Manifesto for Transreal Arithmetic

1. Manifesto for Transreal Arithmetic Manifesto for Education Primary schools throughout the world teach: But this is false, in real arithmetic, because it allows division by zero. Secondary schools and universities throughout the world teach: Where a is positive and finite. But this is false, as this counter example shows: These falsehoods arise because teachers do not teach the guarding clauses that are needed to exclude the exceptional cases where real arithmetic does not apply. Manifesto: conduct research to see if it is more effective to continue teaching arithmetic via falsehoods or whether it is better to teach a total arithmetic. Manifesto for Physics Physical equations that involve division by zero are said to have poles or singularities. Real arithmetic breaks down in these cases, but it is sometimes possible to get useful information by examining the locations where poles and singularities occur. If the physical universe operates according to real arithmetic, how does it know it should break down when dividing by zero, and how does it know what to do in these cases? If physical equations are written in transreal arithmetic, and its generalisations, then thy never break down. The universe can continue to work, even if it divides by zero. Manifesto: conduct research to see if it is useful to re-work physical equations in transreal arithmetic and its generalisations. Manifesto for Computing Transreal arithmetic is a total arithmetic. This means that every arithmetical operation can be applied to any numbers and the result is a number. Transreal arithmetic defines division by: In particular, this defines division by zero. Transreal arithmetic reduces numbers with a zero denominator to least terms like this. Here k is positive: Transreal arithmetic defines addition by: Except that: It has been proved that transreal arithmetic is self consistent and that it contains real arithmetic as a subset. This means that transreal arithmetic computes the same results as real arithmetic, wherever real arithmetic applies, but it also computes consistent results where real arithmetic does not apply. Transreal arithmetic generalises trigonometry, calculus and metric spaces so all practical computations are made total. Transreal arithmetic generalises complex, quaternion, and octonion arithmetics so all arithmetics used in engineering and physics are made total. Transreal arithmetic specialises all of the arithmetics used in computers so all computable functions are made total. Manifesto: use total arithmetics in computers because they always work. Use transreal arithmetic, and its generalisations, because they are the simplest arithmetics that make all practical computations total. • Contact information • School of Systems Engineering, University of Reading, Whiteknights, Reading, England, RG6 6AY • Email: j.anderson@reading.ac.uk • www.reading.ac.uk/sse/about/Staff/j-anderson.asp