The following is an account of a method of addition and some methods of subtraction which may appeal to teachers who like to lace their teaching with enrichment material, but who sometimes refrain from doing so because of a fear that an “off beat” treatment of a topic may confuse rather than clarify.

The set of whole numbers less than and coprime to n, with the operation of multiplication modulo n, forms an Abelian group (try to prove it for yourselves). We'll write Mn for this group, and call all such groups modulo groups or M-groups. M-groups are easy to compute. It is also particularly easy to calculate quotient groups of M-groups, making them a good source of examples and practice for the beginner. There are also some very hard, possibly unsolved, problems in classifying M-groups. In attacking such problems there is an interaction between number theoretical and group theoretical arguments which makes this a rich topic, for projects or personal pleasure.

Except for the circle, for which the true anomaly v is proportional to the time t, the position of a body in orbit about a central body at a given time is simplest to derive for a parabola. The classical determination of the time of flight on a parabolic trajectory is through the integration of the dynamic equations of motion. (See Appendix.)

Some students on first meeting the axioms of a vector space are puzzled by the inclusion of the scalar unit law: 1v = v for all vectors v. It is the purpose of this note to discuss the consequences of omitting this law from the list of axioms.

There are many families of polynomials in mathematics, and they often occur naturally in pairs. The Fibonacci polynomials and the Lucas polynomials, for example, are generated by the same recurrence relation but with different starting values, and there are many identities that link the two families [1]. The same is true for the Chebyshev polynomials of the first and second kinds, Tn (x) and Un (x) [2], respectively. There are two further polynomial families that are less well-known, the Chebyshev polynomials of the third and fourth kinds, Vn (x) and Wn (x) [3], respectively. Each of the four kinds is an example of an orthogonal polynomial family Pn (x), where for some appropriate weight function W (x), whenever n ≠ m. The families Tn (x) and Un (x) in particular are ubiquitous in their mathematical uses, in approximation theory, in differential equations, and in solving the Pell equation, to name but three. There are also many connections between Tn (x), Un (x), Vn (x) and Wn (x), some of which are explored here, and some of which we hope are new.

(a) The foundation of calculus operations on the circular functions is

The traditional steps by which the inequality