3.1 APPEARANCE OF FRACTIONS
Counting is the simplest and most fundamental operation using numbers.
However, just counting was insucient and inconvenient for people even
in ancient civilizations. People at some early stage of their civilizations
realized that arithmetic operations such as addition, subtraction, multi-
plication, and division were necessary and very important. It is probably
more than 4000 years ago that people in Egypt, Mesopotamia, and other
ancient civilized regions could already manipulate arithmetic operations
using their number systems, although it is dicult to specify when and
where people rst discovered such techniques for calculations.
Among the four arithmetic operations, multiplication and division
were dicult compared with addition and subtraction. People invented
doubling and halving operations a long time ago. ese and similar opera-
tions were convenient for their lives, and made multiplication and division
easier. Consequently, the fractions 1/2, 1/4, 1/8, 1/16, and so on became
commonly used numbers in addition to natural numbers. Furthermore,
the use of reciprocals of integers, such as 1/2, 1/3, 1/4, 1/5, 1/6, and so
on, also became common and of great importance. e reciprocal of each
nonzero natural number is called a unit fraction. ey noticed that multi-
plications of unit fractions by integers were also useful for calculations. In
this way, fractions appeared in ancient civilizations .
16 ◾ Computing
Ancient Egyptians used special symbols representing fundamental
unit fractions and some other fractions (e.g., 2/3). ese symbols appeared
in several mathematical tablets and papyri found in Egypt . Akhmim
wooden tablets and Cairo wooden tablets are two ancient Egyptian doc-
uments that contain, for example, descriptions about multiplications
by fractions 1/3, 1/7, 1/10, 1/11, and 1/13. e following calculation of
fractions also appeared in the tablets:
1/2 + 1/4 + 1/8 + 1/16 + 1/64 + 5/320 = 1
Historians suggest that these tablets were probably inscribed at the
beginning of the Egyptian Middle Kingdom around 1950 BC . ese
are now held in the Egyptian Museum in Cairo.
In the British Museum in London, the Rhind Papyrus (also called Rhind
Mathematical Papyrus) is displayed. It is approximately 5 m long and 33 cm
wide, one of the oldest existing texts of Egyptian mathematics. A Scottish
lawyer, Alexander H. Rhind (1833–1863), purchased it in 1858 in Egypt
. It was named the Rhind Papyrus aer him. It was copied by a scribe
named Ahmes around 1650 BC. In the rst paragraph of the papyrus,
Ahmes presents that it is copied from an ancient copy made during the
12th dynasty of Upper and Lower Egypt (c. 1985–1795 BC) [3, 8].
e rst part of the Rhind Papyrus contains a list of the fractions 2/n
for odd n from 3 to 101. e following are examples in the list:
2/3 = 1/2 + 1/6
2/5 = 1/3 + 1/28
2/7 = 1/4 + 1/28
2/9 = 1/6 + 1/18
2/15 = 1/10 + 1/30
2/101 = 1/101 + 1/202 + 1/303 + 1/606
e second and third parts of the Rhind Papyrus consist of geometry
problems, and 84 problems with the solutions, respectively. Below is an
example in the third part:
Rational and Irrational Numbers ◾ 17
Let the sum of 2/3 and 1/10 of an unknown quantity be 10. Calculate
the unknown quantity.
Using modern notation, (2/3 + 1/10)x = 10, where x is the unknown
quantity. en x = 300/23.
e Moscow Papyrus (also called Moscow Mathematical Papyrus) is
now held in the Puskin State Museum of Fine Arts in Moscow . It is
approximately 5.5 m long and 3 ~ 7.7 cm wide, one of the oldest Egyptian
mathematical texts in existence. e Moscow Papyrus probably dates
to the 11th dynasty of Upper and Lower Egypt (c. 1850 BC). It contains
25 problems of arithmetic, algebra, and geometry. e following are
examples of the problems:
Let the sum of 1 and a half of an unknown quantity and 4 be 10.
Calculate the unknown quantity.
Using modern notation, (1 + 1/2)x + 4 = 10, where x is the unknown
quantity. en x = 4.
Let 1/2 + 1/4 of the square of an unknown quantity be 12. Calculate
the unknown quantity.
Using modern notation, (3/4)(3/4)x
= 12(3/4), where x is the unknown
quantity. en x = 4.
3.2 RATIONAL NUMBERS
e set of rational numbers is usually denoted by Q, and the elements
of this set have the property that they can be represented as a ratio
of two integers (hence the name rational). ese numbers are, in a