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Transcomplex Numbers: Properties, Topology and Functions

Tiago S. dos Reis, and James A.D.W. Anderson, "Transcomplex Numbers: Properties, Topology and Functions," Engineering Letters, vol. 25, no. 1, pp90-103, 2017.

We derive the properties of transcomplex arithmetic from the usual definition of transcomplex numbers as a fraction of complex numbers, whose denominator may be zero. This is equivalent to giving an axiomatisation. In particular we characterise the partial associativity of transcomplex addition and the partial distributivity. We describe specifically how the transcomplex numbers depart from field structure and relate this to earlier work on transfields. We review the transcomplex elementary functions and the topology of transcomplex numbers. Thus armed we extend several functional properties of the complex numbers to the transcomplex numbers.

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PhD places in Transmathematics

I can accept self-funded, full-time or part time PhD students in transmathematics:

Students can obtain a degree from Reading University by studying at the university or by studying mainly abroad.

Reading University:

Department of Computer Science:

I can also co-supervise PhD students who study entirely abroad.

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Last week I gave a lecture on Transmathematics and division by zero to the Program of History of Science, Technique, and Epistemology, in Federal University of Rio de Janeiro, Brazil on Wednesday 23th November 2016.

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Naive Set-Theory Without Paradox!

Research seminar at the University of Reading on 23rd November 2016.!.pdf

I would really like to collaborate with a set-theory specialist on this! If you are a set-theory specialist, get in touch! If you know a set-theory specialist then pass this request along!

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Um pouco de transmatemática: números transcomplexos (A little on transmathematics: transcomplex numbers)

REIS, T. S. dos; Um pouco de transmatemática: números transcomplexos. In: Scientiarum Historia VIII, 2016, Rio de Janeiro. Proceedings Scientiarum Historia VIII, 2016.

I presented an oral communication. The paper will be published in the proceedings: .

We present some results obtained in other works. We bring to the table the set of the transcomplex numbers, a set that allows division by zero and is an extension of the usual complex numbers. Numbers such as transcomplexes and transreals are called transnumbers. The mathematics that arises from the transnumbers is called transmathematics. We present the transcomplex arithmetic, that is, we show how one can carry out the arithmetic operations between numbers that allow division by zero. Of course, we even show how to divide any complex number by zero. Finally, we discuss some developments of this new set.

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Transreal truth valued square of opposition

Tiago S. dos Reis, Walter Gomide, and James A.D.W. Anderson, Transreal truth valued square of opposition. In 5th World Congress on the Square of Opposition, Easter Island, Chile. Handbook of Abstracts 5th World Congress on the Square of Opposition, 2016.

I presented an oral communication in the 5th World Congress on the Square of Opposition.

Our transreal truth valued square of opposition is a square made from total semantics of transreal numbers. This square is continuous at some points and discontinuous at other ones. Transreal numbers -∞, 0, ∞ and Φ, in this order, are the vertices. The side between the vertices 0 and -∞ is made of a semi-straight line which begins at 0 and goes indefinitely with all negative real numbers toward to -∞. In same way the side between the vertices 0 and ∞ is made of a semi-straight line which begins at 0 and goes indefinitely with all positive real numbers toward to ∞. The side between the vertices -∞ and Φ is made of a gap. In same way the side between the vertices ∞ and Φ is made of a gap. In this square the vertices are not the only truth values, but the continuous part of the sides are also truth values. Note that -∞, 0, ∞ and Φ play the role of Belnap values F, δ, T and γ respectively, the side between 0 and -∞ plays the role of fuzzy value of falsehood and the side between 0 and ∞ plays the role of fuzzy value of truthfulness.
In transreal truth valued square of opposition, the opposite values are the intersections of the square with lines parallel to the diagonal between -∞ and ∞. In this way, symmetrical real numbers are opposites, -∞ and ∞ are opposites, 0 is opposite to 0 and Φ is opposite to Φ. This is according to trasnreal negation defined elsewhere, where ¬(x)=-x (transreal arithmetic gives -0=0 and -Φ=Φ). Let us consider the triangle –x0x whose hypotenuse is -xx. This triangle represents the “paraconsistent negation”. The “paraconsistent triangle” could be increased, and in the limit, when -x=-∞ and x=∞, we have the “classical triangle” -∞0∞. In this way, negation is a logical operation that “diagonalizes” truth values. Every logic that has contradiction could be seen as an operation in a triangle, such that operation goes from the position x to the antipodal position -x. Contradiction must be defined in a triangle whose origin is a dialetheia – a true contradiction.

There was a new research monograph "An Introduction To S-Structures And Defining Division By Zero" by Brendan Santangelo. Basically defining a mathematical field, the S-structure is from his surname.

The link is

Interesting research in the area of division by zero. I would love to know other comments, insights, and input.

Thanks, my best! Will

Hi everyone:

The General Science Journal has published the research paper "Division by Zero Fallacies using Transmathematics" which is an improved version of an original work. Kudos and thanks to Dr. James Anderson and Ilija Barukčić for their thoughtful comments and feedback.

The paper is here:{$cat_name}/View/6667

I welcome commentary and input. My best and thanks!

William Gilreath

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Construction of the Transnatural Numbers in Transset Theory

Comment and collaboration is invited on the SECOND version of the paper.

We assert that the universe of all classes is partitioned by the universe of all sets and the universe of all antinomies. The usual set operations are applied to all classes, giving a total transset-theory that contains a set theory and an antinomy theory. Transset theory is just naive set-theory with antinomies so it is pedagogically simple.

We define that a transset that contains only itself is an atom and that all other transsets, including the empty transset, are molecules. Transset theory can model all set-theories, including those with atoms, and the whole of category theory, using atoms as category objects and molecules as category relations.

We construct the transnatural numbers such that: the natural numbers are von Neumann ordinals; transnatural infinity, being the greatest ordinal, is built from the universal set; and transnatural nullity, being the only unordered, transnatural number, is built from the universal antinomy.

We extend transnatural arithmetic to transordinal arithmetic and show that the classical paradoxes of set theory are dissolved in transset theory, with counting that is consistent with transordinal arithmetic.

We use the fact that all antinomies have subsets to provide a foundation for paraconsistent logics and to explain how scientific theories can be useful, despite having both internal contradictions in explanations and external contradictions with observations.

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