Mathematics Journal: Division of Zero by Itself - Division of Zero by Itself Has Unique Solution
Issue:
Volume 7, Issue 3, June 2018
Pages:
20-36
Received:
17 April 2018
Accepted:
14 May 2018
Published:
8 August 2018
Abstract: The division by zero has been a challenge over years, which is in two forms: one involves a non-zero numerator while the other involves a zero numerator. This work deals with the second form of division, with the aim of finding a solution to the equation obtained when the expression is equated to, say x, where x is not a quantity but the ‘number of times of one whole’. In this work, zero divided by itself has been exhausted using different approaches and methods to come to a conclusion; that this division has a unique solution, 1. Some of the methods employed include geometric series, logarithm, indices, reciprocals, factorials, self-operations, Euler’s number, binomial expansions, graphical method among others. The conclusion has been made that zero divided by zero is 1. The reverse of division by multiplication is not applicable because zero has been associated with two ‘abnormal’ properties or behaviour that’s not with other numbers.
Abstract: The division by zero has been a challenge over years, which is in two forms: one involves a non-zero numerator while the other involves a zero numerator. This work deals with the second form of division, with the aim of finding a solution to the equation obtained when the expression is equated to, say x, where x is not a quantity but the ‘number of...
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Entropy-Like State Counting Leads to Human Readable Four Color Map Theorem Proof
Robert Luckett Shuler Jr.
Issue:
Volume 7, Issue 3, June 2018
Pages:
37-44
Received:
24 August 2018
Accepted:
7 September 2018
Published:
28 September 2018
Abstract: The problem of how many colors are required for a planar map has been used as a focal point for discussions of the limits of human direct understanding vs. automated methods. It is important to continue to investigate until it is convincingly proved map coloration is an exemplary irreducible problem or until it is reduced. Meanwhile a new way of thinking about surfaces which hide N-dimensional volumes has arisen in physics employing entropy and the holographic principle. In this paper we define coloration entropy or flexibility as a count of the possible distinct colorations of a map (planar graph), and show how a guaranteed minimum coloration flexibility changes based on additions at a boundary of the map. The map is 4-colorable as long as the flexibility is positive, even though the proof method does not construct a coloration. This demonstration is successful, resulting in a compact and easily comprehended proof of the four color theorem. The use of an entropy-like method suggests comparisons and applications to issues in physics such as black holes. Therefore in conclusion some comments are offered on the relation to physics and the relation of plane-section color-ability to higher dimensional spaces. Future directions of research are suggested which may connect the concepts to not only time and distance and thus entropic gravity but also momentum.
Abstract: The problem of how many colors are required for a planar map has been used as a focal point for discussions of the limits of human direct understanding vs. automated methods. It is important to continue to investigate until it is convincingly proved map coloration is an exemplary irreducible problem or until it is reduced. Meanwhile a new way of th...
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