Comparative form of mad: more angry, irritated, or insane than something or someone else.
From Old English 'gemad' (made insane), related to 'mad.' The comparative form adds '-er.' Separately, 'madder' is also a plant producing a red dye historically used in textiles (from Old English 'madere').
English has a clever homonymic accident: 'madder' (more angry) and 'madder' (a red dye plant) are completely unrelated etymologically but sound identical—ancient textile workers had no idea their dye plant would become slang for anger.
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