Problem description:
1/3 = 0.33333 ...(3 cycles)
2/3 = 0.66666 ...(6 cycles)
And 1/3+2/3= 1.
It is the equivalent amount of 0.3333 ... +0.6666 ...= 0.9999 ...(9 cycles).
It will never be equal to 1. Why?
Analysis:
This is a very famous question. Some people will say inequality. But please believe that I am indeed equal to those comrades who say they are equal.
There are many ways to prove this:
The first and simplest:
Let x = 0.9999999999999 ..., then 10x = 9. * * * * * * * * * ..., we get
10x-x=9
X= 1。
Second, it is very simple:
Let x = 0.999999999999 ..., then x/3 = 0.333333333 ... =1/3.
x/3= 1/3
x= 1
Third, a little thinking:
You divide 1 vertically by 1 (vertically, I learned it in primary school), but the difference is that you don't directly quotient 1 at first, but want to quotient 0, so the remainder is 1, add a 0 to become 10, and then quotient 9,/kloc.
Fourth, you can use the limit to do:
The summation formula of proportional sequence is [a1(1-q n)]/(1-q), then when q; At infinity, the limit of this formula is a 1/( 1-q). Self-circulating decimal 0. Aaaaaaaa ... = a/10+a/100+a/1000+a/10000 ..., and each addend of it just constitutes an infinite geometric series, q = 65438. Then you can use a 1/( 1-q) to calculate 0. * * * * * * * ... At this time, a 1=0.9, q =110, you can easily get 0. ..
The above are the common methods to prove 0. * * * * * * * * ... =1.There are many methods. The final result is: 0. * * * * * * * * * * ...= 1.
In addition, I can clearly tell you that the above reasoning process is more rigorous. Don't believe the so-called 0. * * * * * * * * * ... is only equal to1/3,0. * * * * * * * * * * ...
You can also check relevant information on Baidu, especially the history of dynasties.
Finally, I tell you clearly, tell everyone who has read these words, 0.9999...= 1.
Because 0.999999 ... = 3 * 0.33333 ... = 3 *1/3 =1
Use order: (The teacher gave this example when learning order)
0.99999999999999………=0.9+0.09+0.009+0.0009+0.00009+0.000009………
The first term is 0.9, and the common ratio is 0. 1, which is the sum of n terms of geometric series.
The formula s = a1(1-q) for the sum of infinite geometric series terms can be obtained.
s = 0.9+0.09+0.009+0.0009+0.00009+…………= 0.9/( 1-0. 1)= 1