This has bothered me forever! And everyone always asks a slightly different question than me. When I know I'm not going to be using my computer for say, 3 or 4 hours at a time is it better to put it in sleep mode before I leave, or is it better to just let it run all day long until I shut it off for the night? I always put my laptops into sleep mode because there's not much else you can do with it (I don't want to leave it open all day) so this question is involving desktops, my gaming rig in particular. I run two monitors too and it puts more stress on my GPU at idle (I'd assume since it idles higher than with one monitor) would taking that stress of the GPU also be something to consider?
Monday through Thursday I'm always at home so I don't have to worry about this so much, it's just on the weekends when I turn my computer on earlier, and off later, and leave the house for longer periods of time.
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I'd go the other way. I really hate hibernate since sleep wakes up much faster. The power draw is very minimal so much better than leaving them running. My wireless connection comes back up faster than the whole machine does from hibernate. That's just been my experience.
I use sleep mode, no hibernate.
If there is anything essential in ram, I save it first.
Sleep is a very low power state. The electricity draw is minimal, so cost should not be an issue.
As to longevity, one can argue, but so far as I know there is nodifferrence between sleep and a cold boot.