Some neglected niches in the understanding and teaching of numbers and number systems

Cultural questions have attended arithmetic since it began to develop in ancient times. They include possible differences between integers and non-integral numbers and in operating with them, religious and mystical uses and interpretations, the roles of zero, extensions to infinite numbers, and representing numbers by numerals in ways which aid calculation (including the use of algebra). The selection of historical examples given here concentrates on aspects of numbers which are not well known but which could be used in teaching, either at school or undergraduate level. Comments on educational utility are given, mostly at the end of each section; the word “student” refers to learners of all ages.