Republic of the Philippines

LEYTE NORMAL UNIVERSITY

Tacloban City

Name: Lovely A. Booc

Student number: 1500253

ANCIENT BABYLONIAN MATHEMATICS

BRIEF HISTORY OF BABYLON

 Beginning over 4000 years ago, the Babylonians were discovering how to use
mathematics to perform functions of daily life and to evolve as a dominant civilization.
Since the beginning of the 1800s, about half a million Babylonian tablets have been
discovered, fewer than five hundred of which are mathematical in nature. Scholars
translated these texts by the end of the 19th century. It is from these tablets that we gain
an appreciation for the Babylonians’ apparent understanding of mathematics and the
way they used some key mathematical concept.

 Between the years of 3500 B.C. and 539 B.C., various Mesopotamian civilizations
inhabited this “land between the rivers” of the Euphrates and the Tigris. Around 3500
B.C., the Sumerians established the first city-states; one of the best city-states was
called Ur.

 After the Sumerians came the Akkadians, who inhabited the area of the surrounding
desert. The Akkadians were conquered in about 1900 B.C. by the First Babylonian
Empire. Just over 1000 years later, in 885 B.C., the Assyrians took over the land from
the Akkadians and maintained control of the land for nearly 300 years until, in 612
B.C., the Chaldeans conquered the Assyrians and began the Second Babylonian
Empire. Unlike the First Babylonian Empire, the Chaldeans’ reign was short-lived, a
mere 73 years, until the Persians invaded the land in 539 B.C.

 Babylon was a city in Mesopotamia which is now modern-day Iraq. The city was
known by the Acadians as barville which means “Gate of the Gods”.
  The Babylonians, and other cultures including the Assyrians, and Hittites, inherited
Sumerian law and literature and importantly their style of writing which is the
cuneiform pattern of writing.

BABYLONIAN NUMBERS

 Babylonian Mathematics (also known as Assyro-Babylonian mathematics) is the

mathematics developed or practiced by the people of Mesopotamia, from the time of
the earliest Sumerians to the centuries that followed the fall of Babylon in 539 BC.

 There have been 400 clay tablets discovered since the 1850s that provide information
on Babylonian mathematics. Tablets with Cuneiform writing were carved while the
clay was moist, then hardened in an oven or under the sun. The bulk of the clay tablets,
which range in age from 1800 to 1600 BC, contains information on the Pythagorean
theorem, algebra, quadratic and cubic equations, and fractions.

- The scripts were written on moist clay tablets using a stylus, which is a blunt reed.
- “cuneiform,” which literally translates “wedge shaped.”

 Two types of mathematical tablets are generally found, table-texts and problem texts.
- Table-texts are tables of values for some purpose, such as multiplication tables,
weights and measures tables, reciprocal tables, and the like.
- Problem-texts are concerned with the solutions or methods of solution to algebraic
or geometrical problems.

 The Babylonian system of mathematics was a sexagesimal (base 60) numeral system.
From this, we derive the modern-day usage of 60 seconds in a minute, 60 minutes in an
hour, and 360 degrees in a circle. For two reasons, the Babylonians were able to achieve
significant mathematical advancements.
1. The number 60 is a superior highly composite number.
2. The Babylonians had a true place-value system, where digits written in the left
column represented larger values (much as, in our base ten systems, 734 =
7×100 + 3×10 + 4×1).

 Egyptians used body parts as units of measurement, like counting with 10 fingers,
 which influenced the base 10 number system. Years later, from 4000 BCE to 2000
BCE, civilization in the Mesopotamia region introduced new math concepts that we
still use today like arithmetic, such as addition, subtraction, multiplication, and division.
The Babylonians were quite advanced.

 The Babylonians refused to work with numbers with non-terminating decimals, which
are numbers with a decimal that never ends. So, the base 10 system caused issues. For
example, 10 divided into thirds does not give you an even number. You get 0.33333,
which never ends. Dividing by 3 was something people needed to do on a regular basis.

