Zanthra

Do you consider infinity a number. Because for any number X, X != X + 1 infinity does. Anything that is small but has any size at all has a finite size. Infinity is the lack of a bound, so 1 / infinity would equal zero. If you have 1 / X, then X (1 / X) = 1 but in that case 2X ( 1 / X ) = 2. That means that since infinity * 2 = infinity then infinity ( 1 / infinity ) = 1 and infinity ( 1 / infinity ) = 2 a contradiction. Math cannot be done with infinity as a number, however when you use limits you can, in that case 1 / infinity = 0 and 1 / 0 = infinity, meaning that something infinately small is equal to nothing. Infinity is not a number that math can be done with.

I do not believe there are any idiots in the world. Just different forms of Inteligence. Since the real question about the difference between 1 and .999 is what infinity means, I would like to show you this page that I found a few minutes ago. http://diveintomark.org/archives/2003/12/04/infinite-hotel It shows the significant difference between thinking of infinity and thinking of real numbers is that infinity is the lack of a bound. Infinity means there is no number larger than it, even itself, and even itself * itself, or itself to the power of itself.

I have to disagree with the fact that, repeting decimals are not a finite number. I think they are well defined by the integer ratios they turn into through the method I described. It works for simple ratios such as 1/3 = .333, and even more and more complex ones, what is wrong with it working or .999 You are saying if you have X nines after the decimal point, then there will be X - 1 nines if you multiply by ten. That leaves 1 nine behind when you are done with the subtraction. The problem here is that you are saying infinity - 1. For all practical purposes infinity * 50 = infinity. For example there is the problem of the infinately large hotel. There are rooms labeled from 0 ... infinity. One person wants a room but all the rooms are full. So in solution to this, the hotel calls up everyone in the hotel and tells them each to move to their room + 1. This makes room 0 vacant and the new arival sleeps there. If 2 people arrive at the hotel, you can have everyone move to their room + 2. Now what about the case where an infinate number of people want rooms in the hotel. In that case the hotel can once again call up everyone in the hotel and tell them each to move to their room * 2. This frees every odd numbered room in the hotel, and there is room for an infinate number of people. This shows how finite numbers and infinite numbers can have relationships, and how infinate variables can do strange things in word problems. Another problem that was described to me is if you have two buckets. You do the following steps an infinate number of times. Have 2 variables, X and Y both at zero. Place X and X + 1 in the first bucket, and increment X by 1. Take Y from the first bucket and place it in the second bucket and increment Y by 1. Repeat. When you are done, what numbers are in each bucket? Since you do it an infinate number of times, there are no numbers that can be in the first bucket, and all numbers will be in the second bucket, dispite the fact that logic will state that there should be the same number of numbers in the first bucket as the second. This shows how logical assupmtions and resonable truths can be bent by the rules of infinity. In the case of the repeating decimal, we are not talking about a finite number. Instead think of it as saying that after each repetition there is another repetition. For example if you take 9999 and you take off a nine from the front, if there is always another repetition it is still 9999 you take off 100 million, or more and you still get 9999 All fractions are exactly equal to some number with repeating decimals. Many of them have repeating zeroes however, and it that case the zeroes are simply not written. Infinity is a fickle thing and hard to understand. When I was first posed with the bucket example, I tried to find some way to get an equal size of each of the buckets, when I did so I got it wrong. When the answer was revealed however it made perfect sense. I think that is where I really began to understand infinity.

The point of a hash code is to turn a string of bits into another string of bits in a way that can check weather two hashed strings of data only have a very small chance of beeing the same. MD5 is one common hashing algorithm. It is a 128 bit hash, meaning that any 2 hashed datasets have a 1 in 340282366920938463463374607431768211456 chance of beeing the same. This can be used to check a password for authenticity. When the hash it cannot be turned back into a string of binary data that will hash into that particular hash code. When the password is entered it can be hashed again, and if the password matches the origonal one, the hash code will also match. Not only are hashes used for passwords however. When transfering a very large file over the internet, lets say 100MB, there is a possibility that a bit may end up reversed, and the file becomes corrupted. If the file is not meant to be used, but instead to be served out again to someone else, having such a corrupt file can be a very bad thing. Since hashes have such a rare chance to come out the same, you can run the entire file through the hash algorithm on one side, and get 128 bits more data to send along with the file. On the other end the recieved file gets hashed, and if the hash matches the one passed from the provider, there is only a astronomically small chance that the file was corrupted in any way through the transfer. Once again there is no way to turn the hash code back into anything. If there was, then the hashing method would be broken, and there would be a gaping security problem.

I think of it by the fact that any number with a repeating decimal is a rational number. In the case of .333 it is equal to 1/3. In the case of .131313 it is equal to 13/99. For such numbers there is an easy algebraic way to turn it into a fraction. Take X to be the repeating decimal: x = 3.723969696 Break X into two parts the repeating portion and the nonrepeating portion: x = y + z y = 3.723 z = .000969696 First find the fraction of z: z = .000969696 : Multiply both sides untill you have one complete repetion to the left of the previous repetition, equal to 10 times the number of digits in the repetition, in this case 100. 100z = .0969696 : Subtract one of the variable from both sides. On the left use the representative variable, and on the right use the repetitive value. 99z = .096 : Now that you have gotten rid of the repetition, you can find a fraction by simplyfying. z = .096 / 99 z = 96 / 99000 Now go back and simplify the same fraction for the origonal nonrepetitive portion: y = 3.723 y = 3.723 / 1 y = 3723 / 1000 Multiply them to have the same denominator: z = 96/99000 y = 368577/99000 x = y + z : Put them back together and simplify. x = 368673/99000 x = 122891/33000 122891/33000 = 3.723969696 This works for any number with a repeating decimal, including .999: x = .999 x = y + z y = 0 z = .999 10z = 9.999 9z = 9 z = 1 x = y + z x = 0 + 1 x = 1 .999 = 1 There is no number with a repetative decimal that does not work in this method. There are also no numbers with repetative decimals that do not exqual an integer ratio. Meaning that .999, 1.999 etc. all are integer ratios 1/1, 2/1 etc...

Windows XP uses a hashing algorithm to encrypt the password into the file. Hashing algorithms are one way algoriths that turn a hunk of data into another hunk of data. When the password is first created the OS turns the password into a hash code then stores that in hexadecimal values in the password file. Whenever the user attempts to log in, it will again hash the entered password and check the hexadecimal code against the one on file. Once you enter the password the OS has no way to turn the hash code back into a readable password. Programs like the one offered in this thread will do what is called brute force. They will take the hash algoriths and go through each and every password possible and hash them and check them against the code in the password file. This can take a very long time depending on the hashing algorithm and the speed of the processors that it is running on, but can find the password. Many such programs also have a dictionary of common passwords that it can check against the user password, and is the most common attempts to hack computers. Two of the more common hashing algorithms are MD5 and SHA, and microsoft does have a proprietary hashing algorithm that can be used to store passwords on windows machines.