brief introduction
0.999 ... is a number in the decimal system, and some simplest proofs of 0.999 ... = 1 depend on the convenient arithmetic properties of this system. Most decimal arithmetic-addition, subtraction, multiplication, division and size comparison-is similar to integers. Like integers, any two finite decimals must have different values as long as they have different numbers. Especially any number with the shape of 0.99 ... 4, only a few of which are strictly less than 1.
Misunderstanding the meaning of "..." (ellipsis) in 0.999 ... is one of the reasons for misunderstanding 0.999 ... = 1 ... The usage of ellipsis here is different from everyday language and the ellipsis in 0.99 ... 9 means that a limited part is omitted. However, when used to represent cyclic decimals, "…" means omitting the infinite part, which can only be explained by the mathematical concept of limit. In this way, the real number represented by "0.999 ..." is the limit of the convergent sequence (0.9, 0.99, 0.999, 0.9999, ...). "0.999 ..." is the limit of a sequence. In this respect, the equation of 0.999 ... = 1 is very intuitive.
Unlike integers and finite decimals, a number can be expressed in many other ways. For example, if you use fractions, 1? 3=2? 6。 However, a number can only be expressed in two infinite decimal ways at most. If there are two methods, one of them must contain an infinite number of 9s, and the other must start with all zeros.
0.999...= 1 There are many proofs, and the rigor of each proof is different. A strict proof can be simply stated as follows. Consider that two real numbers are equal if and only if their difference is equal to zero. Most people think that the difference between 0.999 ... and 0, even if it exists, is very small (approaching zero). Considering the above convergent sequences, we can prove that this difference must be less than any positive number, and we can also prove (see Archimedes principle for details) that the only real number with this property is zero. Since the difference is zero, we can see that 1 and 0.999 ... are equal. Similarly, why is it 0.333...= 1? 3,0. 1 1 1...= 1? Nine, wait.
certificate
imagine
0.999 ... is it 1? If subtraction is used for direct calculation (only five digits are listed after the decimal point and omitted after five digits):
1.00000
— 0.99999
——————
0.00000
The result is 0.000 ... which is a finite cycle of 0.0. Because you will always fill in 0 after five decimal places, you can't find the last digit to fill in 1. 1.(0)-0.(9)=0.(0), so 1=0. (9).
mark
Infinite decimal is a necessary extension of finite decimal, and one of the reasons is that it is used to represent fractions. With long division, a word like 1? The simple integer division of 3 becomes a cyclic decimal, 0.333 ..., in which there are infinitely many numbers 3. Using this decimal, we can quickly get a proof of 0.999 ... = 1.0.333, each 3 times 3 ... you get 9, so 3×0.333 ... equals 0.999 ... and 3× 1? 3 equals 1, so it is 0.999...= 1.
Another form of this proof is multiplied by 1/9=0. 10 1 ... before eight o'clock. mathematics
decimal
The early form was based on the following equation: Mathematics
Because both equations are correct, 0.999 ... must be equal to 1 according to the transitivity of the equation relationship. Similarly, 2/2= 1, 2/2=0.999 ... So, 0.999 ... must be equal to 2.
Bit operation
Another proof is more suitable for other cyclic decimals. When a decimal number is multiplied by 10, its number remains the same, but the decimal point moves one place to the right. Therefore, 10×0.999 ... equals 9.999 ... 9 more than the original figure.
Consider subtracting 0.999...9.999 cases. We can reduce them one by one; Every digit after the decimal point, the result is 9-9, which is 0. Both numbers after the decimal point are 0.999 ... so they can cancel each other out, and the result is zero after the decimal point. The last step is algebra. Let 0.999...=c, then 10c? C=9, which means 9c=9. Dividing the two ends of the equation by 9 can be proved: d= 1. Represented by a series of equations, it is mathematics.
The correctness of the digital operation in the above two proofs need not be blindly believed or regarded as axioms; It comes from the basic relationship between decimal system and the number it represents. This relationship can be expressed in several equivalent ways. It has been stipulated that 0.999 ... and 1.000 ... both represent the same number.
Real number analysis
Because the problem of 0.999 ... does not affect the formal development of mathematics, we can suspend our research until the standard theorem of real number analysis is proved. One of the requirements is to describe all real numbers that can be expressed as decimals. These real numbers consist of an optional symbol, a finite number of digits that make up the integer part, a decimal point and a series of digits that make up the decimal part. In order to discuss 0.999 ..., we can generalize the integer part as b0 and ignore the negative sign, so that the decimal expansion has the following form: Mathematics.
The fractional part is different from the integer part. The integer part can only have a finite number of digits, and the decimal part can have an infinite number of digits. This is crucial. This is a carry system, so 4 in 400 is 10 times of 50, and 5 in 0.05 is one tenth of 0.5.
Infinite series and sequence