Cloud computing has dramatically reshaped the whole IT industry in recent years. With the transition from IPv4 to IPv6, services running in Cloud computing will face problems associated with IPv6 addressing: the notation is too long (39 bytes), there are too many variants of a single IPv6 address and a potential conflict may exist with conventional http_URL notation caused by the use of the colon (:). This paper proposes a new scheme to represent an IPv6 address with a shorter, more compact notation (27 bytes), without variants or conflicts with http_URL. The proposal is known as dot-base62x as it is an IPv6 address with Base62x and uses the well-known period (or dot) as a group delimiter instead of the colon. The relative merits and demerits of other works that predate this paper have been reviewed and critically evaluated. Cloud computing, as a continuously emerging mainstream of network-based applications, is likely to be a forerunner in the use of IPv6 as the base protocol. As a result, Cloud computing will benefit most from the new, compact and user-friendly textual representation of IPv6 address proposed by this paper.