Counting the Floats

Book excerpt: Dear Reader, Let’s start with an important disclaimer !!! Please note that this offering is the first incarnation of what is a very important subject matter. It is also my “maiden voyage” into the wonderful world of self-publishing to online devices. Over 127 years or about six generations ago the famous Cantor’s Diagonal Argument (CDA for short ) entered set theory history. For some inexplicable reason, and in spite of steadfast opposition from giants of mathematics ( Gauss for example ) CDA was universally accepted. two of its predominant conclusions were published and taught ever since, without encountering any significant dissent. The first stated that floating point numbers ( floats for short ) constitute a higher order of infinity than integers. The second derivative claim insisted that it is not possible to count floats using integer counting agents. This was the status-quo up until the day when I got involved, armed with my trusted companion, Maplesoft’s mathematical development tool. CDA, with its vast supporting literature, proved no match for the two of us. After some false starts and a lot of contemplating, the body of set theory relevant to CDA and the countability of the floats came into focus. I managed to prove beyond any doubt the claim that the floats can be put into one-to-one correspondence with the positive integers. But I had no idea if it was possible to actually generate such a correspondence and whether it can be done on-demand for an arbitrary float no matter how large. Then I struck gold. A single serendipitous insight showed me the one-and-only-one PERFECT way floats could be counted. From that moment on it was only a clever sequence of steps which produced the float-to-integer algorithm and soon after its inverse as well. (Note : The first table on the cover of the book gives a strong hint what this magical insight was. ) Can you guess it ? If not, then within this book you will find the fascinating detailed analysis including unassailable proofs and demonstration of the one-to-one correspondence between floats and integer counting agents. Some of the topics and results we will present : 1. Dual Complete Tree structures 2. Level & Bracket pyramid design 3. The only possible way floats can be ordered for counting purposes 4. The detailed step-by-step construction of a very elegant two-way counting algorithms. 5. Many examples of actual counting sequences. 6. A claim that the integer sequence number for a single float representing a 50,000-digit approximation for PI was computed. Conclusion : It is wonderful that floating point numbers with a strict set of syntax constraints will finally rejoin the big happy family of countable sets taking their place next to integers. However the vast, unpredictable, truly unlimited cavalcade of algorithms produced by intelligent agents remain forever uncountable. This book was written at the level of difficulty usually found in quality recreational mathematics publications. No, I did not deliberately adjust the level. Instead, I was aiming to find the most elegant and streamlined treatment possible. I believe I succeeded in spades and it is this quality of the book I am most proud of. I hope that you will enjoy the book as much as I enjoyed creating it. Your feedback will be greatly appreciated as it will help me to make the next incarnation a better product. Tamas Varhegyi, author