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Full Idea
One reason for introducing the real numbers is to provide answers to square root problems.
Gist of Idea
Real numbers provide answers to square root problems
Source
A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3)
Book Ref
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.70
A Reaction
Presumably the other main reasons is to deal with problems of exact measurement. It is interesting that there seem to be two quite distinct reasons for introducing the reals. Cf. Ideas 10102 and 10106.
Related Ideas
Idea 10102 The integers are answers to subtraction problems involving natural numbers [George/Velleman]
Idea 10106 Rational numbers give answers to division problems with integers [George/Velleman]