Math History

Math Department

Pi

Table of computation of Pi from 2000 BC to now

```Babylonians                 2000? BCE           1   3.125  = 3 + 1/8
Egyptians                   2000? BCE           1   3.16045
China                       1200? BCE           1   3
Bible (1 Kings 7:23)         550? BCE           1   3
Archimedes                   250? BCE           3   3.1418 (ave.)
Hon Han Shu                  130 AD             1   3.1622  = sqrt(10) ?
Ptolemy                      150                3   3.14166
Chung Hing                   250?               1   3.16227 = sqrt(10)
Wang Fau                     250?               1   3.15555  = 142/45
Liu Hui                      263                5   3.14159
Siddhanta                    380                3   3.1416
Tsu Ch'ung Chi               480?               7   3.1415926
Aryabhata                    499                4   3.14156
Brahmagupta                  640?               1   3.162277 = sqrt(10)
Al-Khowarizmi                800                4   3.1416
Fibonacci                   1220                3   3.141818
Otho                        1573                6   3.1415929
Viete                       1593                9   3.1415926536 (ave.)
De Lagny                    1719              127   (112 correct)
Rutherford                  1824              208   (152 correct)
Shanks                      1874              707   (527 correct)
```

## The evolution of notation for decimal fractions

Modern methods of writing decimals were invented less than 500 years ago. Some use of decimal fractions was made in ancient China, medieval Arabia and in Renaissance Europe. By about 1500, decimals were well accepted by professional mathematicians but not widely used.

Thousands of years earlier, the Babylonians had used a place-value number system based on 60 (whereas ours is based on 10) and had been extended to deal with numbers less than one. This was still in use in the 1500s in Europe, but the advantages of a thoroughly base ten system were becoming apparent. In his book Canon-mathematicus (1579) the Italian/French mathematician Francois Viete called for the use of base ten decimal fractions rather than base 60 sexagesimal fractions when he wrote:

"Sexagesimals and sixties are to be used sparingly or never in mathematics, and thousandths and thousands, hundredths and hundreds, tenths and tens, and similar progressions, ascending and descending, are to be used frequently or exclusively"

Viete used decimals in this work, but wrote them differently to how we now write them. For example, for the length of the side of a square inscribed in a circle of diameter of 200 000 he wrote which we would write as 141421.35624. (This is a very accurate approximation to 100 000 times the square root of 2). Later he used for 100 000 times pi, the number we would write as 314 159.26535. He also used boldface type to indicate the whole number part of this number, writing it as 314,159,265,36. Sometimes he also included a vertical stroke to separate the whole number and fractional parts. For example, he wrote 99946.45875 as 99,946|458,75.

The person who is credited with making decimal fractions widely known and understood among common people and practical users of mathematics in Europe is Simon Stevin of Bruges, in the Flemish Netherlands. He sought to teach everyone "how to perform with an ease unheard of, all computations necessary between men by integers without fractions ". In a book entitled De thiende ("The Tenth") published in 1585, he wrote decimal expressions for fractions by writing the power of ten assumed as a divisor in a circle above or after each digit.

For example, the approximate value of pi which we write as 3.1416 appeared as

3 (0) 1 (1) 4(2) 1(3) 6(4)

or as

This notation indicated to the reader that the number was

Which is the same as

The use of the decimal point to separate the whole number and fractional parts in decimal numbers seems to first occur in a 1593 table of values for sines of angles constructed by a friend of Kepler, either G.A. Magini (1555-1617), a map maker or Christoph Clavius (1537-1612).

Decimal fractions as they look today were used by John Napier, a Scottish mathematician who developed the use of logarithms for carrying out calculations. The modern decimal point became the standard in England in 1619.

Place value numeration had been in full use for many centuries before its ability to handle fractions was recognised. Even then, a range of different notations and symbols for separating the whole number from the fractional part of the number were tried before an accepted method of writing decimal fractions without the use of unneeded symbols was finally stabilized. The following table, adapted from Tobias Dantzig's book, Number the Language of Science, illustrates different ways that decimal fractions were shown prior to the use of our modern notation.

Fractions

### The History of Fractions

The history of fractions began with human observations of nature. The divisions of the day, the month, the seasons and the patterns in nature. The use of fractions increased as growing societies needed ways to measure goods and merchandise. Often the mathematicians interest in studying and predicting planetary movements lead to mathematical progress.

Numbers representing parts of a whole are called rational numbers or fractions. Fractions can be expressed as the quotients of two integers (a and b).

