This discussion focuses on recommended texts for understanding the construction of number systems, specifically the transition from natural numbers to complex numbers. Two primary books are highlighted: "Grundlagen der Analysis" by Edmund Landau, which employs Dedekind cuts, and "Mathematical Analysis" by H. A. Thurston, which utilizes Cauchy sequences. Landau's work is noted for its dryness and lack of exercises, while Thurston's book balances motivational content with formal definitions and includes exercises without solutions. Both texts provide valid approaches to defining real numbers.
PREREQUISITESMathematics students, educators, and anyone interested in the foundational aspects of number systems and mathematical analysis.
Thanks, I think I'll pick up Thurston's book eventually.Petek said:Here are two books that cover what you asked for:
https://www.amazon.com/dp/082182693X/?tag=pfamazon01-20 by Edmund Landau
https://www.amazon.com/dp/0486458067/?tag=pfamazon01-20 by H. A. Thurston
Landau's book is well-known. However, as some of the Amazon reviews point out, it is extremely dry. It contains little, if any, motivational material and no exercises.
Thurston, on the other hand, consists of approximately one-half motivational material and one-half formal definitions, theorems and proofs. It also contains exercises, but no solutions.
Landau uses Dedekind cuts to define the real numbers, while Thurston uses Cauchy sequences. Both approaches are valid, but the Dedekind cuts approach is probably more commonly seen elsewhere.
So, you might want to see if your library has both books and decide which one you like the most.
Petek