As a noun, it is a piece of material made of crossed threads with holes, used for catching or separating things. As an adjective or verb, 'net' means the final amount after everything extra has been added or taken away, like net income after taxes.
From Old English 'nett' meaning 'mesh, net', related to other Germanic words for woven structures. The financial sense of 'net' grew from the idea of what remains after all 'entanglements' are removed.
A fishing net catches fish by letting water pass through; in math and money, 'net' catches the important part by letting extras pass away. 'Internet' literally means a 'network between networks', built from the same simple idea of connections.
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