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Foundations of Analysis

Edmund Landau
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This book, first published in German in 1930, gives a concise and rigorous development of the complex numbers, starting from the Peano postulates. As the title says, the book sticks strictly to the foundations and does not do any analysis. The treatment is symbolic, and there is no topology of the real line (in fact, there’s no real line). The real numbers are developed from the rationals by Dedekind cuts and not by Cauchy sequences, so there’s no need to introduce ideas of convergence.

There’s not much that’s original or startling in this book; it’s a clear and concise treatment of a standard subject. Most of the conceptual work is in the beginning of the book, and the development gradually shifts from concept to calculation as we move to more complicated types of numbers. The book does have a very interesting proof (p. 67) of the irrationality of the square root of 2. The standard proof depends on the concept of even numbers, but Landau gives a proof by infinite descent: if a fraction were equal to the square root of 2, then we could construct a fraction with a smaller denominator that would also be equal to the square root of 2.

The book was written as a college textbook, but by present-day standards it is a reference, as it has no pictures or examples, and in fact does not even define the number 2. (The proof mentioned above shows the irrationality of 1′, the successor of 1.) It’s awe-inspiring and slightly appalling that you can write a whole book on numbers without exhibiting more than about three specific numbers.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.


  • Natural Numbers: 1.1 Axioms; 1.2 Addition; 1.3 Ordering; 1.4 Multiplication
  • Fractions: 2.1 Definition and equivalence; 2.2 Ordering; 2.3 Addition; 2.4 Multiplication; 2.5 Rational numbers and integers
  • Cuts: 3.1 Definition; 3.2 Ordering; 3.3 Addition; 3.4 Multiplication; 3.5 Rational cuts and integral cuts
  • Real Numbers: 4.1 Definition; 4.2 Ordering; 4.3 Addition; 4.4 Multiplication; 4.5 Dedekind's fundamental theorem
  • Complex Numbers: 5.1 Definition; 5.2 Addition; 5.3 Multiplication; 5.4 Subtraction; 5.5 Division; 5.6 Complex conjugates; 5.7 Absolute value; 5.8 Sums and products; 5.9 Powers; 5.10 Incorporation of the real numbers into the system of complex numbers