Brian Clegg - Introducing Infinity: A Graphic Guide

WHAT WOULD YOU CALL A REALLY, REALLY BIG NUMBER (SAY 1 WITH 100 NOUGHTS AFTER IT)?
A GOOGOL!

To deal with any number we need symbols that represent numerical values. The symbol equivalents of the words one, two, three and so on (1, 2, 3) arrived in the West from India via the Arabic world. The oldest known ancestors of the modern system were found in caves and on coins around Bombay dating back to the 1st century AD.

The numbers 1 to 3 were based on a line, two lines and three lines, like horizontal Roman numerals, though they can still be seen with some imagination in the main strokes of our modern numbers. The markings for 4 to 9 are closer ancestors of the symbols we use today.

The Indian symbols were adopted in the Arabic world, coming to the West in the 13th century thanks to two books, written by a philosopher in Baghdad and a traveller from Pisa. The earlier book, lost in the Arabic original, was written by alKhwarizmi (c. 780850) in the 9th century. The Latin translation of this, Algo-ritmi de numero Indorum, was produced around 300 years later, and is thought to have been considerably modified in the process.

The version of al-Khwarizmis name in the title is usually given as the origin of the term algorithm, though its sometimes linked to the Greek word for number, arithmos.

The traveller from Pisa was Leonardo Fibonacci (c. 11701250). (His father, a Pisan diplomat, was Guglielmo Bonacci, and Fibonacci is a contraction of filius Bonacci, son of Bonacci.) He travelled widely in North Africa and became the foremost mathematician of his time, his name inevitably linked to the Fibonacci numbers (see ), which he popularized but didnt discover. Although Numero Indorum was translated into Latin a little before Fibonaccis book Liber abaci came out in 1202, it seems that Liber abaci (The Book of Calculation) had the bigger influence in introducing the Indian system to the West.

ON MY TRAVELS I WAS INTRODUCED TO THE ART OF THE INDIANS NINE SYMBOLS.

0, a powerful tool

The symbols we use for numbers are arbitrary. , , , , would do as well as 1, 2, 3, 4, 5. However, the new Indian numerals brought with them a very powerful tool. Earlier systems from Babylonian through to Roman were tallies, sequential marks to count objects. Were most familiar with Roman numerals the tally sequence is obvious in I, II, III, IV, V where V is effectively a crossed through set of IIII and IV is one less than V. But the trouble with such systems is that theres no obvious mechanism to add, say, XIV to XXI.

THE NEW SYSTEM USED COLUMNS WITH A PLACE-HOLDER O FOR EMPTY SPACES, TRANSFORMING ARITHMETIC.

Archimedes: The Sand Reckoner

But whatever symbols are used, big numbers kept their appeal. In a book called The Sand Reckoner, ancient Greek philosopher Archimedes (c. 287-212 BC) demonstrated to King Gelon of Syracuse that he could estimate the number of grains of sand it would take to fill the universe.

We dont know a lot about Archimedes, but we do have a number of his books, which show him to be a superb mathematician and a practical engineer. He is said to have devised defence weapons for Syracuse ranging from ship-grabbing cranes to vast metal mirrors to focus sunlight and set ships on fire.

Unlike many of Archimedes other works, The Sand Reckoner wasnt exactly practical. But there was a serious point behind this entertaining exercise. What Archimedes set out to do was to show how the Greek number system, which ran out at a myriad myriads (100 million), could be extended without limit. He first estimated the size of the universe at around 1,800 million kilometres (just outside the orbit of Saturn).