 The Babylonians created their own base 60 number system with their own scripture and
symbols. There were symbols for 1 and 10. Numbers beyond 1 or 10 consisted of
multiples of the symbols. To write 60, the symbol for 1 was reused. 60 has an advantage
because the numbers 1 through 6, including 3, evenly divide into it.

 The Square Root of 2

- Scholars found a number on the Old Babylonian Tablet which is a very high
precision estimate of √2.
- Engraved in the tablet is the figure of a square, with one side marked with the
number 30.
- Scholars agree on transliterating it as 1; 24,51,10, which is approximately √2.

- 1; 24, 51, 10 is equal to 1 + + + , the sum of which is 1.41421296.

 Pythagorean Mathematics
- Of all the tablets that reveal Babylonian mathematics, the most famous is arguably
one that has been named “Plimpton 322”—a name given to it because it possesses
the number 322 in G.A. Plimpton’s Collection at Columbia University.
- Many scholars argue that the numbers on this particular tablet serve as a listing of
Pythagorean triples.
 - It appears to have the left section broken away. The tablet contains 4 columns and
15 rows.

 Summary of Babylonian Numbers

- Their mathematical notation was positional but sexagesimal.
- They had no zero.
- More general fractions were admitted but not all fractions.
- They could extract square roots.
- They could solve linear systems.
- They worked with Pythagorean triples.
- They solved cubic equations with the help of tables.
- They studied circular measurement.
- Their geometry was sometimes incorrect.

ANCIENT MATHEMATICS IN INDIA

 Archaeological excavations at Mohenjo Daro and Harappa give evidence of an old and
highly cultured civilization in the Indus Valley during the era of the Egyptian pyramid
builders (ca. 2650 BCE), but we have no Indian mathematical documents from that age.
 The roots of Indian mathematics are held within Vedic literature. Vedic literature is
nearly 4000 years old. Indian mathematics was founded between the years 1000 B.C.
and 1000 A.D.
 Vedas - groups of ancient, essentially religious texts, include references to large
numbers and decimal systems. Especially interesting are dimensions, shapes, and
proportions given for bricks used in the construction of ritual fire altars.
 Sulba (or sulva) refers to cords used for measurements.
 Sutra means a book of rules or aphorisms relating to a ritual or a science.
 Sulvasūtras are compositions aimed at providing instruction on the principles involved
 and procedures of construction of the vedis (altars) and agnis (fireplaces) for the
performance of the yajnas, which were a key feature of the Vedic culture.
 Yajnas refer to the offerings to the gods based on rites prescribed in the earliest
scriptures of ancient India.
 There are a number of Sulbasutras; the major extant ones, all in verse, are associated
with the names of Baudhayama, Manava, Katyayana, and the best-known, Apastamba.
 The word siddhanta means a system.
 Five different versions of the Siddhantas are known by the names: Paulisha Siddhanta,
Surya Siddhanta, Vasisishta Siddhanta, Paitamaha Siddhanta, and Romanka
Siddhanta.
 Surya Siddhanta (System of the Sun)
- written about 400 CE
- it is the only one that has survived and remained intact
 Paulisha Siddhanta
- The Paulisha Siddhanta, which dates from about 380 CE, was summarized by
the Hindu mathematician Varahamihira.
- it was referred to frequently by the Arabic scholar Al-Biruni.
- was derived in considerable measure from the work of the astrologer Paul,
who lived in Alexandria.
- uses the value 3 177 / 1250 for π, which is in essential agreement with the
Ptolemaic sexagesimal value 3;8,30.

HINDU-ARABIC NUMERALS
 The Indians used the Brahmi system. This number system has been found on cave walls
and on coins.
 The decimal system of numeration, in which 10 symbols are used and the value of a
symbol depends on its physical location relative to the other symbols in the
representation of a number, came to the modern world from India by way of the
medieval Muslim civilization.
  These symbols have undergone some changes in their migration from ancient India to
the modern world, as shown in the photo.

