Construction of the Number Systems ... Natural, Integers, etc

Ethan D Bloch's "The Real Numbers and Real Analysis" for a detailed and rigorous treatment of number systems, but notes that some explanations may not be clear. Jambaugh recommends Conway's "On Numbers and Games" as an alternative approach. In summary, the conversation discusses the various options for understanding number systems, with Bloch's book being recommended for its detail and rigor, but also acknowledging potential clarity issues. Conway's book is also mentioned as a different approach to consider.
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At present I am trying to understand the construction of the number systems ... natural, integers, rationals and reals ...

What do members of PFs think is the clearest, most detailed, most rigorous and best treatment of number systems in a textbook or in online notes ... ?

NOTE: I am currently using Ethan D Bloch: "The Real Numbers and Real Analysis" ... ... where the coverage is detailed ... and proofs in particular are detailed and in full ... but some of the explanations are not particularly clear ... ...Peter
 
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I think most any analysis text will be sufficiently clear and rigorous. The only odd step is the filling out of the real numbers as Dedekind cuts on the set of all rational numbers.

Now an interesting approach which extends much further is Conway's "On Numbers and Games".
 
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Thanks Jambaugh ... appreciate your help ...

Peter
 

1. What is the purpose of constructing number systems?

The purpose of constructing number systems is to have a consistent and organized way of representing and manipulating numbers. It allows for a deeper understanding of mathematical concepts and facilitates problem-solving in various fields such as science, engineering, and finance.

2. What is the difference between natural and whole numbers?

Natural numbers are the counting numbers starting from 1, while whole numbers include 0 in addition to the natural numbers. In other words, whole numbers are a superset of natural numbers.

3. How are integers constructed?

Integers are constructed by including all natural numbers, their negatives, and 0. This creates a number line that extends infinitely in both directions.

4. What is the significance of rational numbers?

Rational numbers are significant because they represent the ratio of two integers and can be used to describe quantities and values that are not whole numbers. They are also crucial in understanding fractions, decimals, and percentages.

5. How is the real number system constructed?

The real number system is constructed by including all rational and irrational numbers. Irrational numbers are those that cannot be expressed as a ratio of two integers, such as pi and square root of 2. This system is continuous and can represent all possible values on a number line.

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