READING: II §3 DECIMALS
Back to reading:
II §3 LUB5
begin quote ita"
CHAPTER II [:] The Real Number System [...] §3. THE LEAST UPPER BOUND PROPERTY. "end quote
Please get the book and read along. topic index contents
We now discuss the decimal representation of real numbers.
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.
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
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.
If we add to this last number a lot of cancellation occurs, resulting in
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.
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.
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