CONCEPT OF ZERO
 The discovery of zero was said to be an odd discovery due to the fact that when other
discoveries were made they represented a tangible object. Zero is unique because it
represents something in which we do not have.
 Two uses of zero
1. One use is as an empty place indicator in our place-value number system.
2106
The zero is used so that the positions of 2 and 1 are correct.
2. The second use of zero is as a number itself in the form we use it as 0.
 We can see from this that the early use of zero to denote an empty place is not really
the use of zero as a number at all, merely the use of some type of punctuation mark so
that the numbers had the correct interpretation.
 a round goose egg for zero
 The new numeration, which we generally call the Hindu system, is merely a new
 combination of three basic principles, all of ancient origin:

1. A decimal base
2. A positional notation
3. A ciphered form for each of the ten numerals
 The term for zero was pujyam.
 Another term for zero was shunyam which means blank.
 Zero now made it possible to note higher numerals with limited character.

TRIGONOMETRY
 The sines of angles up to 90° are given for twenty-four equal intervals of 3 ° each.

 Aryabhata used the value √10 for 𝜋, which appeared so frequently in India that it is
sometimes known as the Hindu value.

MULTIPLICATION
 Indians seem at first to have preferred to write numbers with the smaller units on the
left, hence they work from left to right.
 Among the devices used for multiplication was one that is known under various names:
lattice multiplication, gelosia multiplication, or cell or grating or quadrilateral
multiplication.
 In the first example (Fig. 10.2), the number 456 is multiplied by 34. The multiplicand
has been written above the lattice and the multiplier appears to the left, with the partial
products occupying the square cells. Digits in the diagonal rows are added, and the
product 15,504 is read off at the bottom and the right.
 To indicate that other arrangements are possible, a second example is given in Fig. 10.3,
 in which the multiplicand 537 is placed at the top, the multiplier 24 is on the right, and
the product 12,888 appears to the left and along the bottom.

ANCIENT INDIAN MATHEMATICIANS

 Apastamba - He was born on 600 B.C. and died on 500 B.C. He knew that the square
on the diagonal of a rectangle is equal to the sum of the squares on the two adjacent
sides. He is well known for his Sulbasutra wherein he gave the accurate value for the
square root of 2.
 Aryabhata I - Lived in the late fifth and early sixth centuries at Kusumapura. He wrote
a book entitled Aryabhatiya. Aryabhatiya is a slim volume, written in verse, covering
astronomy and mathematics. It is a brief descriptive work, in 123 metrical stanzas,
intended to supplement rules of calculation used in astronomy and mensurational
mathematics, with no appearance of deductive methodology. The position of the
Aryabhatiya of Aryabhata in India is somewhat akin to that of the Elements of Euclid
in Greece some eight centuries earlier. Both are summaries of earlier developments,
compiled by a single author.
 Brahmagupta – He is an ancient Indian astronomer. He was born in 598 A.D. at
Bhillamala. He is credited for the concept of zero. He was the first mathematician to
provide the formula for the area of a cyclic quadrilateral. Brahmagupta’s best-known
work, the Brahmasputa Siddhanta (Correctly Established Doctrine of Brahma), was
written in Bhinmal, a town in the Jalore district of Rajasthan, India. He was the first to
use zero as a number. He gave rules to compute with zero. Besides positive numbers,
he used negative numbers and zero for computing. The modern rule that two negative
numbers multiplied together equals a positive number first appears in Brahmasputa
Siddhanta.
 Bhaskara II - Approximately 500 years after Brahmagupta, in the twelfth century, the
mathematician Bhaskara, the second of that name, was born on the site of the modern
city of Bijapur, in southwestern India. Bhaskara II apparently wrote the Lilavati as a
textbook to form part of what we would call a liberal education. Bhaskara II (ad 1150)
is so much charmed of Brahmagupta’s intellect that he respectfully refers to him as
“Mahamatiman” (very intelligent person) and even confers the unique “Ganita Chakra
Chudamani” (the gem of the circle of mathematicians) title on Brahmagupta.
References
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