begin quote ita"
CHAPTER II [:] The Real Number System [...] §3. THE LEAST UPPER BOUND PROPERTY. "end quote
topic index     contents

DECIMALS

begin quote ita-2-3-11-pg26 "
We now discuss the decimal representation of real numbers.
"end quote

This is the place where we should reconstruct old fashioned arithmetic from Rosenlict's axiomatic approach to the real number system. However, that is a project for a future date - the date of this writing being 26 July 2013.

begin quote ita-2-3-11-pg26"
First consider finite decimals. If     is any integer, and     any integers chosen from among   0, 1, 2, ..., 9,   the symbol     will mean, as usual, the rational number

.

If   m   is a positive integer less than   n,   then

.
"end quote

Note immediately above that if   n = m+1   then there will be only one term beyond the mth. Note also that     should be a positive integer, because for example   -3 + 0.7 = -2.3   which is not of the form,   ,   given above.

begin quote ita-2-3-11-pg26"
If we add     to this last number a lot of cancellation occurs, resulting in

.

This last inequality is at the base of most rounding-off procedures in approximate calculations and in addtion shows that two numbers in the above decimal form are equal only if (except fo the possible addition of a number of zeros to the right, which doesn't change the value of the symbol) they have the same digits in corresponding places. It also enables us to tell at a glance which of two numbers in the given form is larger. The ordinary rules for adding and multiplying numbers in this form are clearly legitimate.
"end quote

Clearly, for Rosenlicht's purposes, it would be outside the scope of the book to look in detail at the "ordinary rules" for addition and subtraction and multiplication and division of numbers in decimal form. But, for my purpose, such exercises will be very important. However, I will not do them here. I will wait to do these exercises with other people as collaborators.

Rosenlicht now considers infinite decimals.

begin quote ita-2-3-12-pg27 "
By an infinite decimal we mean a formal expression     (this is just another way of writing a sequence) where     is an integer and each of     is one of the integers   0, 1, ..., 9.   The set     is nonempty and bounded from above (for any integer ,     is an upper bound) hence has a least upper bound.
"end quote

When I first looked at the set which Rosenlicht defines in the previous paragraph, I was still thinking of   n   as a fixed positive integer. A little reflection soon reminded me that Rosenlicht means something different, and I present in the image immediately below what I think Rosenlicht intended.

Larger versions of the above image are here: 500by606, 600by727, 700by848, 800by970, and 920by1115. Versions of the image in shades of gray, rather than orange, are here: 500, 600, 900. The original images are orange here and gray here.

on to §4: The Existence of Square Roots