The DNS root system names 13 logical root name servers, identified from A-root through M-root. These are logical identities and service addresses, not merely 13 physical machines.
The number arose from early DNS and networking constraints. Traditional DNS responses using UDP were limited to 512 bytes, so designers needed to fit a practical list of root-server addresses into a single response. Thirteen became the established maximum for the root-server set.
Modern DNS uses anycast routing, allowing each logical root-server identity to be served by many distributed instances around the world. That means the answer is 13 logical servers or clusters, while the number of physical and globally distributed instances is far larger and changes over time.
A common mix-up is confusing the 13 named root identities with the number of machines operating them. Root servers do not usually resolve ordinary domain names directly; they refer resolvers to the authoritative servers for top-level domains such as .com or .org.