A lot of books devoted to the development of the real number system (and other number systems) were published in the 1960s. During the past few years I've been keeping a list of such books (regardless of when they were published) when I happen to see one in a library, and the ones I've come across so far are listed below.
Note that if we expand the list to books in real analysis, "transition to advanced mathematics", abstract algebra, metric spaces and/or topology, etc. that include such a treatment, the list of such books would be at least several hundred. Thus, I've restricted the list below to only those books that are primarily devoted to this topic. The links I've provided are not necessarily to the same edition that I've given information about.
Besides what others have recommended, you can try visiting a nearby college or university library and look for some of these books. Some of these books will probably be more useful to you than others for what you want, and you will likely find other books in the same general shelving location that are not in the list below.
 Leon Warren Cohen and Gertrude Ehrlich, The Structure of the Real Number System, The University Series in Undergraduate Mathematics, D. Van Nostrand Company, 1963, viii + 116 pages.
 Solomon Feferman, The Number Systems. Foundations of Algebra and Analysis, Addison-Wesley Publishing Company, 1964, xii + 418 pages.
The 2nd edition was published by Chelsea Publishing Company in 1989 (xii + 418 pages).
 Norman Tyson Hamilton and Joseph Landin, Set Theory and the Structure of Arithmetic, Allyn and Bacon, 1961, xii + 264 pages.
 Edmund Jecheksel Landau, Foundations of Analysis, 1951, Chelsea Publishing Company, 1951, xiv + 134 pages.
Translation by Fritz Robert Steinhardt of the 1930 German edition (xiv + 134 pages).
 Elliott Mendelson, Number Systems and the Foundations of Analysis, Academic Press, 1973, xii + 358 pages.
Reprinted by Robert E. Krieger Publishing Company in 1985 (xii + 358 pages). Reprinted by Dover Publishers in 2008 (xii + 308 pages).
 John Meigs Hubbell Olmsted, The Real Number System, Appleton-Century Monographs in Mathematics, Appleton-Century-Crofts, 1962, xii + 216 pages.
 Francis Dunbar Parker, The Structure of Number Systems, Teachers. Mathematics Reference Series, Prentice-Hall, 1966, xiv + 137 pages.
 Joseph [Joe] Buffington Roberts, The Real Number System in an Algebraic Setting, A Series of Undergraduate Books in Mathematics, W. H. Freeman and Company, 1962, x + 145 pages.
 Hugh Ansfrid Thurston, The Number-System, Interscience Publishers, 1956, viii + 134 pages.
Reprinted (slightly corrected) by Dover Publications in 1967 and 2007.