I picked up the basics of LED science from the LEDs for Beginners
Instructable, reading not only the Instructable itself but the many associated comments. They supply a wealth of theory and important links to everything you want to know about LEDs.
The back of the LED package provides the information we need to properly build the working circuit. Use this information to determine which type and quantity of battery and what size resistor to use.
This blue LED requires a 4.0 Forward Voltage Drop (Vf) to light.
It will pass 25 milliamps of Current (If).
Three 1.5 volt batteries in series will supply 4.5 volts.
Any combination of batteries that add up to the required voltage will do. For instance, AAA batteries are 1.5 volts and 3 in series will give you 4.5 volts.
I found these tiny 1.5 volt button batteries inside of an A23 battery (see the second picture). Three of these work nicely. See this [https://www.instructables.com/id/12-Volt-Battery-Hack!-You_ll-be-Surprised.../ 12 Volt Battery Hack Instructable] for more about that.
Obeying Ohm's Law and using this Current Limiting Resister Calculator of LEDs
, a 20 ohm resistor should be placed inline in the circuit.