James Anderson (computer Scientist) - Transreal Arithmetic - Transreal Arithmetic and Other Arithmetics

Transreal Arithmetic and Other Arithmetics

"Transreal arithmetic" closely resembles IEEE floating point arithmetic, a floating point arithmetic commonly used on computers. IEEE floating point arithmetic, like transreal arithmetic, uses affine infinity (two separate infinities, one positive and one negative) rather than projective infinity (a single unsigned infinity, turning the number line into a loop). Division of any non-zero finite number by zero results in either positive or negative infinity.

However, in IEEE arithmetic, division of zero by zero is still considered indeterminate. The reason for this is simple: A statement about the quotient of two numbers is understood in mathematics as another statement about multiplication. Specifically, if

this is understood as simply another way of saying that

Thus, if for some number

then this is just another way of saying that

But in fact this is true for all real numbers . And that is precisely the reason that mathematicians do not assign a single value to but rather label it "indeterminate". Assigning a value to, even a newly fabricated "number", misses the point entirely.

In IEEE arithmetic, the value of is therefore represented by the symbol Not a Number (NaN) (Not a Number). NaN is not meant to be a number, but rather an error message conveying the fact that the arithmetical operation the computer just attempted cannot be assigned a single number as an answer – even if and are considered numbers. Because is an error message and not a number, it is not considered equal to anything, even itself. That is, the comparison evaluates to false.

Here are some identities in transreal arithmetic with the IEEE equivalents:

Transreal arithmetic IEEE standard floating point arithmetic
(i.e. applying unary negation to NaN yields NaN)

The main difference between transreal arithmetic and IEEE floating-point arithmetic is thus that nullity compares equal to nullity, whereas NaN does not compare equal to NaN.

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

Due to the more expansive definition of numbers in transreal arithmetic, several identities and theorems which apply to all numbers in standard arithmetic are not universal in transreal arithmetic. For instance, in transreal arithmetic, is not true for all, since . That problem is addressed in ref. pg. 7. Similarly, it is not always the case in transreal arithmetic that a number can be cancelled with its reciprocal to yield . Cancelling zero with its reciprocal in fact yields nullity.

Examining the axioms provided by Anderson, it is easy to see that any term which contains an occurrence of the constant is provably equivalent to . Formally, let be any term with a sub-term, then is a theorem of the theory proposed by Anderson.

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