I would appreciate any constructive ideas on the most economical way to run my computer. For example, if I turn it on and then need to leave it for a short time, is it wiser to turn
it off altogether or leave it running and turn the modem off?
Similarly, if the computer is turned on in the morning, but I don't intend to return to it until mid-afternoon, what is my best option? Thanks in advance.

I don't too much about computers but here's my opinion which should be wrong and which I've learn from some of my uncles.

Well as far as I know, if I have to leave the computer for some time, I switch off the monitor. That's what my uncle suggests, though I'm not sure.

If you want to come back in the afternoon, I suggest you hibernate your computer or standbye it. Stand bye puts your computer in a low-power state and you can quickly do what you want. Another thing, you can do, is hibernate your computer (I know it works on Windows XP, I don't know about other OS's). In hibernating, the computer shuts down but remembers how it was before the shut down. So if you had a document open, and hibernated the computer, the computer would open with the document.

Different people have different ideas, my computer supplier says that more damage is done to a computer on booting with the surge than wear and tear by leaving it switched on but allowing it to go into powersaving mode. I read somewhere that a huge amount of power is wasted by computers being left on, so what does one so, waste electricity or ruin the computer? I leave my computert switched on unless I am leaving home for a few days, then I switch it off.

As for the monitor, screensavers and powersaving modes should take care of that.

I've always been told that turning off your computer every time you don't intend on using it for a while is not the best idea. Switching it on and off will do more harm than it will do good, your pc's power supply will be worn down very fast (it's just like a striplight really ).

I usually turn on my computer when I get up and turn it off when I go to bed at night. I only turn it off when I'm going away for long during the day, because I'm a little bit afraid of it short-circuiting and causing a fire in my absence. I may also turn it off because of the noise it makes. My power bill is divided between all 500 people in my apartment building, so I can't tell the difference I guess.

In the end I think it comes down to balancing the pros and cons, the higher power bill or the worn down pc...

I do much the same as Sue and Leau ...and for the same reasons.

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I may also turn it off because of the noise it makes.

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You're not alone in that. I hate the constant whirring of the computer. I have recently upgraded the innards of mine. The parts meant 2 extra cooling fans adding to the noise. I've built a cupboard around the computer to shut the whirring in a bit. Of course, this has to be balanced with allowing enough ventilation to let the heat out.

One of the requirements of the next computer I get will be that it should be quiet. There are computers built without fans and some have quiet fans (not sure how much quieter though). I'm going to be looking into this.

Although I have put in a stronger fan than needed just to be safe, I think my pc is relatively quiet. But not quiet enough. My living room is also my bedroom and my study, so no escaping from it. On very rare occasions I have to let my pc run all night long to defrag and I just can't sleep without earplugs!

After a Mother board failure a few years ago I was advised to not turn it off. I really do not know if on and off hurts them or not. But it made sense to me and it was more practical to not turn it off. The only time I have turned mine off since is just to reboot after the memory is all attached. ( Why Another good question)
You can go to the control panel and find your power management. There you can set when you want you monitor and hd to power down. I think I have mine set at 15 or 20 minutes. I always turn my monitor and speakers off too because the cat has been known to power it back up in the middle of the night.
Once the monitor and HD are powered down, the electrical draw should be miniscule. With the power saver set it will power down any time you are away for the set time. This way it is ready to go when you are instead of having to wait for the program load before checking your e-mail.
Leaving it on also allows you to schedule maintenance during the middle of the night when you are not using it. The fans did bother my wife for the first few months but now we don't even hear it.
Jax