Why is my iPhone 4 only working on a dead battery?

The screen faded out a few minutes ago and seems to only turn on when the battery is less than 20%. I kno it works then cuz i left it off of the charger or an entire day, plug it up to the charger, and it would work for a few hrs. Its out again.y could this b?

then i tried restoring it on itunes, but there was a restore 1602 error, then a 1600 error.

Added (1). The charger is fine b/c wen i use it w/ my ipod, that charges normally.could it b i used my phone while letting it charge millionssss of times (it became hot a lot) and it burned the battery out? @Ting. Hope you c this.

Added (2). I simply can't access ANY data in my phone while its doing this (running out the battery on its own). Its going in and out.working for 1 day, then blacking out. Then working for a few hrs, and blacking out again. Like i can't use it if it isn't charged at all but it seems like wen it gets to 60% or sumthin, thats wen things stop flowing.hmm… My mom is gonna try and c if sprints repair center can fix it tomorrow but i was just in so many deep conversations so i'm pretty upset that it went out ONCE AGAIN
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Added (3). I simply can't access ANY data in my phone while its doing this (running out the battery on its own). Its going in and out.working for 1 day, then blacking out. Then working for a few hrs, and blacking out again. Like i can't use it if it isn't charged at all but it seems like wen it gets to 60% or sumthin, thats wen things stop flowing.hmm… My mom is gonna try and c if sprints repair center can fix it tomorrow but i was just in so many deep conversations so i'm pretty upset that it went out ONCE AGAIN
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Added (4). Ohh that makes sense.

It could be your charger. Not properly charging your phone. I know I broke a bunch of iPhone 4 chargers. If not then try resetting it by holding the home button and the top button for like 30 sec. If that doesn't work the. It might just be the phone. I would replace it.

Yes, this is true with all apple products that have a dead battery. The small reserve on the battery lets them stay on for a few min when they're plugged in. And the restore won't work because iTunes needs the iPhone to be running, so it can download iOS to your device, and since your device is out of battery (the battery can't hold voltage anymore) it won't restore. GEt a new iPhone or ask Apple if they can do anything for you. Be sure to show Apple your receipt of purchase.

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