> But some random "mined" numbers? Why do they have value?

The random mined numbers are not where the value is stored. Those numbers just confirm that the miner who finds/guesses that number did the necessary work to find it, and in turn, validate the authenticity of the current block of transactions. Their reward being Bitcoin for doing the work (how Bitcoin comes into existence).

That Bitcoin has value because it represents energy or work expended, just like any currency—dollars, vodka, whatever. That's the whole reason any currency exists: as a medium to trade value generated by one person's work for the results of other people's work. Instead of physical labor, though, the work being done is computation.

It also has value because of how the underlying system that backs it works. In fiat systems, the rules about how much money is in existence, who can or cannot access the system, and the ability to accumulate savings is dictated by a central authority. With Bitcoin, there is no central authority. The amount of Bitcoin produced cannot be changed without fundamental changes to its halving algorithm which would result in a fork of the network, creating a brand new currency (with the original network continuing on uninterrupted by people who disagree with the new rules).

This is another reason Bitcoin is valuable: everything is driven algorithmically by consensus, meaning, humans can't alter it on a whim (like a fiat system). Instead, the only way to make a change is to propose it in the form of a code patch and then have the rest of the network accept that change. If the network doesn't accept it, the change doesn't occur. In terms of money, there's never been a more valuable form (there is no analog for a form of money that doesn't require human trust—Bitcoin is unique in that category).

Beyond that, it also has value because it's digital and permisionless. I don't have to get a bank's permission to do business or transact with someone else. I can just _do it_. And I can do that on a global scale, instantly. Contrast that with the existing system, any amount over a few thousand dollars triggers red flags, transactions get blocked, and insane amounts of time and energy get wasted just for you to get access to your money (that most people don't view that as patently insane proves how well the conditioning works).

Even further, Bitcoin isn't something that can be confiscated (like fiat, precious metals, or other assets). This means that corrupt governments and individuals can't just blindly steal from you. The only way to get your Bitcoin is to get access to your private keys. The only way to get those is, ultimately, by force/violence (i.e., they can't just dip their hand into your bank account and clean you out because of a "law" they invented). And that only works if your private keys are accessible—though difficult, you could memorize them creating the ultimate security system.

The best part and why Bitcoin is extremely valuable: its base cannot be inflated. Meaning, when I work and earn money, that money either retains its current value, or, increases in value. There's no potential for a government to randomly print off trillions on a whim and devalue the money I've already earned (essentially, stealing my life from me covertly).

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All of the above is radically different from fiat money. Again, we've never had a currency like Bitcoin in the history of humanity (in part, why it's so difficult for people to understand it). It's like showing fire to cavemen.