Newbie laptop questions

So I just bought a nice refurbished thinkpad T40, with a new battery. I like it.

I've never owned a laptop before, so I have no idea what I'm doing. How long should I charge the battery for before its first use? How long would it take to charge completely from empty?

And subsequently, is it best to use the battery until it's empty before charging? How many charges is this kind of battery good for?

Most laptop stores tell you that you should charge for ~8 hours (full night, won't hurt if you leave it longer) first, to "format" the bat. After that, it depends on the capacity. As for draining completely: do it every 2 week - 1 month, if you are a carebear. It might prolong it's capacity and lifetime.

It depends on the battery type. for NiCad, NiMH based battery what >>2 suggested. Li-Ion doesnt suffer from the memory effect. But if you want to know more how to care your battery, read this article:
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I'm the OP, the battery is whatever comes with this one -

http://www.laptops.com.au/products_details.php?id=188&mark=1

Can I ask though, is there any way to run the laptop off the mains power without it recharging the battery? What if the battery is only like half drained but you're at home and don't want to charge it or drain it further?

When the battery is still physically plugged with the laptop? Unless your laptop's BIOS supports the feature you got no way of doing it while the battery is still plugged with the laptop, you'll have to take out the battery physically from the laptop. Btw your battery seems to be the Li- variety.

Oh, so is it OK to take the battery out and just have it running from power?

I actually charged it for 14 hours and the battery was still totally dead. Rang up the store and the guy told me to "warm up" the battery by rubbing it against my body, which is the most intimate I've ever been with a machine. However, it now seems to be charging (the LED is steady rather than blinking). Has anyone else heard of this method or is the clerk jacking off to a mental picture of me having machine sex?

Yeah, read the article if your laptop is going to be desk bound all day might as well just charge it up to 40%, take it out and put into a fridge (after dble wrapping it of course). Never heard of warming up a battery.... it might be defective ask for a replacement.

Another question - when people talk about batteries having a finite amount recharge cycles, what counts as a "cycle"? If I charge it from 80% to full, is it as degrading on the batt as charging it from 0 to full?

From what is out on the internet the definition of cyle varies. But at best it's described as from 0% to 100%. As to your question of is it any more harmfull from 80-100 I don't think it's harmfull unless you're doing it to often,but I reccomend not to do it.