If you have the private key, it restores the balance and the address.
If you do not have the private key, there is no way for you to recover either, unless you randomly generate the same address again, which is like hitting the powerball ten thousands times in a row.
Thank you very much for the answer, however will I recover the address too even if I take the private key to another host (a different wallet website then the original one)?
yes, the private key corresponds to the address. They are forever linked.
ok, now I did read at many different places that this random generation of the address from the private key is very very hard to be figured by anyone.
what I read is: the address can be generated from the private key, however the opposite is invalid, and I understand that too, but if this is true then the private key should always generate a different address based on this theory.
Cause if the private key will always generate the same address, and this is made by a machine (I mean coding), which means the machine will generate the address in a math way from the private key. Then after looking at so many private keys and addresses then someone eventually will be able to predict the private key from the address.
Unless the machine will randomly chose an address for the private key (not based on math) like for example a human will just brain pick a random address for his key, but in this case there is a big risk of generating the same address for 2 wallets. Kind of we all have different email address but some of us might have the same password (cause this is something we just brain make).
I do understand I read everywhere that this is impossible but I am just guessing based on some simple math.
And to avoid any 2 people from having the same address, then all the bitcoin wallet website should run off the same machine (which in case I believe is the bitcoin open source or mother unit or whatever) to prevent the double address from happening.
All in all, if the private key generate the address based on math (regardless how complicated) eventually someone will figure out a way to generate the private key from the address. UNLESS Like I said if the private key generates the address completely random then there is the chance to generate the same address twice or more in which if all wallet websites run of the same machine, then the address will be checked (to make sure it is not doubled) and then the key will keep trying randomly until it generate a unique address, however in the later case there is the theory of the key will always generate a different address.
Now that you said the key is link to that address forever, then it will bring me to the only option left:
1- when you create a wallet, the machine will create a completely random private key that is never used before, and that by checking the bitcoin mother unit that is used by every single wallet website to make sure it is not doubled.
2- then the machine will also randomly pick an address for the wallet that is never used before (same thing above by checking the mother coding unit for bitcoin or whatever it is that is used by every wallet website)
3- the machine will link those (the address and private key) together forever.
and that can explain why if you always have the private key then you always have the address with it on the same paper or file and can always be both recovered (balance and address).
But one question will be left, if the wallet.dat file will only have the private key, and then private key will generate the address AGAIN, then it means it is based on math.Otherwise the private key will generate a new unique address (never used by anyone BUT new address, NOT same address).
IF the answer is still that wallet.dat file will carry ONLY private key and that when you export the key into a wallet then it will ALWAYS generate the SAME address, then the way for that is: the private key and the address are both LINKED on the bitcoin MOTHER UNIT MACHINE, which means if it is ever hacked then all the 12 million coins will be gone!
I understand the guy who came up with all this is very smart, but I am just an idiot who is trying to understand a little so please be patient with me.