This book brings reader the fully-fledged type of thinking used by professional mathematicians. It develops the more formal approach as a natural outgrowth of the pattern of underlying ideas, building on a school-mathematics background to develop the viewpoint of an advanced practicing mathematician. It covers the nature of mathematical thinking; a review of the intuitive development of familiar number systems; sets, relations, functions; an introduction to logic as used by practicing mathematicians, methods of proof (including how mathematical proof is written); development of axiomatic number systems from natural numbers and proof by induction to the construction of the real and complex numbers; the real number as a complete ordered field; cardinal numbers; foundations in retrospect.It is indeed an interesting and helpful book for readers in transition from ‘school mathematics’ to be a professional mathematician.