Completely and fully; in every respect; to the exclusion of everything else.
From Middle English, derived from 'entire' plus '-ly'. 'Entire' comes from Old French 'entier', which traces back to Latin 'integer' meaning 'whole, complete, untouched'. The Latin 'integer' combines 'in-' (not) with 'tangere' (to touch), literally meaning 'untouched' or 'whole'.
The word 'entirely' shares its root with 'integer' from mathematics, both stemming from the concept of wholeness and completeness. This mathematical connection gives 'entirely' a precision that other synonyms lack - when something is done entirely, it's as complete as a whole number, with no fractions or remainders.
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