Arginine has a positive side chain.
Doesn't this mean it wants to lose a proton H+? Isn't something that wants to lose or donate a proton an acid?
Yet my book states, "arginine has a positively charged side chain, i.e it is basic" .
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Your textbook is imprecise. We consider whether a sidechain is acidic or basic based on the action of the sidechain when it is neutral.
Because arginine's sidechain is basic, it preferentially picks up a proton, and hence becomes positively charged. (In the same vein, carboxylic acid sidechains are acidic because they preferentially lose a proton, and exist as the negatively charged carboxylate anion.)