No advanced mathematics or calculation is involved. You may have heard that miners solve complex mathematical problems – that’s not true at all. What they do is try to be the first miners to offer a 64-digit hexadecimal number (a “hash”) less than or equal to the target hash. It’s basically riddles. Because it’s a guess, you need a lot of computing power to reach the first goal. To succeed in mine, you must have a high “hash rate”, measured in terms of megahash per second (MH / s), gigahashes per second (GH / s) and terahashes per second (TH / s). The job is only “wasted” in that it does not help to solve a problem other than the one for which Bitcoin was designed; namely, provide cryptographic assurance that funds can only be held by one party at a time, can not be duplicated or spent twice. “Mining” by Bitcoin solves a very specific problem with Bitcoin – how to get consensus on the order of transactions added to a ledger of distributed transactions. Is it useful for humans? A decentralized digital currency that resists censorship brings value to humans, yes. For example, WikiLeaks had no way to get funding other than bitcoin. So, Bitcoin is useful this way. When a block has been resolved, no bit can be changed without invalidating it. This means that all transactions in the block have been processed and are “final” in the sense that they are in the blockchain and will not change. (assuming this block is part of the best / longest block chain). Imagine how deplorable it would be for you to modify the general ledger of transactions to indicate that you are no longer paying, or the amount you have declared not to be paid. Requiring billions of mathematical steps to solve a block means that it is very difficult to accomplish. Honest miners who mine at the top of the blockchain are rewarded in the form of bitcoins. But attackers who try to go back in history and replace old blocks with new ones are at a disadvantage, because that requires a huge amount of hash power. network has continued its exploitation activities on the tip of the blockchain, making the attacker even more difficult to catch up. Therefore, to create blocks faster than the rest of the network, it must control more than 50% of the network’s hash power. It would take millions of dollars now.