The document provides a history of mathematics from ancient times through its development in various regions. It discusses: 1) Early counting methods and the origins of numerals in places like ancient Egypt, Mesopotamia, and India. 2) The mathematical advances of early civilizations like the Greeks, Chinese, Hindus, Babylonians and Egyptians - including concepts like zero, algebra, trigonometry, and geometry. 3) The transmission of mathematics from these early civilizations to medieval Islamic mathematics and eventually to European mathematics during the Renaissance, leading to modern developments.

1. Maths project
History of Mathematics
Done by
Pavan 9E
J.H.P.S

2. The area of study known as
the history of mathematics is
primarily an investigation into
the origin of discoveries
in mathematics and, to a lesser
extent, an investigation into
the mathematical methods and
notation of the past.

3. The first method of counting was counting on fingers. This
evolved into sign language for the hand-to-eye
communication of numbers. But this was not writing.
Tallies by carving notches in wood, bone, and stone were
used for at least forty thousand years. Stone age cultures,
including ancient Native American groups, used tallies for
gambling with horses, slaves, personal services and trade-
goods.
Roman numerals evolved from this primitive system of cutting
notches .It was once believed that they came from alphabetic
symbols, or from pictographs like the hand, but these
theories have been disproved.

4. Before the modern age and the worldwide
spread of knowledge, written examples of
new mathematical developments have
come to light only in a few locales. The most
ancient mathematical texts available
are Plimpton 322 Babylonian mathematics
Greek mathematics greatly refined the
methods (especially through the
introduction of deductive reasoning
and mathematical rigor in proofs) and
expanded the subject matter of
mathematics.

5. Egyptian mathematics c. 2000-1800 BC and
the Moscow Mathematical Papyrus Egyptian
mathematics c. 1890 BC. All of these texts concern
the so called Pythagorean theorem which seems to be
the most ancient and widespread mathematical
development after basic arithmetic and geometry.
The study of mathematics as a subject in its own right
begins in the 6th century BC with the Pythagoreans
who coined the term "mathematics" from the ancient
Greek word (mathema), meaning "subject of
instruction.

6. Chinese mathematics made early
contributions, including a place value
system.The Hindu-Arabic numeral
system and the rules for the use of its
operations, in use throughout the world
today, likely evolved over the course of
the first millennium AD in India and was
transmitted to the west via Islamic
mathematics Many Greek and Arabic texts
on mathematics were then translated into
latin which led to further development of
mathematics in medieval Europe.

7. From ancient times through the Middle
Ages, bursts of mathematical creativity
were often followed by centuries of
stagnation. Beginning
in Renaissance Italy in the 16th century,
new mathematical developments,
interacting with new scientific
discoveries, were made at anincreasing
pace that continues through the
present day.

8. Indian mathematics
The earliest civilization on the Indian subcontinent is
the Indus Valley Civilization that flourished between
2600 and 1900 BC in the Indus river basin. Their cities
were laid out with geometric regularity, but no known
mathematical documents survive from this civilization
The oldest extant mathematical records from India
are the Sulba Sutras (dated variously between the 8th
century BC and the 2nd century AD), appendices to
religious texts which give simple rules for constructing
altars of various shapes, such as squares, rectangles,
parallelograms, and others

9. zero
Zero was invented independently by the Babylonians, Mayans
and Indians (although some researchers say the Indian number
system was influenced by the Babylonians). The Babylonians got
their number system from the Sumerians, the first people in the
world to develop a counting system. Developed 4,000 to 5,000
years ago, the Sumerian system was positional — the value of a
symbol depended on its position relative to other symbols.
Robert Kaplan, author of "The Nothing That Is: A Natural History
of Zero," suggests that an ancestor to the placeholder zero may
have been a pair of angled wedges used to represent an empty
number column. However, Charles Seife, author of "Zero: The
Biography of a Dangerous Idea," disagrees that the wedges
represented a placeholder.

