That's not naive, @niallrobinson, it's not obvious at all.
_"In physics, a dimensionless physical constant ... is a universal physical constant that is dimensionless –  **having no unit attached** , so its numerical value is the same under all possible systems of units."_ [http://en.wikipedia.org/wiki/Dimensionless_physical_constant]

I'm not sure why we use 1 but it's clear it needs to be made more clear - if possible!
Can anyone elaborate on this?