Overview

What is Number? explores the hitherto long taboo topic of division by zero. This book is suitable for college-level students majoring in mathematics, although the best effort was given to make it suitable for the everyday person. The book is concise and provides practice problems and solutions to increase accessibly at the level of independent study, while also includes a method of teaching the content to guide the reader in attaining understanding. We explore division by zero by asking first precisely what numbers are. We then develop this system by using rectangles and their properties. What we end up with is a unification of arithmetic and basic geometry. Then we take what we have established and applied it to studies of collections of objects, initially finite collections, then expand our understanding to infinite collections. Immediately after, we attack a long-standing problem in set theory that has troubled mathematicians, logicians, and the layman - the problem that revolves around the logical soundness of the proof of there existing infinitely many infinities each larger than the previous and our intuition of this notion. What this book is not is a definite solution to the problem of division by zero; rather, What is Number? is a development that pushes the boundaries of our understanding infinity and zero, highlighting their duality. Several key problems must be resolved before we can move forward. Essentially these problems can be reduced to two. The first is stated simply by asking, What connection exists with calculus? The second requires we consider PEMDAS, a version of an accepted order of operations for arithmetic calculations. As many mathematicians are aware, our accepted order of operations is arbitrary. So we ask, Does there exist a natural order of operations, and if there does, what is it? If you are considering purchasing What is Number?, I encourage you to do so now. This is the 3rd and final edition of this book. For the reader, I leave open problems in probability theory and a promising conjecture regarding the numerosity of the set of prime numbers.

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