10. India: Where zero became a number
Some scholars assert that the Babylonian concept wove its way down
to India, but others give the Indians credit for developing zero
independently.
The concept of zero first appeared in India around A.D. 458.
Mathematical equations were spelled out or spoken in poetry or chants
rather than symbols. Different words symbolized zero, or nothing, such
as "void," "sky" or "space." In 628, a Hindu astronomer and
mathematician named Brahmagupta developed a symbol for zero — a
dot underneath numbers. He also developed mathematical operations
using zero, wrote rules for reaching zero through addition and
subtraction, and the results of using zero in equations. This was the first
time in the world that zero was recognized as a number of its own, as
both an idea and a symbol. By the 1600s, zero was used fairly widely
throughout Europe. It was fundamental in Rene Descartes’ Cartesian
coordinate system and in Sir Isaac Newton’s and Gottfried Wilhem
Liebniz’s developments of calculus. Calculus paved the way for physics,
engineering, computers, and much of financial and economic theory.

11. A Persian mathematician, Mohammed ibn-Musa al-
Khowarizmi, suggested that a little circle should be
used in calculations if no number appeared in the
tens place. The Arabs called this circle "sifr," or
"empty." Zero was crucial to al-Khowarizmi, who used
it to invent algebrain the ninth century. Al-
Khowarizmi also developed quick methods for
multiplying and dividing numbers, which are known
as algorithms — a corruption of his name.
Zero found its way to Europe through the Moorish
conquest of Spain and was further developed by
Italian mathematician Fibonacci, who used it to do
equations without an abacus, then the most
prevalent tool for doing arithmetic. This development
was highly popular among merchants, who used
Fibonacci’s equations involving zero to balance their
books.

12. Maths in differrent
countries
• The Ishango Bone, found in the area of the headwaters of the
Nile River (northeastern Congo), dates as early as 20,000 BC.
One common interpretation is that the bone is the earliest
known demonstration[7] of sequences of prime numbers and
Ancient Egyptian multiplication. Predynastic Egyptians of the
5th millennium BC pictorially represented geometric spatial
designs. It has been claimed that Megalithic monuments in
England and Scotland from the 3rd millennium BC, incorporate
geometric ideas such as circles, ellipses, and Pythagorean
triples in their design
1 Early mathematics
2 Ancient Near East (c. 1800-500 BC)
2.1 Mesopotamia
2.2 Egypt
3 Ancient Indian mathematics (c. 900
BC—AD 200)
4 Greek and Hellenistic mathematics
(c. 550 BC—AD 300)
5 Classical Chinese mathematics
(before c. 4th century BC— AD 1300)
6 Classical Indian mathematics (c.
400—1600)
7 Islamic mathematics (c. 800—1500)
8 Medieval European mathematics (c.
500—1400)
8.1 The Early Middle Ages (c. 500—
1100)
8.2 The Rebirth of Mathematics in
Europe (1100—1400)
9 Early Modern European mathematics
(c. 1400—1600)
The earliest known mathematics in ancient India
dates back to circa 3000-2600 BC in the Indus Valley
Civilization (Harappan civilization) of North India and
Pakistan, which developed a system of uniform
weights and measures that used the decimal system,
a surprisingly advanced brick technology which
utilised ratios, streets laid out in perfect right angles,
and a number of geometrical shapes and designs,
including cuboids, barrels, cones, cylinders, and
drawings of concentric and intersecting circles and
triangles. Mathematical instruments discovered
include an accurate decimal ruler with small and
precise subdivisions, a shell instrument that served
as a compass to measure angles on plane surfaces or
in horizon in multiples of 40–360 degrees

13. A shell instrument used to measure 8–12 whole sections of the
horizon and sky, and an instrument for measuring the positions
of stars for navigational purposes. The Indus script has not yet
been deciphered; hence very little is known about the written
forms of Harappan mathematics. Archeological evidence has led
some historians to believe that this civilization used a base 8
numeral system and possessed knowledge of the ratio of the
length of the circumference of the circle to its diameter, thus a
value of ?. Dating from the Shang period (1600—1046 BC), the
earliest extant Chinese mathematics consists of numbers
scratched on tortoise shell . These numbers use a decimal
system, so that the number 123 is written (from top to bottom)
as the symbol for 1 followed by the symbol for a hundred, then
the symbol for 2 followed by the symbol for ten, then the
symbol for 3. This was the most advanced number system in the
world at the time and allowed calculations to be carried out on
the suan pan or Chinese abacus. The date of the invention of the
suan pan is not certain, but the earliest written reference was in
AD 190 in the Supplementary Notes on the Art of Figures
written by Xu Yue.

14. Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (modern Iraq) from the
days of the early Sumerians until the beginning of the Hellenistic period. It is named Babylonian
mathematics due to the central role of Babylon as a place of study, which ceased to exist during the
Hellenistic period. From this point, Babylonian mathematics merged with Greek and Egyptian mathematics
to give rise to Hellenistic mathematics. Later under the Arab Empire, Iraq/Mesopotamia, especially Baghdad,
once again became an important center of study for Islamic mathematics.
In contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is
derived from more than 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets
were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these
appear to be graded homework.
The earliest evidence of written mathematics dates back to the ancient Sumerians, who built the earliest
civilization in Mesopotamia. They developed a complex system of metrology from 3000 BC. From around
2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical
exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.
The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics which include fractions,
algebra, quadratic and cubic equations, and the calculation of Pythagorean triples (see Plimpton 322).[11]
The tablets also include multiplication tables, trigonometry tables and methods for solving linear and
quadratic equations. The Babylonian tablet YBC 7289 gives an approximation to ?2 accurate to five decimal
places.
Babylonian mathematics was written using 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 (60 x 6) degrees in a circle.
Babylonian advances in mathematics were facilitated by the fact that 60 has many divisors. Also, unlike the
Egyptians, Greeks, and Romans, the Babylonians had a true place-value system, where digits written in the
left column represented larger values, much as in the decimal system. They lacked, however, an equivalent
of the decimal point, and so the place value of a symbol often had to be inferred from the context.
Egypt.

15. Egyptian mathematics
The Rhind papyrus (c. 1650 BC [3]) is another major Egyptian
mathematical text, an instruction manual in arithmetic and geometry.
In addition to giving area formulas and methods for multiplication,
division and working with unit fractions, it also contains evidence of
other mathematical knowledge (see [4]), including composite and
prime numbers; arithmetic, geometric and harmonic means; and
simplistic understandings of both the Sieve of Eratosthenes and perfect
number theory (namely, that of the number 6)[5]. It also shows how to
solve first order linear equations [6] as well as arithmetic and geometric
series [7].
Also, three geometric elements contained in the Rhind papyrus suggest
the simplest of underpinnings to analytical geometry: (1) first and
foremost, how to obtain an approximation of ? accurate to within less
than one percent; (2) second, an ancient attempt at squaring the circle;
and (3) third, the earliest known use of a kind of cotangen

16. Ancient Indian mathematics (c. 900
BC—AD 200)
Vedic mathematics begins in the early Iron Age, with the Shatapatha Brahmana (c. 9th century
BC), which approximates the value of ? to 2 decimal places and the Sulba Sutras (c. 800-500 BC)
were geometry texts that used irrational numbers, prime numbers, the rule of three and cube
roots; computed the square root of 2 to five decimal places; gave the method for squaring the
circle; solved linear equations and quadratic equations; developed Pythagorean triples
algebraically and gave a statement and numerical proof of the Pythagorean theorem.
Between 400 BC and AD 200, Jain mathematicians began studying mathematics for the sole
purpose of mathematics. They were the first to develop transfinite numbers, set theory,
logarithms, fundamental laws of indices, cubic equations, quartic equations, sequences and
progressions, permutations and combinations, squaring and extracting square roots, and finite
and infinite powers. The Bakshali Manuscript written between 200 BC and AD 200 included
solutions of linear equations with up to five unknowns, the solution of the quadratic equation,
arithmetic and geometric progressions, compound series, quadratic indeterminate equations,
simultaneous equations, and the use of zero and negative numbers. Accurate computations for
irrational numbers could be found, which includes computing square roots of numbers as large
as a million to at least 11 decimal places.