The Egyptians were one of the first groups to study fractions. They were the first to use sums of unit fractions, fractions with one in the numerator. For example, the fraction 3/4 was written in hieroglyphic as 1/2 + 1/4. This method did not allow numbers such as 2/7 to be represented except as sums of unit fractions and so they kept prepared tables for such fractions. To divide 3 loaves equally among 5 men, each man would be given three separate portions, a 1/3, a 1/5 and a 1/15 portion1. The Egyptians were able to calculate the areas of geometric shapes and volume including the planning and building of the pyramids and kept detailed accounting of land and goods:

"The Egyptians' concern for the accurate dealing with fractions almost certainly originated from practical problems such as the division of food, supplies....in a country which had no metallic currency or money, and in which payments were made in kind."

Although different, the Babylonian system was closely connected to their alphabet. For example, a one wedge indicated the number 1 and an arrow like wedge stood for 10. It is thought that the adoption of Sumerian script by conquering Akkadians resulted in symbolization of the written language. For example, technical terms for several operations were expressed by a single cuneform sign3. Therefore, numbers were formed by adding symbols. The Akkadians used base ten for calculations; the Sumerians used base 60 which may have derived from the ratio of 60-1 that was used for measuring silver. The word for one, "gesh" is 60; "gesh-u"=60x10; "shar"=60x60. They also used multiplication and division tables as well as the Pythagorean Theorem (Pythagoras, 572-497BC). The Babylonian tablet of Pythagorean triples is the oldest number theory in existence.

Euclid's (300 BC) Algorithm used continuous fractions to help solve mathematical equations that contained fractions in the problem. For example this is used to find the greatest common denominator of two numbers which, if the sequence is continued, the remainder will end in zero. Fractions were used in Greek astronomy, architecture and music theory for describing musical intervals and the harmonic progression of string lengths.

In 550 AD, an Indian mathematician named Aryabhata used continued fractions to study and solve linear equations. He was interested in predicting eclipses of the sun and the moon and described astronomical rules.

Two men from the city of Bologna, Italy named Rafael Bombelli and Pietro Cataldi studied repeating continuous fractions such as the square root of 13 and later on the square root of 18. Although the were good mathematicians, they did not study the properties of repeating continued fractions. In 1585 Ladisme used decimal fractions to unify the systems of measurements on a decimal base; he was a student of Johann Kepler.

Christiaan Huygens (1629) was the first to use continued fractions in a practical application for approximating gear ratios. Like the mathematicians before him. he was very interested in the movement of the planets and needed gear works to build a mechanical planetarium. He was also a Dutch astronomer, physicist, and mathematician. Toward the end of the 17th century clocks were invented further dividing time and the circle of the day.

John Wallis, a professor of geometry at Oxford in 1757, developed the study of continued fractions. He wrote a book called Arithemetica Infinitorium where he developed and presented the identity 4 over Pi. In this book, he talked about the first steps to generalizing continued fraction theory. He also wrote a book called Opera Mathematica. In this book he dealed with convergents and some of the properties of convergents. He was also the first to use the term "continued fraction," although people had studied them before but never actually called them that. Continued fractions form a fractional expansion such as:

Pi = 2x2x4x4x6x6x8x8

2 1x3x3x5x5x7x7x9

He was also the first to use the symbol ( ¥ ) to represent infinity.

In the next century De Fractionlous Continious (1737) written by Leonard Euler, expressed the modern theory of fractions, stating that every rational number can be expressed as a simple terminating fraction. Euler also adopted the symbol of for Pi. His colleague, Joseph Louis Lagrange, proved that a real root of a quadratic irrational number is a periodic continued fraction. In his memoir (1767) called Sur la resolution des equations numeriques LaGrange described the uses of continued fractions.

The nineteenth century has been called "the golden age of continued fractions." The theory of continued fractions now embraced convergents and complex variables as terms to work with. With the industrial revolution practical application of fractions allowed for precision instrumentation and, in the twentieth century, continued fractions became used within computer algorithms for solving the advanced physic and astronomical questions of the present day.

The study of fractions, through experimentation and observation, continues to progress as we divide the atom to its smallest components. Although observations in nature and desires to predict were the first driving force to the human imagination's study of mathematics and fractions, possibly commerce became the purpose for perfect divisions and predictable fractions. The history of fractions is not finished, however, since they continued to be studied in the present and used now.

Bibliography:

Gillings, Richard, Mathematics in the Time of the Pharaohs, MA, MIT Press, 1972

Wilder, Raymond, Evolution of Mathematical Concepts-An Elementary Study, NY, John Wiley and Sons Inc., 1968

Struik, Dirk, A Concise History of Mathematics, NY, Dover Publications, 1967

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