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Division by Zero and Transreal Numbers: The Computing Giving
Rise to a Conceptual Discussion in Mathematics
The current computing machines have a limitation of processing that are the arithmetic
division by zero exceptions. This causes overspending of time and energy since that is necessary
to configure exceptions. A novel computer which has no exceptions have been developed. This
computer is based on transreal numbers (ANDERSON, 1997, 2005). The set of transreal
numbers, proposed by James Anderson1 , is an extension of the set of the real numbers. In
this new set, division by zero is allowed. James Anderson postulates, in addition to the real
0
−1 1
, and and he calls the real numbers added
numbers, the existence of three new numbers,
0 0
0
T
these three new elements as transreal numbers, R (ANDERSON, VÖLKER, ADAMS, 2007).
The idea of the transreal numbers has not been easily accepted among mathematicians.
Possibly one reason for this resistance is the fact that the transreal numbers allows division by
zero which is ingrained in the human mind not be possible. It is interesting to note that this
is a recurring event in the history of science. Anderson introduces a novel concept of number
in axiomatic way. Remember that this is a common process in the mathematics history. For
example, when Bombelli2 found the square root of a negative number, he had the audacity
to operate this object assuming that the usual arithmetic properties were valid. Bombelli did
not give the rigor of the current mathematics, but subsequent mathematicians have established
in constructive and rigorous way the, today so accepted and important in mathematics and
various sciences, set of complex numbers. The real numbers themselves were initially conceived
intuitively. Until a certain time there was only the idea that each point on the line corresponds
to a number. However a formalization of the real numbers was given by Dedekind3 via the
notion of cut in the set of rational numbers. Another emblematic example in the history of the
evolution of the numbers are the infinitesimals. Leibniz4 developed his calculus under the idea
of infinitesimal numbers. Without a rigorous definition of infinitesimal, Leibniz suffered hard
1
James A. D. W. Anderson currently is teacher and research at School of Systems Engineering, University
of Reading in England.
2
Rafael Bombelli: Italian mathematician who lived in the sixteenth century.
3
Richard Dedekind: German mathematician who lived in the nineteenth century.
4
Gottfried Wilhelm Leibniz: philosopher, mathematician and German diplomat who lived in the seventeenth
and eighteenth centuries.
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criticism and just in 1960s, Robinson5 gave a construction of infinitesimal numbers from the
real numbers (EVES, 1990).
Similarly currently happens with transreals. In the history of mathematics new notions of
number have frequently been required to deal with advances in physics. The computer, and
the shift from analytic to an algorithmic conception of mathematics, has had, is having, a
similar impact. In this text we consider an extension of the notion of number motivated by
the desire to give a valid result to all possible arithmetical operations on a computer, such
0
as , which are considered invalid in mathematical, but which can arise very naturally due
0
to overflow and underflow on a computer. Since the emergence of these new numbers, the
transreal mathematics has been developed. Some studies on this subject have already been
published and others are in preparation. For example, transreal functions, transreal topology, transcomplex numbers and transmetric have been proposed (ANDERSON, 2007, 2008,
2011), (GOMIDE, REIS, 2013), construction of the transreal numbers from the real numbers
and construction of the transcomplex numbers from the complex numbers are under review for
publication, furthermore, transreal analysis and trans-Newtonian physics operating at singularities are in preparation. We propose the algebraic structure transfield. In the ordinary algebra,
the structure field generalizes the notion of real number. A field is an set provided with two
arithmetic operations called as addition and multiplication such that both are total, satisfy the
commutative, associative and distributive properties and have the inverse operation, except
multiplication by the inverse of zero. We propose the algebraic structure transfield which both
is a superset of the field and generalizes the concept of transreal number.
We define a transfield T as a set provided with two binary operations + and × and two unary
operations − and
−1
, such that: (a) T is total for +, ×, − and
−1
; (b) + and × are associative
and commutative; (c) There exists F ⊂ T such that (F, +, ×) is a field and F and T have
common additive and multiplicative identities; (d) For each x ∈ F , −x coincides with additive
inverse in F and for each x ∈ F \{0}, x−1 coincides with the multiplicative inverse in F (0 denote
the additive identity). In a suitable sense, we define complete ordered transfield, absorptive
element and extremal infinity and we show that the set of transreal numbers is the smallest
5
Abraham Robinson: American mathematician who lived in the twentieth century.
2
complete ordered transfield with absorptive element and extremal infinity which contains the
set of real number as a complete ordered field. That is, if T ⊃ R is a complete ordered transfield
with absorptive element and extremal infinity, then T contains a set isomorphic to RT .
It is interesting to note that a non-mathematician was required for an idea of definition
for division by zero arise. We conjecture that, as mathematicians are already so entrenched
with their rules, they fail to envision possibilities that are beyond the already established
status. Mathematics is known as an exact science. However, with the exposed here, we realize
that even a mathematical concept that basic is mutable, not unchanging. The study of the
transreals is still at an early stage, this allows us ask what are the philosophical and conceptual
impacts of introducing new numbers. As previously mentioned, several issues are being studied
in transmathematics. Remember that this new conception of number was provoked by the
computer. It illustrates the revolutionary conceptual, philosophical impact of the computer.
References
ANDERSON, J. A. D. W. “Representing geometrical knowledge”. Philosophical Transaction
of The Royal Society B, 352, 1997.
ANDERSON, J. A. D. W. “Perspex Machine II: Visualisation”. Vision Geometry XIII Proceedings of the SPIE, 5675, 2005.
ANDERSON, J. A. D. W., VÖLKER, N., ADAMS, A. A. “Perspex Machine VIII: Axioms of
transreal arithmetic”. Vision Geometry XV Proceedings of the SPIE, 6499, 2007.
ANDERSON, J. A. D. W. “Perspex Machine IX: Transreal Analysis”. Vision Geometry XV
Proceedings of the SPIE, 6499, 2007.
ANDERSON, J. A. D. W. “Perspex Machine XI: Topology of the Transreal Numbers”. In:
IMECS 2008: international multiconference of engineers and computer scientists. International
Association of Engineers, Hong Kong, 2008.
ANDERSON, J. A. D. W “Evolutionary and revolutionary efects of transcomputation”. In:
2nd IMA Conference on Mathematics in Defence, Institute of Mathematics and its Applications,
2011.
EVES, H. An Introduction to the History of Mathematics. Saunders, 1990.
GOMIDE, W., REIS, T. S. “Números transreais: Sobre a noção de distância”. Synesis, Universidade Católica de Petrópolis, 5(2), 2013.
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