17. Greek mathematics
Pythagoras of Samos Greek mathematics refers to mathematics written in Greek between about
600 BCE and 450 CE.[12] Greek mathematicians lived in cities spread over the entire Eastern
Mediterranean, from Italy to North Africa, but were united by culture and language. Greek
mathematics is sometimes called Hellenistic mathematics.
Thales of MiletusGreek mathematics was much more sophisticated than the mathematics that
had been developed by earlier cultures. All surviving records of pre-Greek mathematics show
the use of inductive reasoning, that is, repeated observations used to establish rules of thumb.
Greek mathematicians, by contrast, used deductive reasoning. The Greeks used logic to derive
conclusions from definitions and axioms.[13]
Greek mathematics is thought to have begun with Thales (c. 624—c.546 BC) and Pythagoras (c.
582—c. 507 BC). Although the extent of the influence is disputed, they were probably inspired
by the ideas of Egypt, Mesopotamia and perhaps India. According to legend, Pythagoras
travelled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests. Some
say the greatest of Greek mathematicians, if not of all time, was Archimedes (287—212 BC) of
Syracuse. According to Plutarch, at the age of 75, while drawing mathematical formulas in the
dust, he was run through with a spear by a Roman soldier. Ancient Rome left little evidence of
any interest in pure mathematics.

18. Chinese mathematics (before c. 4th
century BC— AD 1300)
From the Western Zhou Dynasty (from 1046 BC), the oldest
mathematical work to survive the book burning is the I Ching,
which uses the 8 binary 3-tuples (trigrams) and 64 binary 6-
tuples (hexagrams) for philosophical, mathematical, and/or
mystical purposes. The binary tuples are composed of broken
and solid lines, called yin 'female' and yang 'male' respectively
(see King Wen sequence).
The oldest existent work on geometry in China comes from the
philosophical Mohist canon of c. 330 BC, compiled by the
followers of Mozi (470 BC-390 BC). The Mo Jing described
various aspects of many fields associated with physical science,
and provided a small wealth of information on mathematics as
well.

19. Classical Indian mathematics (c. 400—
1600)
AryabhataThe Surya Siddhanta (c. 400) introduced the trigonometric functions of sine, cosine, and inverse sine, and laid
down rules to determine the true motions of the luminaries, which conforms to their actual positions in the sky. The
cosmological time cycles explained in the text, which was copied from an earlier work, corresponds to an average sidereal
year of 365.2563627 days, which is only 1.4 seconds longer than the modern value of 365.25636305 days. This work was
translated to Arabic and Latin during the Middle Ages.
Aryabhata in 499 introduced the versine function, produced the first trigonometric tables of sine, developed techniques and
algorithms of algebra, infinitesimals, differential equations, and obtained whole number solutions to linear equations by a
method equivalent to the modern method, along with accurate astronomical calculations based on a heliocentric system of
gravitation. An Arabic translation of his Aryabhatiya was available from the 8th century, followed by a Latin translation in
the 13th century. He also computed the value of ? to the fourth decimal place as 3.1416. Madhava later in the 14th century
computed the value of ? to the eleventh decimal place as 3.14159265359.
In the 7th century, Brahmagupta identified the Brahmagupta theorem, Brahmagupta's identity and
Brahmagupta's formula, and for the first time, in Brahma-sphuta-siddhanta, he lucidly explained the use of
zero as both a placeholder and decimal digit and explained the Hindu-Arabic numeral system. It was from a
translation of this Indian text on mathematics (around 770) that Islamic mathematicians were introduced to
this numeral system, which they adapted as Arabic numerals. Islamic scholars carried knowledge of this
number system to Europe by the 12th century, and it has now displaced all older number systems
throughout the world. In the 10th century, Halayudha's commentary on Pingala's work contains a study of
the Fibonacci sequence and Pascal's triangle, and describes the formation of a matrix.

20. Islamic mathematics (c. 800—1500)
The Islamic Arab Empire established across the Middle East, Central
Asia, North Africa, Iberia, and in parts of India in the 8th century
made significant contributions towards mathematics. Although most
Islamic texts on mathematics were written in Arabic, they were not all
written by Arabs, since much like the status of Greek in the Hellenistic
world, Arabic was used as the written language of non-Arab scholars
throughout the Islamic world at the time. Some of the most important
Islamic mathematicians were Persian.

21. Done by
Battula pavan