THE SIEVE OF ERATOSTHENES

A useful rule for making  a table of prime numbers is attributed  to another famous Greek astronomer and mathematician, Erastosthenes (c.275-194 B.C.) To find all the prime numbers not greater than n, we write all the integers from 1 to n in a convenient tabulated form; from this table we delete every second number after 2, because they all are divisible by 2, every third number after 3, because they all are divisible by 3, every fifth number after 5, and so on; the numbers which remain after all the deletions are prime numbers less than or equal to n.

SETS

A mathematical set must be defined either by a list, a rule or a formula, so that its elements can be recognized without any doubt or ambiguity. If N is a set, we write n E N meaning that 'n is a member of the set N'. The set of all elements that are being considered in a problem is called the reference set (or universal set), and it is denoted by E or U. The empty set has no elements (compare it with the empty drawer of a desk); it is denoted by O. Any collection of elements of E, ranging from O to E it self, is called a subset of E.

ROTATIONAL SYMMETRY IN A PLANE

When we say that a figure has rotational symmetry we mean that it is possible to rotate it about a center in such a way that the whole figure is unchanged but the parts of it have been rotated. The number of ways in which this can be done is called the order of rotational symmetry, and it is usual to count the identity operation (rotation through 0 degree or 360 degree, and so on) as one of the symmetries. Any figure has the trivial rotational symmetry of order one about any point. The letter N, the parallelogram and many other familiar figures, have rotational symmetry of order two.

THE TRAPEZIUM

There is one more special type of quadrilateral which frequently occurs in practical work, the trapezium; this is a quadrilateral with just one pair of opposite sides parallel.

In the side figure, this is a trapezium ABCD with an axis of symmetry. Such a figure is called an isosceles trapezium.

BUSINESS CALCULATIONS

The custom of charging interest on loans must be as old as money itself. Merchants and moneylenders have always tried to make a profit and to relate the profit to the size of transaction; a big deal calls for a big profit.

Roman taxes were usually 1/20, 1/25, or 1/100 of the quantity involved, and these fractions may have led to the practice of working in hundredths, or percentages. During the middle ages calculation of profit or interest as so much in a hundred became the common practice and the phrase per cent was established. This phrase was abbreviated in many different ways; one of the abbreviations was P0c and it is thought that the modern symbol % may have been derived from this.

One of the earliest examples of the phrase ‘per cent’occurs in the Liber Abaci by Leonardo of Pisa, published in 1202, where the author uses it in a problem concerning a merchant who is to sell wool in Florence at a profit of 20 per cent

SCALES

The scale of a map is often given in the form 1 : 100.000, or 1 : 25.000, that is as a ratio 1 : n. The scale 1 : 25.000 means that 1 cm on the map represents 25.000 cm or 0,25 km on the ground. When 1 : n is given the scale of a map it is called the representative fraction (R.F.) on the map.

TRIANGLES AND QUADRILATERALS (Part 1)

Thales of Miletus, knew that the measurements of a triangle were fixed when the length of its base and the sizes of its basebase angles were known. He used this fact to determine the distance of a ship at sea. Pythagoras of Samos, who settled eventually in Crotona, develoved  the theory of congruent triangles. It was probably in his school at Crotona that the congruence theorems were first analysed and related to one another in some kind of logical order.

During the life time of Plato (429-348 B.C.) Athens became the chief center of mathematical studies. It held supremacy for about 150 years, when the first university in the world, at Alexandria, displaced it. Plato was born near Athens of rich and noble parents. He studied in Egypt, Cyrille, and Italy, returning to Athens about 380 B.C. About a mile outside Athens was a beautiful walled garden called Academia, after its owner Academos. Here Plato formed a school of students which came to be known as the Academy. Plato was a philosopher who believed in geometry as one of the foundations of a liberal education. The inscription over the entrance to the academy ran 'Let none ignorant of geometry enter my door'. Plato did great service to mathematics in making its foundations logical and secure. We still tend to regards the compasses and ruler as the chief mathematical instruments largely because he would allow no other instruments to be used.

CURRENCY

Most countries have a decimal currency, with a main unit divided into one hundred minor units. In the United Kingdom the main unit is the pound, United States is the dollar, Kingdom of Saudi Arabia is the real, and Indonesian Republic is Rupiah.