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Resistance
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Resistance is a type of defence that reduces damage of a particular type by a percentage of the total damage. Players and enemies all have a resistance stat to each of Fire, Cold, Lightning and Chaos damage, and these are raised or lowered independently. Increases to 'Elemental resistance' are applied to fire, cold and lightning but not chaos.
Resistance defaults to a value of 0, which means it does not reduce damage at all. Resistance is capped at 75% by default, but this can be increased as high as 90%. Having a negative resistance percentage in a damage type means you will take that much extra damage from that type.
Calculation
Final Damage = D * (100-R)/100
Example: being hit by 1000 fire damage, with a fire resistance of 30%, means you would take: 1000 * (100-30)/100 = 1000 * 0.7 = 700 damage
The higher your resistance gets, the more effective an increase in resistance is in reducing damage taken. This makes increases to maximum resistance a particularly valuable stat. For example, increasing resistance by 10 from 0% to 10% will reduce damage taken from 100% to 90%, or 10% less. But increasing by 10 from 60% to 70% would be reducing damage taken from 40% to 30%, reducing damage taken by a quarter.
Resistance Penetration
Having Penetration in a damage type reduces the effective resistance of enemies you damage with that type, thus increasing the damage they take. It only applies against positive resistances; it has no effect once the effective resistance is 0 or negative.