"Number" study--0.999 of the fans ...

Six months ago, accidentally noticed a strange phenomenon 1/3*3=1 and 0.333...*3=0.999 ..., a very small number 0.000...1. have been baffled by the solution.

And recently discovered more bizarre is, 2/5+3/5=1,0.4+0.6=1, this is perfect in decimal, but after conversion to binary (manual conversion, there is no computer character points inaccurate problem)

Decimal Score	Decimal decimal	Binary Fractions	Binary decimals
1/5	0.2	1/101	0.001100110011 ...
2/5	0.4	10/101	0.0110011001100 ...
3/5	0.6	11/101	0.1001100110011..
4/5	0.8	100/101	0.110011001100 ...
5/5	1.0	101/101	0.111 ... or 1.0??

The same number in different binary is actually not the same?

This is a very strange question, why the same number converted to decimal, different binary actually will appear different results. Does that mean that human beings, if they are in octal or hexadecimal, are different from the current technology development of the decimal? We know that in decimal, except the number of 2 and 5 factors can be removed, that is, no infinite loop decimal. In the case of binary, the addition of only 2 can be done. (Here the reason is with the shift of the decimal point, please do not specifically Baidu) such words different into the system, because some have become limited, some into infinity, the same number combined with the result is not the same.

The same reason we go back to 1/3 and 0.333 ... Problem, if the use of the three -way, 1/3 is directly equal to 0.1, 3*0.1=1 is completely correct. Where does the problem really go?

1/3 and 0.333 ...

Step by step to debug, let's first look at 1/3 to decimal 0.333 ... There is no problem with this step. What do you think? Our routine division conversion step is too much to learn, and it's hard to see if there are any bugs. Then push the results back.

0.333 ... = 0.3+0.03+0.003+...+0.00...3

= 3/10+3/100+...+3/10n

= * * * (1/10+1/100+1/1000+ ... 1/10N)

= * * (10n-1/10n + 10n-2/10n + 10n-3/10n + ... +100/10n)

= (((10n-1)/9)/10n summation formula

= 1/3 * (1-1/10n)

= 1/3-1/(3*10N) (n→∞)

Accidentally found that the original 0.333 ... In fact, is not equal to 1/3, that is, we in the fractional conversion to decimal time, for the infinite cycle of the province of the decimal notation is actually error!

Where is the specific error? Then carefully divide the primary school division calculus once again:

1÷3 = 0.9÷3 + 0.1÷3

= 0.3 + 0.1÷3 = 0.3+ (0.09+0.01)÷3

= 0.33 + 0.01÷3 = 0.33+ (0.009+0.001) ÷3

= 0.333 + 0.001÷3

= ...

When we divide the infinite cycle of decimals, in fact, just take the first half, the back of the item is ignored, what is the back? Careful observation you will find in fact 1/10n÷3 (n→∞).

By pushing back and forward, we find that there are bugs. So 3*0.333 ... = 0.999 ... actually 3*[1/3-1/(3*10n)] = 1-1/10n (n→∞)

This indirectly explains the 0.999...≠1, and 0.999 ... The difference between the 1 and the 0.00...1 is actually 1/10n (n→∞)

One might ask 1/10n (n→∞) is it equal to 0 ? The simplest rough 1/10n * 10n = 1, and 0 times any number equals 0 cannot be equal to 1, so 1/10n is not 0. Of course, there are thousands of rigorous methods to prove that 1/10n is not 0, this should be consulted on their own.

Looking back at the different problems of different numbers, there is no problem, it is the error that the man produced when converting to decimal.

other comments and my rebuttal

Interesting to find some other views on the Internet, and give a seemingly unassailable counter-example.

1) David Foster Wallace (David Foster Wallace) introduced another notable proof in his book "Everything and More":

make x = 0.999 ... so 10x = 9.999 ... two subtract 9x = 9 so x = 1

This one at a glance, most people feel that there is no rebuttal, but in fact is blinders. I am in the classmate Li Yuankang the reminder under the Awakening, ask 10x=9.999 ... How did this come out? Maybe it's 10x=9.99...0? It is a language to wake up the dream people, directly to use the limit write: 10 * 0.999 ... = ten * (1-1/10n) =10-10/10n

and 9.999 ... is equal to 10-1/10n , the two are obviously unequal, the second so is already wrong, the final result of course also obtained.

2)"x is not equal to Y equivalent to the existence of a z strictly between x and Y"

Is there a value between 0.999 ... and 1 between the two? Into the limit expression to see,1/10n and 1 There is no, readily can be lifted out (the denominator increased), in the middle there is an infinite. Then why didn't you see it in the form of small numbers? Because this is already constrained by the precision of decimal decimals . Why do you say that? And if it's hex, there's a to F. Hex of 0. FFF ..., there is no more than the decimal 0.999 ... Closer to 1? What if the 100 is a larger system?

If...

If you are pretty sure 1/10n (n→∞) is 0 , then the above discussion is not convincing to you. But that would be equivalent to confirming that the human error between conversions from fractions to infinite decimals (such as 1/3--and 0.333 ...) has no effect on the actual results. And that is to say, 1/0 such number is present, also can appear 0 * 1/0 = 1 such multiplication calculation. Crazy world, who knows?

References

1, Baidu Encyclopedia "0.999 ..."

2, Shell net "the most tangled equation: 0.999...=1"

"Number" study--0.999 of the fans ...