History of Mathematics/Early Math

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Ancient Egypt[edit]

It is believed that the earliest use of a base 10 numerical system was that of the ancient Egyptians. The system employed hieroglyphs, shown in figure 1, in a manner similar to the later Roman numerals, and also allowed the Egyptians to describe fractions. 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.

D21
Z1 Z1 Z1
= \frac{1}{3}
D21
V1 V1 V1
V20 V20
V20 Z1
= \frac{1}{331}
Aa13
= \frac{1}{2}
D22
= \frac{2}{3}
D23
= \frac{3}{4}

Figure 3: Fractions

1 10 100 1,000 10,000 100,000 1 million, or
infinity
Z1
V20
V1
M12
D50
I8
or
I7
C11

Figure 1: Numeric Hieroglyphs

Z1 Z1 Z1 V20 V20 V1 M12
Z1 Z1 Z1 V20 V20 V1 M12
Z1 Z1 Z1 V20 V20 V1

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.

D54 and D55
Z1 Z1 V1
Z1 Z1 V20
D54
Z1 Z1 Z1 V20
Z1 Z1 V20
D21
Z1 Z1 V20
D55
Aa13
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:

Z1 Z1 V20
Z1
12 - 1
Z1 Z1 V20
Z1 Z1 V20
Z1 Z1
24 - 2
Z1 Z1 Z1 Z1 V20 V20
Z1 Z1 Z1 Z1 V20 V20
Z1 Z1
Z1 Z1
48 - 4
Z1 Z1 Z1 V20 V20 V20
Z1 Z1 Z1 V20 V20 V20
V20 V20 V20
Z1 Z1 Z1
Z1 Z1 Z1
Z1 Z1
96 - 8
   
Z1 Z1 Z1 V20 V20 V20
Z1 Z1 Z1 V20 V20 V20
V20 V20 V20

96

D55

+

Z1 Z1 Z1 Z1 V20 V20
Z1 Z1 Z1 Z1 V20 V20

48

D55

+

Z1 Z1 V20

12

 =
Z1 Z1 V20 V20 V1
Z1 Z1 V20 V20

156

Z1 Z1 Z1 V20

13

D54

-

Z1 Z1 Z1
Z1 Z1 Z1
Z1 Z1

8

D54

-

Z1 Z1
Z1 Z1

4

D54

-

Z1

1

 

Figure 7: Egyptian Multiplication of 12 by 13

Rhind papyrus[edit]

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[edit]

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[edit]

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[edit]

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 refered 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. These tablets

Arithmetic[edit]

Figure 1: Babylonian Numerals

Babylonian mathematics was based on a sexigesmal 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 sexigesmal, 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 sexigesmal numbers using modern arabic numerals, places are often seperatted 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 sexigesmal, divide the number by the highest sexigesmal 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\times1/b. An example of 144/6, using sexigesmal notation (1,14/6):

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


Algebra[edit]

Geometry[edit]

Pythagrean Theorem[edit]

Ancient India[edit]

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.

Review[edit]

Questions[edit]

Contents[edit]

  1. Early Math Prehistoric and ancient maths, from Egypt, India, and Mesopotamia
  2. The Greeks Hellenistic math, including the accomplishments of Pythagoreans, Eudoxus, Aristotle, Euclid, and Archimedes
  3. Post Greco Maths Maths during the Medieval times (c 300-1100), in Europe, India, the Middle East, and China
  4. Rebirth and the Dawn of Modern Mathematics