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I'm a computer science graduate with interest in mathematics, and I'm trying now to read some textbooks as self learning.

I want to read about the extension of natural numbers to integers, fractions, real numbers and up to the complex numbers. I thought this belongs to abstract algebra, but I read the table of contents of some abstract algebra books and it contained nothing about that.

So what field of mathematics should I study to understand these issues? I always wanted to really understand complex numbers for example, but I'm not sure where to start. Is it complex analysis, abstract algebra or what?

Loai Ghoraba
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  • What's used in each step is the concept of equivalence relation and (induced by it) equivalence classes. – emacs drives me nuts Oct 04 '23 at 07:03
  • @emacsdrivesmenuts can you recommend a book? Or better, to what subfield do these concepts belong to? – Loai Ghoraba Oct 04 '23 at 07:43
  • The first 20 pages of these notes are worth a look: https://www.math.uni-konstanz.de/~krapp/research/Constructions_of_the_real_numbers.pdf This material should be in some number theory or analysis books. Maybe baby Rudin. “Constructing the real numbers from the integers” is a phrase to search for. Calculus by Spivak has a good explanation of constructing complex numbers from real numbers. – littleO Oct 04 '23 at 07:50
  • If you want to start at the very beginning, you might consider "Foundations of Analysis" by Landau. – awkward Oct 04 '23 at 13:03

2 Answers2

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There are numerous youtube videos and book sections on how N, Z, Q, R, and C all came to be. One book that contains this info is Integers, Polynomials, and Rings, by Ron Irving. I believe you may be able to view it in pdf form online for free by using the books option of Google search. All of these sets are introduced by the end of Algebra 2 in high school so I'd say it's really just part of standard high school algebra, and not necessarily any more advanced level of algebra. But again: youtube is your friend! I'm sure you can find videos that answer your question satisfactorily.

Nate
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Hint: The book Numbers by H.-D. Ebbinghaus, R. Remmert and five other authors provides a nice and readable introduction to numbers, which on the first 80 pages leads from natural numbers to complex numbers and then on to some other extensions. The text is enriched with a lot of historical information. You might also want to take a look at the table of contents.

Markus Scheuer
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