READING:     II     2     O9                    Back to reading:     II     2     O8
begin quote ita"
CHAPTER II [:] The Real Number System [...] 2. ORDER. "end quote
Please get the book and read along.    
topic index     contents

O9: THE RULES OF ARITHMETIC etc

In this reading, I will not try to improve on either Rosenlicht's presentation or my earlier readings. To me, these topics represent many decades of learning what I call number crunching, which is distinct from the mathematics which is being presented here. The mathematics is a logical foundation. If we have the foundation first, it can be used as a very powerful aid in the process of learning the number crunching by seeing problems solved and then doing similar problems.

THE NATURAL NUMBERS, SUMS AND PRODUCTS:   When we had only the field properties, we had only two real numbers,   0   and   1   as discussed on my field of two elements page from my first reading of Rosenlicht's second chapter. Also in the first reading and after the order property consequences, I wrote a page to define the natural numbers and to describe how the set of real numbers consequently becomes infinite here. This included a discussion of Rosenlicht's first two field consequences, F1 and F2. In accordance with my chapter two preview in this reading, F1 and F2 were postponed until the present page. On re-reading, I am satisfied simply to link the earlier work. I think it was necessary in this reading completely to postpone F1 and F2 because I am removing the prerequisites as described at the top of the contents page.

FRACTIONS, INTEGERS, AND RATIONAL NUMBERS:   Rosenlicht presents fractions after F9 under the field properties but uses F01 and F02 which, in this reading, have just been covered above. Then it seems to me that my earlier work presenting Rosenlicht's results can be used here. Then the integers and rational numbers are derived here.

INTEGRAL EXPONENTS:   Please go here.

ABSOLUTE VALUES:   For previous readings, please go here. New readings:

 1 |a|0,  2 |ab|=|a|·|b|,  3 |a|²=a²,  4 |a+b||a|+|b|,  5 |a-b|||a|-|b||,
generalized triangle inequality
ε-neighborhoods

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