I don't think they would be surprised if we call it offset or index.

>To me, 0.0.0.63 didn't say that I'd get something ending in .63.

Perhaps this is just a difference in backgrounds then. Even though I'm work
on network stuff all of the time, when I see that it's not obvious that it
will be masked with the inverse subnet mask to get the host bits and then
combine it with the network to get the real address. Do you have some
examples where this syntax is used elsewhere?

Ignoring the usability concern since that might just be an issue with me,
there are two other reasons that still make an integer preferable to me.
First, the dotted-quad notation unnecessarily ties it to IPv4. If this is
going to work for IPv6 subnets at some point, we would need to support both
formats, each with their own validation logic, even though they are
ultimately representing the same idea. Second, using an integer could allow
negatives to make asking for the last address in the subnet as simple as
'-1' instead of making the user calculate it out.
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