Foundations of Analysis by Edmund Landau is a rigorous and celebrated mathematical text that constructs the number system from the ground up. Originally published in German as Grundlagen der Analysis, this English translation by F. Steinhardt of Columbia University makes Landau's precise and elegant work accessible to English-speaking readers.

The book systematically develops the arithmetic of the four fundamental number systems, building each from first principles with the logical clarity and economy of style for which Landau was renowned:

• Whole (natural) numbers, constructed via the Peano axioms
• Rational numbers, derived rigorously from the integers
• Irrational numbers, introduced through Dedekind cuts or equivalent constructions
• Complex numbers, completing the classical number hierarchy

Landau's approach is famously terse and axiomatic — every theorem is proved from scratch, with no hand-waving or appeals to intuition. The result is one of the most logically airtight introductions to real analysis ever written. The book is subtitled A Supplement to Text-Books on the Differential and Integral Calculus, reflecting its role as a rigorous foundation for students who wish to understand why calculus works, not merely how to apply it.

Key themes include:

• Axiomatic construction of the number systems
• Formal proof techniques in pure mathematics
• The logical foundations of real and complex analysis
• Rigorous definitions of arithmetic operations and order relations

This third edition, published by Chelsea Publishing Company in 1966, is an essential reference for mathematics students, graduate researchers, and anyone seeking a deep, uncompromising grounding in the foundations of mathematical analysis.