The Little Book of Mathematical Principles, Theories & Things. Robert Solomon
The Seven Millennium Prize Problems
HOW TO USE THIS BOOK
The Little Book of Mathematical Principles, Theories, & Things is an easy-to-use comprehensive guide to mathematics. It features over 130 entries on key principles or theories essential to understanding the subject. Written in an easily accessible manner, the Little Book explains sometimes very difficult concepts and theories, putting them in their historical context, giving background information on the experts who proposed them in the first place, analysing influences and proposing, where relevant, links to other related entries. The book also features tables, equations and illustrations, and ends with a glossary, and a comprehensive index.
The Little Book of Mathematical Principles, Theories, & Things is arranged chronologically and the country of origin is listed, where appropriate. Each entry includes a clear main heading, the person or people responsible for the discovery, birth and death dates, where relevant, followed by a short introductory paragraph explaining the concept concisely. In some cases, the main essay is also cross referenced to linked subjects. The key on the opposite page explains the order of information in each entry.
3000 BC Global
Writing Numbers
The place value system uses only a finite number of symbols to write any number.
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In some number systems, there are different symbols for each power of 10. In a place value system, only a small number of symbols are used.
In the Ancient Egyptian number system, dating from about 3000 BC, there were symbols for units, symbols for 10s, and so on. The number 365 was written:
where | represents a unit,
The Chinese system writes numbers much as we say them. We say “three hundred and sixty-five:” in other words, so many hundreds, so many tens, and so many units. The number 365 is written as shown below.
It represents 3 x 100 + 6 x 10 + 5.
In both these systems there is no limit to the number of symbols required. We need a different symbol for millions, another symbol for 10 millions, and so on. The modern system uses precisely 10 symbols: the digits 0 to 9.
The value of each digit is shown by its place in the number. In 365, for example, the digit 5 on the right represents 5, the digit 6 represents 60, as it is one place to the left, and the 3 represents 300. This system came to the West from India via the Arab countries and is known as the Indo–Arabic system.
The ancient Babylonian place value system was even more economical. It used only two symbols:
represents 3 x 602 + 21 x 60 + 43 = 12 103.
3rd millennium BC Egypt & Babylonia
Fractions
There are different systems for writing fractions. This has always been the case, even in ancient times. For example, the Egyptian system was very limited, while the Babylonian system is still in use today.
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Any advanced civilization has a system of writing fractions. Despite their renowned technological prowess, the Ancient Egyptians had a system of fractions that was comparatively clumsy.
With the exception of 2/3, the only fractions recognized by the Ancient Egyptians were those with 1 on the top, called aliquot fractions, such as 1/2, 1/3, 1/4. Any other fraction had to be written in terms of these aliquot fractions. Furthermore, they were not allowed to repeat a fraction. If they wanted to write 2/5, for example, they could not write it as 1/5 + 1/5. For the second 1/5, they had to find aliquot fractions with sum 1/5,