Transreal Limits Expose Category Errors in IEEE 754 Floating-Point Arithmetic and in Mathematics

<p>The IEEE 754 standard for floating-point arithmetic is widely used in computing. It is based on real arithmetic and is made total by adding both a positive and a negative infinity, a negative zero, and many Not-a-Number (NaN) states. The IEEE infinities are said to have the behaviour of limits. Transreal arithmetic is total. It also has a positive and a negative infinity but no negative zero, and it has a single, unordered number, nullity.</p> <p>We elucidate the transreal tangent and extend real limits to transreal limits. Arguing from this firm foundation, we maintain that there are three category errors in the IEEE 754 standard. Firstly the claim that IEEE infinities are limits of real arithmetic confuses limiting processes with arithmetic. Secondly a defence of IEEE negative zero confuses the limit of a function with the value of a function. Thirdly the definition of IEEE NaNs confuses undefined with unordered. Furthermore we prove that the tangent function, with the infinities given by geometrical construction, has a period of an entire rotation, not half a rotation as is commonly understood. This illustrates a category error, confusing the limit with the value of a function, in an important area of applied mathematics -- trigonometry. We briefly consider the wider implications of this category error.</p> <p>Another paper proposes transreal arithmetic as a basis for floating-point arithmetic; here we take the profound step of proposing transreal arithmetic as a replacement for real arithmetic to remove the possibility of certain category errors in mathematics. Thus we propose both theoretical and practical advantages of transmathematics. In particular we argue that implementing transreal analysis in trans-floating-point arithmetic would extend the coverage, accuracy and reliability of almost all computer programs that exploit real analysis -- essentially all programs in science and engineering and many in finance, medicine and other socially beneficial applications.