Transreal number

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Transreal numbers are a concept created by James Anderson of the University of Reading. They were first mentioned in a 1997 publication^[1], and made well-known on the Internet in 2006, but not recognized by the math community. These numbers are used in his concept of transreal arithmetic. This idea of Mr. Anderson was introduced to public by BBC with their report^[2], and their subsequent follow-up in response to many claims that the theory is flawed.^[3]

According to Anderson, transreal numbers make up the set of numbers used in “transreal arithmetic”. They include all of the real numbers, plus three others: infinity ( $\infty$ ), negative infinity ( $-\infty$ ) and nullity ( $Φ$ ), a numerical representation of a non-number.

1 What transreal arithmetic is
2 What transreal arithmetic is not
3 Differences between transreal and standard arithmetic
4 References
5 Further reading
6 See also

[edit] What transreal arithmetic is

The axioms of transreal arithmetic are intended to complement the axioms of standard arithmetic; they are supposed to produce the same result as standard arithmetic for all calculations where standard arithmetic defines a result. In addition, they are intended to define a consistent numeric result for the calculations which are undefined in standard arithmetic, such as division by zero.^[4]

Transreal arithmetic resembles IEEE floating point arithmetic, a floating point arithmetic commonly used on computers. In IEEE floating point arithmetic, calculations such as zero divided by zero can produce a result, NaN, to which the standard arithmetic axioms do not apply (as it is not a number).

[edit] What transreal arithmetic is not

In December 2006, Anderson was featured on a BBC television segment teaching schoolchildren about his concept of "nullity".^[2] The report implied that Anderson had discovered the solution to division by zero, rather than simply attempting to formalize it. The report also suggested that Anderson was the first to solve this problem, when in fact the result of zero divided by zero has been expressed formally in a number of different ways (for example, see NaN).

[edit] Differences between transreal and standard arithmetic

In transreal mathematics, several calculations including infinity are axiomatically defined. There is also a strictly transreal number, nullity. Represented by $Φ$ , nullity lies outside of the extended real number line: that is, it is a numeric representation of a non-number or an indeterminate form. Here are some identities in transreal arithmetic:

$1 \div 0 = \infty$
$0 \div 0 = \Phi$
$\infty \times 0 = \Phi$
$\infty - \infty = \Phi$
$\Phi + a = \Phi \$
$\Phi \times a = \Phi$
$\Phi = \Phi \$ is True (contrast this with NaN $\neq$ NaN in IEEE floating-point arithmetic)
$\Phi = -\Phi \$

Anderson's analysis of the properties of transreal algebra is given in his paper on "perspex machines". ^[5]

However, due to the more expansive definition of numbers in transreal arithmetic, several identities which apply to all numbers in standard arithmetic are not universal in transreal arithmetic. For instance, in transreal arithmetic, $a - a = 0$ is not true for all $a$ , since $Φ - Φ = Φ$ . That problem is addressed in ref. 3 pg. 7.

Division by zero (or other similiarly undefined operation) may yield many different results, varying from 0 to $\infty$ , when it occurs as a result of taking limit of a function or limit of a sequence.

[edit] References

^ Representing Geometrical Knowledge.
^ ^a ^b "1200-year-old problem "easy"", BBC News, 2006-12-06. “Schoolchildren from Caversham have become the first to learn a brand new theory that dividing by zero is possible using a new number — "nullity". But the suggestion has left many mathematicians cold.”
^ "Nullity is a number, and that makes a difference", BBC News, 2006-12-12.
^ James Anderson (2006). Longin Jan Latecki , David M. Mount, and Angela Y. Wu. "Perspex Machine VIII: Axioms of Transreal Arithmetic" (PDF). Vision Geometry XV: Proceedings of SPIE.
^ James Anderson (2006). Longin Jan Latecki , David M. Mount, and Angela Y. Wu. "Perspex Machine IX: Transreal Analysis" (PDF). Vision Geometry XV: Proceedings of SPIE.