The Doctor approves of Billy. He's clearly intelligent and that's a plus. Eleven likes people who bother to use their heads and ask the intelligent questions, it saves him a lot of time having to exposition about it if they're (mostly) keeping up. He might think Billy's a little on the shy side but he was most helpful during the TARDIS mapping and on top of that, he's Jamie's bro, so Eleven's got pretty positive feelings even on a first impression. Besides, Billy did indirectly save Jamie's life in both physical and mental ways, so he's thankful for that. So basically Eleven has goodwill toward Billy despite them just meeting.
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beloved Jamie what is thisOH FINEThe Doctor approves of Billy. He's clearly intelligent and that's a plus. Eleven likes people who bother to use their heads and ask the intelligent questions, it saves him a lot of time having to exposition about it if they're (mostly) keeping up. He might think Billy's a little on the shy side but he was most helpful during the TARDIS mapping and on top of that, he's Jamie's bro, so Eleven's got pretty positive feelings even on a first impression. Besides, Billy did indirectly save Jamie's life in both physical and mental ways, so he's thankful for that. So basically Eleven has goodwill toward Billy despite them just meeting.