Five geometrical tasks demonstrating the interesting property of conservation of area that may be used in the mathematics classroom are presented. The initial premise is proved and then changes in attributes are investigated using dynamic geometric software to show that in the cases demonstrated there is conservation of area or specific ratios. These demonstrations illuminate the hidden beauty in mathematics and constitute an intellectual stimulus for discovering additional properties.

An infinite simple continued fraction representation of a real number α is in the form

$$\eqalign{& {a_0} + {1 \over {{a_1} + {1 \over {{a_2} + {1 \over {{a_3} + {1 \over {}}}}}}}} \cr & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ddots \cr} $$

$${a_0}$$

$${a_i}$$

$$i \ge 1$$.

$${a_0} + {1 \over {{a_1} + }}{1 \over {{a_2} + }}{1 \over {{a_3} + }} \ldots \;{\rm{or}}\;\left[ {{a_0};\;{a_1},\;{a_2},\;{a_3} \ldots } \right]$$

In the May 1954 issue of the Gazette Daniel F. Ferguson challenged readers to devise their own proof for what he described as a curious and somewhat pleasing sum (see [1])

Figure 1 below shows a triangle ABC and three cevians AX, BY and CZ with intersection points

We aim to produce formulae for a quadrilateral which correspond to the sine and cosine rule for a triangle. Subsequently, we apply these results to solve problems involving a quadrilateral.

Readers will most likely be aware of the works of Johann Sebastian Bach in the field of music, particularly of his Goldberg variations and the changes that can be rung, wherein aesthetically appealing alterations to structure produce a raft of colourful sounding themes. Similarly in the field of quadrature it is possible to revisit and re-establish well-known formulae by developing variations on the theme of a three-point interpolating quadratic formed to represent a function that is to be integrated.

It was my nine-year-old daughter who got me interested in the title question. As she appreciates, multiplying an integer by a power of 10 is a cinch. To multiply 34 by 100, simply add two zeros at the end: 34 × 100 = 3400. Dividing 3400 by 100 is the reverse process: remove two zeros to obtain 34. More generally, to multiply an integer by 10N, for non-negative N, add N zeros to the end of its decimal notation, and to divide an integer by 10N remove N zeros from its end — so long as it has them. Easy-peasy; my daughter knows all that.

Some favourite topics of both mathematics teachers and mathematics students are the golden ratio, Fibonacci numbers, groups, and Pythagorean triples. The material of this paper involves the first three, and there are some analogies with the fourth.

This Article explores how root multiplicity and polynomial degree influence the structure of the roots of a univariant polynomial. After setting up the notation, we draw upon a result derived in [1], and show that all polynomial roots have a common underlying structure comprising just five parameters. Finally we present some examples involving the lower polynomials.

The concept of isomorphism is central to group theory, indeed to all of abstract algebra. Two groups {G, *} and {H, ο}are said to be isomorphic to each other if there exists a set bijection α from G onto H, such that $$\left( {a\;*\;b} \right)\alpha = \left( a \right)\alpha \; \circ \;(b)\alpha $$ for all a, b ∈ G. This can be illustrated by what is usually known as a commutative diagram:

Martin Josefsson [1] has coined the term ‘bisect-diagonal quadrilateral’ for a quadrilateral with at least one diagonal bisected by the other diagonal, and extensively explored some of its properties. This quadrilateral has also been called a ‘bisecting quadrilateral’ [2], a ‘sloping-kite’ or ‘sliding-kite’ [3], or ‘slant kite’ [4]. The purpose of this paper is to explore some more properties of this quadrilateral.

In [1], the following conjecture was proposed concerning the distribution of ages in a closed interval [0, A]

Aesop's Fables is an enduring collection of short stories with morals that is credited to Aesop, a slave who lived in early Ancient Greece about 2600 years ago. Undoubtedly many later ancient Greek philosophers such as Pythagoras, Socrates, Aristotle and Archimedes were told Aesop's fables in their youth. In a race described in ‘The Tortoise and the Hare’, one of the most famous of Aesop's fables, a tortoise, running in a steady constant manner, beats a hare that is racing irregularly. The lesson of the fable is often interpreted as ‘slow but steady wins the race’ or ‘consistent, effective effort leads to success’ (see [1]) and is applicable to many human activities. The fable illustrates the general problem of working toward an objective when the rate of work is either constant or varies randomly.

Mathematical calculation at times may produce erroneous results because of ignoring particular features of the functions involved, such as restricted domains, ranges and discontinuities. In this Article, to highlight the significance of conceptual understanding, we choose six examples and for each we give a flawed solution followed by a correct solution.

The game of tennis has provided mathematicians with many interesting problems. In [1], the problem of finding the probability that a certain player wins a tennis tournament was studied. Gale [2] determined the best serving strategy in tennis. First, we assume Alice and Bob play a game of tennis using the standard (or Deuce/Ad) scoring system, without a tiebreaker, and that Alice serves the game. We also assume that the probability that Alice wins any point she serves is . Stewart [3] proved that the probability that Alice wins is

$${{15{p^4} - 34{p^5} + \;28{p^6} - 8{p^7}} \over {1 - 2p\; + \;2{p^2}}}$$