As the world’s monetary systems teetered on the edge in late 2008, some computer whizzes sought a digital solution. As Lehman Brothers lay dying, a math-based, completely digital means of exchange known as bitcoin was born.

A few months after the white paper that defined it, bitcoin went quietly into operation. As with the internet itself, bitcoin’s first adopters were mostly the highly computer-literate and the highly criminal, followed by speculators. The image of the “digital gold” was first tarnished by association with...