Mathematics

The word “mathematics” comes from the Greek “mathema” which means in ancient Greek “what one learns”, “what one gets to know” also “study”, “knowledge”, “learning” and “science” and in modern Greek just “lesson”.

In English until 1700 the term “mathematics” meant “astrology”, “astronomy” rather than “mathematics” as it is now. Mathematics is the study of quantity, space, structure and change.

Through the use of abstraction and logical reasoning, maths developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical maths has been a human activity for as far back as written records exist. The earliest uses of Maths were in trading, land measurement, painting. In addition to recognizing how to count physical objects, prehistoric peoples also knew how to count abstract quantities, like time – days, seasons, years. Elementary arithmetic (addition, subtraction, multiplication and division) naturally followed.

The systematic study of maths in its own right began with the Ancient Greek between 600 and 300 BC.

Maths continued to develop, for example, in China in 300 BC, in India in 100 A.D. and in the Muslim world in AD 800 until the Renaissance when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the present day.

Mathematics is used throughout the world as an essential tool in many fields, including natural science engineering, medicine, and the social sciences.

Nowadays, all sciences suggest problems studied by maths and many problems arise within Maths itself. Often Maths inspired by one area proves useful in many areas. A distinction is often made between pure maths and applied Maths. However pure Maths topics often turn out to have applications, e. g. number theory in cryptography and computer science.

Many mathematicians talk about the elegance of maths, its inner beauty. Simplicity and generality in Maths are valued.

UNIT 8

HIP TO BE SQUARE: RUBIK'S CUBES AND SUDOKU

Magic squares may seem esoteric, but their cultural impact is evident whenever you open the newspaper or walk into a toy shop. The two most popular puzzles of recent years – Sudoku and the Rubik's Cube – are both consequences of a centuries-long preoccupation with them.

In the 18th century, Leonhard Euler, the greatest mathematician of his day, was devising ways to create magic squares. In order to do this he started looking at another type of square that could be used as a kind of template for producing magic squares.

Euler's new concept was a square in which every number, or symbol, would appear once and only once in each row and column. While these squares had been known about since at least a few centuries before, Euler was the first mathematician to analyse them systematically and he coined their name "Latin square".

He also invented the sister concept in which two Latin squares are superimposed on each other, and such that each cell in the grid is unique. This he called a "Graeco-Latin Square".

In 1782, Euler set the "36 officers problem", a frivolous puzzle that led to much deep academic work and discoveries. Can you make a 6x6 Graeco-Latin made up of six regiments of six officers each of different ranks so that no rank and regiment is repeated in any row or column? Only in 1901 was it proved that this was impossible.

Unlike magic squares, Latin and Graeco-Latin squares have found many uses and applications in non-mathematical settings, for example in cryptography and biological experiments, experimental design.

The best-known occurrence of Latin squares now, however, is in newspapers and puzzle books. Sudoku is a puzzle to complete a partially completed 9x9 Latin square that contains the digits one to nine in each column and row, with the added specification that the 3x3 sub-squares also contain the numbers from one to nine.

The previous puzzle craze to Sudoku was the Rubik's Cube, whose history can also be traced back to the magic square. In the mid-19th century in upstate New York, Noyes Palmer Chapman, an amateur puzzle enthusiast, made a physical model of a magic square such that the numbers from 1 to 16 were on small wooden squares that could be fit in a 4x4 box. He realised that if he left out one of the squares, it was possible to slide the other 15 squares around. This became known as the "15 Puzzle", which was an international fad in 1880 – and is the original sliding block puzzle, versions of which you can still find in toyshops.

In the 1970s Hungarian designer Ernö Rubik was trying to reinvent the 15 Puzzle in three dimensions when he came up with the idea of the Rubik's Cube.

From the magic square to Sudoku we seem to have always liked our puzzle crazes to come in squares – although this is a matter not for mathematicians, but for psychologists.

UNIT 9