History of Mathematics/Early Math

Ancient Egypt

The Egyptian system employed hieroglyphs, shown in figure 1, in a manner similar to the later Roman numerals, that also allowed the Egyptians to describe fractions. While each symbol represents a power of 10, the Egyptian system did not have a base like modern number systems because the position of a digit within an Egyptian number didn't give you information about its value. The Egyptians would express numbers by drawing a given symbol as many times as was needed to describe the multiple it represented. Thus the number 2369 could be written as shown in figure 2.

$={\frac {1}{3}}$ $={\frac {1}{331}}$ $={\frac {1}{2}}$ $={\frac {2}{3}}$ $={\frac {3}{4}}$ Figure 3: Fractions

1 10 100 1,000 10,000 100,000 1 million, or infinity or Figure 1: Numeric Hieroglyphs Figure 2: 2369 Fractions were described using a symbol that looks somewhat like a mouth. This symbol would be placed over a set of glyphs that described a whole number, and thus obtain a unit fraction. Other rational numbers would be described using a sum of unit fractions, however, egyptians would "round-off" rational numbers by not considering more than 6 unit fractions. There were special glyphs for ½, ⅔, and ¾, and numbers could be written phonetically, much as we can write nine instead of 9, but rarely did this occur for numbers other than 1 or 2. Some examples are given in figure 3. The Egyptians used glyphs that looked like a pair of legs to show addition and subtraction. Depending on the direction of the "feet" and the flow of the text, the symbols showed which operation was being performed; flowing into the text meant addition, otherwise subtraction. and Figure 4: Add & Subtract Figure 5: 114 - 25 Figure 6: 7/12ths

Multiplication was done by employing a form of what we now call binary arithmetic. The multiplicand would be written next to the glyph 1, then the multiplicand would be doubled and written next to the glyph 2. Then both numbers would be doubled and this process would be continued until the glyph was at or greater half the value of the multiplier. Finally, the glyphs would be subtracted from the multiplier, while the corresponding multiplicands would be added together. Thus, an example of multiplying $12\times 13$:

12 - 1 24 - 2 48 - 4 96 - 8

96 + 48 + 12 $=$ 156 13 - 8 - 4 - 1

Figure 7: Egyptian Multiplication of 12 by 13

Rhind papyrus

A portion of the Rhind Papyrus

Much of our understanding of Egyptian mathematics comes from the deciphering of a few papyri that were actually written in hieratic script. The most famous such papyrus is the Rhind Mathematical Papyrus, which dates from about 1650 BC, but its author, Ahmes, identifies it as a copy of an older Middle Kingdom papyrus. The Rhind papyrus contains 84 word problems, and a table of 101 Egyptian fraction expansions for numbers of the form 2/n. It also includes formulas and methods for the addition, subtraction, multiplication and division of sums of unit fractions, and contains evidence of other mathematical knowledge, including arithmetic, geometric and harmonic means; composite and prime numbers; and simplistic understandings of both the Sieve of Eratosthenes and perfect number theory. It also shows how to solve first order linear equations as well as summing arithmetic and geometric series.

An example of these word problems, asks the reader to find x and a fraction of x such that the sum of x and its fraction equals a given integer. Problem #24 from the papyrus asks the reader "A quantity added to a quarter of that quantity become 15. What is the quantity?" In modern notation the problem is to solve x + x/4 = 15. The problem is worked out by guessing that x = 4, which will remove the fraction from the problem. With x = 4, x + x/4 = 5, which is not the correct answer, however 5 x 3 = 15, so, 4 x 3 = 12 = x, which is correct. Some historians believe that the method was to divide x into 4 equal pieces, ie x = 4y, so that the solver could find the number of these pieces required to equal 15, being 5, or 4y + 4y/4 = 4y + y = 5y = 15. Thus the size of each piece is 3, being 4 of them to make x, x = 12.

Moscow papyrus

The Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenishchev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Golenishchev. Golenishchev bought the papyrus in 1892 or 1893 in Thebes. It later entered the collection of the Pushkin State Museum of Fine Arts in Moscow, where it remains today.

Based on the palaeography and orthography of the hieratic text, the text was most likely written down in the 13th dynasty and based on older material probably dating to the Twelfth dynasty of Egypt, roughly 1850 BC. Approximately 18 feet long and varying between 1½ and 3 inches wide, its format was divided into 25 problems with solutions by the Soviet Orientalist Vasily Vasilievich Struve in 1930. It is a well-known mathematical papyrus along with the Rhind Mathematical Papyrus. The Moscow Mathematical Papyrus is older than the Rhind Mathematical Papyrus, while the latter is the larger of the two.

Berlin papyrus

The Berlin papyrus, written around 1300 BC, shows that ancient Egyptians had solved two second-order, one unknown, equations that some have called Diophantine equations. The Berlin method for solving x2 + y2 = 100 has not been confirmed in a second hieratic text, though it has been confirmed by a second Berlin Papyrus problem.

Mesopotamia

Another ancient people with respectable mathematics were the peoples of Mesopotamia. From the time of the Sumerians until the fall of Babylon, the mathematics used by these people is referred to as Babylonian mathematics. We know much about these peoples capabilities from several hundred clay tablets that have been uncovered since the mid-nineteenth century.

Arithmetic

Figure 1: Babylonian Numerals

Babylonian mathematics was based on a sexagesimal numeral system, and unlike the Egyptians, they employed a place value system to describe numbers larger than 59. Individual digits were described using a sub-base-ten system, where a "<" would be used to describe the value of what we would call the "tens" place, and a "Y" would describe the "ones" place. So the number 12 could be written <YY and the number 31 would be <<<Y, see figure 1. Also, a number such as 82 would be written Y<<YY; Y or 1 for the high order digit, <<YY or 22 in the lower order digit.

The Babylonians did not initially have a figure to represent the number 0, and simply assumed the idea of zero to simply be the lack of a number. Thus, a number like 364, which would be 1,0,4 in sexagesimal, would be Y YYYY. Note the space, which later would be replaced by a place holder and would be the only approximation to zero the Babylonians ever bothered to use.

To simplify writing sexagesimal numbers using modern Arabic numerals, places are often separated by a comma, and a decimal place described by a semicolon, thus the number 398.44 would be written 6,38;44. To convert from decimal to sexagesimal, divide the number by the highest sexagesimal place value that will go into the number, then repeat with the remainder. (7299 → 20 × 360 + 99 → 20 × 360 + 1 × 60 + 39 → 7299 = 20,1,39)

For multiplication, the Babylonians used tables of squares and the equation

$ab={\frac {(a+b)^{2}-a^{2}-b^{2}}{2}}$ or $ab={\frac {(a+b)^{2}-(a-b)^{2}}{4}}$

Thus, to find 9 x 3, a babylonian could find the squares of 9 + 3 (12) and 9 - 3 (6), or 144 and 36, respectively, subtract the smaller from the larger 144 - 36, or 108, and take a fourth of this number, 27.

To perform division, a table of reciprocals was used, basing the method on the fact that $a/b=a\times 1/b$. An example of 144/6, using sexagesimal notation (1,14/6):

${\frac {2,14}{6}}=2,14\times ;6={\frac {(2,14+;6)}{hi}}$

Ancient India

In India lot of Jain granth (Scriptures) contain complicated mathematics. Those books are very old and contains all forms of mathematics. It will be a great help to world of Mathematics if some body do the proper study of Jain Books which contains mathematics.