Mathematics at all levels depends on infinite processes. This is a “fact of life”. But, like those other “facts of life”, this one has traditionally been hushed up—especially when teaching elementary mathematics to beginners. Of course, in our liberated age, adult mathematicians are no longer embarrassed by infinite processes, and even discuss them openly. But it is not easy to see the connection between the way they discuss infinite processes (as in, say, an undergraduate analysis course) and those parts of elementary mathematics which depend on infinite processes, and which should provide the soil in which the beginner’s seedling intuitions about infinite processes can take root. So even today most children, adolescents, and their teachers must either make do with the traditional smutty versions, or work things out for themselves.

Although it is an old one, the fascinating world of Fibonnaci numbers and Lucas numbers continues to provide rich areas of investigation for professional and amateur mathematicians. We revisit divisibility properties for t0hose numbers along with the closely related Pell numbers and Pell-Lucas numbers by providing a unified approach for our investigation.

For non-negative integers n, the recurrence relation defined by

with initial conditions

can be used to study the Pell (Pn), Fibonacci (Fn), Lucas (Ln), and Pell-Lucas (Qn) numbers in a unified way. In particular, if a = 0, b = 1 and c = 1, then (1) defines the Fibonacci numbers xn = Fn. If a = 2, b = 1 and c = 1, then xn = Ln. If a = 0, b = 1 and c = 2, then xn = Pn. If a =b = c = 2, then xn = Qn [1].

In the literature we find several different ways of introducing elementary functions. For the exponential function, we mention the following ways of characterising the exponential function:

(a)

(b) , also for complex values of x;

(c) x → exp (x) is the unique solution to the initial value problem [4]

(d) x → exp (x) is the inverse of

(e)x → exp (x) is the unique continuous function satisfying the

functional equation f (x + y) = f (x) f (y) and f(0) = 1 [6]; the corresponding definition is done for the logarithm in [7];

(f) Define dr for rational r, and then use a continuity/density argument [8].

All of them have their advantages and disadvantages. We like (a) and (c), mostly because they have natural interpretations, (a) in the setting of compound interest and (c) being a simple model of many processes in physics and other sciences, but also because they are related to methods and ideas that are (usually) introduced rather early to the students.

Space does not allow much room for “reporting” in The Mathematical Gazette, though that deficiency is greatly offset by the Newsletter. As it happens, however, two very contrasting items arrived while this Number was in the proof stage and so they are included, as typical of much else, to give some information about mathematical activities in general.

It has been known for some time that when the sun is reflected in a circular cylinder, the illuminated part of the cylinder's base is bounded by part of a nephroid (the curve obtained by rolling a circle around another circle with twice the radius).

A ferry starting on one side of a river is to cross to the adjacent point on the opposite bank travelling at a constant speed f but moving against a variable river sneed We will ignore starting and stonninc difficulties which only operate for a short interval of time. The ferry moves so that it is always aiming at the target position T. The river is assumed to be straight and the speed of the river is given by the function r(x) which is non-negative, continuous and differentiable for 0 ≤ x ≤ a, where a is the width of the river. We shall show that the solution reduces to two integrations and the condition which determines whetIer the derry can reach its destination will be given.

We have already explained pp. (25-31) that there are two types of uniform three-dimensional space, viz., finite or elliptic space, in which the geodesies or straight lines are closed and of constant length, and infinite or hyperbolic space, in which the geodesies are unclosed and of infinite length. Euclidean or parabolic space may be regarded as a special third type, or as the dividing case between the two general types; but it is best to regard it as a variety of hyperbolic space, owing to the reality of the infinite. In elliptic space all geodesies through a point either return without intersecting again, or have at most one other fixed point of intersection. In hyperbolic space it does not appear necessary that a geodesic should consist of only a single branch, though this is of course the simplest case; if the geodesies have several branches, all geodesies through a point will have a fixed point on each branch, and two geodesies which intersect will have as many points of intersection as they have branches. When we consider two-dimensional space we find that, even in the elliptic type, the geodesies may have any number of intersections, and may be unclosed and infinite. The proof to the contrary on p. 31 does not apply when the geometry is confined to a surface.