical. For example, we really think that 0.999 ... is greater than 0.666 ... and this phenomenon can no longer be explained by the theory of four operations in elementary mathematics, so it is necessary to introduce the concept of boundary interval. Suppose there are two points on the coordinate axis, and an interval is formed between these two points. When the distance between these two points is infinitely close, the interval is infinitely reduced, the values of these two points are getting closer and closer, and the boundary values represented by these two points tend to be the same value. Whether this value is finite or infinite, its value belongs to this interval. Because the number on the coordinate axis is increasing in one direction, generally speaking, the number on the left side of the X axis is smaller than the number on the right side, so there can be a finite number A and an infinite number B, and B belongs to the interval of (c, d). When c and d are finite values, if a-c >；; 0，a-d & gt； 0, the finite amount A is on the right side of the interval (c, d), and A is greater than any value in the interval (c, d), so the finite amount A is greater than the infinite amount B; If a-c

The comparison between infinity and infinity can also follow this method. There are infinity A and infinity B, where A belongs to the interval (c, d) and B belongs to the interval (e, f). When c-f >; 0, all values in the interval (c, d) are greater than all values in the interval (e, f), so the infinite quantity A is greater than the infinite quantity B. ..

I don't need to prove the limit and convergence interval in detail. Those are the minimum ranges that describe the existence of the value of infinity, so that infinity can get the approximate finite value as accurately as possible with a certain precision, and infinity can indirectly participate in the operation of addition, subtraction, multiplication and division. The concept of accuracy is 10 order of magnitude to the n power. The common expression is how many digits are reserved after the decimal point, for example, two digits are reserved after the decimal point, indicating that the accuracy is 0.0 1, that is, the negative power of 10. What we usually call rounding is to define an infinite interval. For example, if you divide 1 by 3 with a calculator, you get 0.333 ... in fact, you can't get an infinite cycle decimal, because it is impossible to calculate the value of an infinite cycle decimal. Calculators usually keep 9 decimal places, so they will not continue to display, and then they usually choose 2 decimal places, which is 0.33. If the last 0.0033 ... is omitted, it means that the circular decimal 0.333 ... is accurate to the interval (0.33, 0.34), because this interval is not very accurate, so when you compare the finite number 0.33 with this interval, you will find that 0.333 ... is rounded to two digits to get 0.33, 0.33-0.33=0. At this time, it is necessary to increase the reserved decimal places, that is, to improve the accuracy, and make it 0.333 ... accurate to (0.333, 0.334), so it is 0.33-0.333.

Cyclic decimal is an infinite quantity, and its value is uncertain. We can't directly apply the four operations to infinite quantities such as cyclic decimal or infinite acyclic decimal. Instead, we should take the limit value or boundary interval value of infinity as a finite value under the premise of a certain accuracy order, and then use these approximations to calculate. LimX= 1, which doesn't mean that X= 1, lim(0.999...)= 1, can't be directly equivalent to cyclic decimals 0.999 ... and 1. The correct description is that the limit value of cyclic decimal is 0.999 .. equal to 1. Suppose a=0.999 ..., then 10a=9.999 ...,10a-a = 9.999 ... = 9a = 9, so a= 1, so the calculation method of counterpoint subtraction is wrong. In fact, it is impossible for the most advanced supercomputer in modern times to calculate the exact value of 1 divided by 3 = 0.3333333 ... even if it can be accurate to one trillion decimal places, the calculation is far from over. If you don't artificially set an order of magnitude to make the computer accurate to how many decimal places, the computer will fall into an infinite loop state because of calculating an infinite quantity, leading to a crash.