Strange dotless decimal notation of IP address... How does it work?

This is quite an interesting question, and took me a little while to figure out. The short answer is the last 32-bits of the number are 3660944368 (in decimal, which can be found by 714687644714805209715128610715964400 mod 2^32)

This is the decimal value of the IPv4 address 218.53.147.240, which can be found by converting it to base-256 3660944368 = 218*(256^3)+53*(256^2)+147*(256)+240 analogous to writing out a number in decimal (base-10). For example 234 = 2*10^2+ 3*10 + 4.

As @chritohnide points out, each section of dotted IPv4 addresses is called an octet as it represents 8 binary digits. It is also worth noting that the various formats of IPv4 addresses (such as the dotted decimal, or the pure decimal) are just different ways of representing the 32-bit binary number for out benefit.

Since IPv4 addresses are 32-bit numbers, so only last 32-bits of the number are used to resolve the address. Why this is true is not as obvious. As others have pointed out, the full number looks strikingly similar an IPv6 address in decimal, but is not a valid address.

Looking at the Teredo specification (see 4. Teredo Addresses), the Client IPv4 occupies the last 32-bits of the IPv6 address, but the prefix of the number does not match the Teredo specification (Also see wikipedia).