Full Idea
Every generalisation of number has presented itself as needed for some simple problem. Negative numbers are needed to make subtraction always possible; fractions to make division always possible; complex numbers to make solutions of equations possible.
Gist of Idea
New numbers solve problems: negatives for subtraction, fractions for division, complex for equations
Source
Bertrand Russell (Introduction to Mathematical Philosophy [1919], VII)
Book Reference
Russell,Bertrand: 'Introduction to Mathematical Philosophy' [George Allen and Unwin 1975], p.74
A Reaction
Doesn't this rather suggest that we made them up? If new problems turn up, we'll invent another lot. We already have added 'surreal' numbers.