How does solving math problems create bitcoins? If I understand correctly, Bitcoin mining software "creates" bitcoins by solving complex mathematical equations. Is there a need for these problems to be solved, is there a value to solving these equations? If not, where does the solving of the math problems fit into the whole Bitcoin scenario?
The problem that mining solves is the problem of providing secure transactions without a central authority. There is value to solving these problems because otherwise, there would be no way to securely exchange Bitcoins.

Bitcoin uses proof of work as its means of solving the double spend problem without a central authority. If I try to send the same Bitcoin to two different accounts, the miners choose one transaction or the other and use it as the basis of the computations they do. Because nobody else can outcompute all the Bitcoin miners, this makes it possible for others to determine which transaction "wins". Without this mechanism, there would be no way to resolve conflicting transactions.

Because this proof of work costs money to do, it's not likely that it could be accomplished without some incentive. Mining rewards provide this incentive.