> It's actually possible to finish a train of thought or conversation and then look up the facts, keeping everybody on the same page. It's even more fun that way.
I'd be careful about "keeping everybody on the same page". If you know they'd appreciate it, then sure. But a significant portion of the population do not like being fact checked. They assume malicious intent (not realizing you factcheck everyone and not just them).
Below is an email I once got. The context: A bunch of us were having a social conversation after an event. A professor made a claim about how hot it would get in his country. It struck me and another one as off because the number was a bit higher than the world record. We hinted at it but he insisted we were wrong.
Some hours later, at home, my friend fact checked and sent a polite email pointing out that the highest ever recorded temperature in his country was a few degrees lower than his claim. His response:
"If I were you i would not have spent a minute doing that unless
you want to prove a point: I was a liar. ... At last, that's why I do not hang around with you guys."
People will jump to conclusions about your motives.