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A Class A IP address is an IP address in which the first bit of the octet is set to zero, which provides values from 1 to 127. By contrast, a Class B address has the first bit set to one, and the second bit set to zero to utilize the remaining address set.
Address classes and other parts of the IPv4 IP address structure are administrated by the Internet Assigned Numbers Authority (IANA). Class A addresses are to be utilized for addresses with larger networks, with a large number of hosts. Class B addresses are for medium-sized to large-sized networks, and Class C addresses are for small networks.
The philosophy behind this and other aspects of IANA administration is to provide an orderly process of identifying machines and host networks. In many ways, the internet has been a "Wild West" that officials try to regulate, and IP addresses are an important part of this regulation.