why does my battery died before the end of the day i have notheing running

Here's one big battery drainer:

http://support.apple.com/kb/HT5467

Others are frequent email pushes, bluetooth or wifi on when not being used, game center apps checking on status (you say nothing is running ... but are you sure? Apps run in the background unless you explicitly turn them off.).

I have to respectfully disagree about BlueTooth or WiFi. BlueTooth uses no power if it is on but no BT device is connected; it just listens for a device to connect to; the receiver uses unmeasureably small amounts of power. After connecting it still uses a very small amount of power (about 3 mw). WiFi also uses very little power; 30 mw when actually in use, but it turns off if the phone is asleep. And the 30 mw when in use is a lot less than cellular data would use if you had any apps using data while WiFi is off.

The biggest users of energy are MS Exchange email accounts, iCloud email if you get a lot of messages, streaming audio or video and social networking apps (Facebook being the largest consumer of energy, but any that send you notifications). And all of these can run in background.

Notifications in themselves use very little power, however, when an app receives a notification it wakes up to process it, and usually downloads data updates. The same is true for Location services.

I thought the biggest single user of power was a geofenced reminder of the form "Remind me to ... when I leave home."

I wouldn't call it the biggest, but it does consume a lot of power.