The IEEE 754 standard for
oating-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.
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 con-
struction, 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 mathe-
matics { trigonometry. We brie
y consider the wider implications of this
category error.
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 theo-
retical and practical advantages of transmathematics. In particular we
argue that implementing transreal analysis in trans-
floating-point arith-
metic 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