Download The area of study known as the history of mathematics is primarily

The area of study known as the history of mathematics is primarily an investigation into the
origin of new discoveries in mathematics and, to a lesser extent, an investigation into the
standard mathematical methods and notation of the past.
Egyptian and Babylonian mathematics were then further developed in Greek and Hellenistic
mathematics, which is generally considered to be one of the most important for greatly
expanding both the method and the subject matter of mathematics. The mathematics
developed in these ancient civilizations were then further developed and greatly expanded in
Islamic mathematics. Many Greek and Arabic texts on mathematics were then translated into
Latin in medieval Europe and further developed there.

The Ishango bone, dating to perhaps 18000 to 20000 B.C.
Long before the earliest written records, there are drawings that indicate some knowledge of
elementary mathematics and of time measurement based on the stars. For example,
paleontologists have discovered in a cave in South Africa, ochre rocks about 70,000 years old,
adorned with scratched geometric patterns. Also prehistoric artifacts discovered in Africa and
France, dated between 35,000 and 20,000 years old, suggest early attempts to quantify time.
There is evidence that women devised counting to keep track of their menstrual cycles; 28 to
30 scratches on bone or stone, followed by a distinctive marker. Moreover, hunters and
herders employed the concepts of one, two, and many, as well as the idea of none or zero,
when considering herds of animals.
The Ishango bone, found near the headwaters of the Nile river (northeastern Congo), may be
as much as 20,000 years old. One common interpretation is that the bone is the earliest known
demonstration of sequences of prime numbers and of 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, dating from
the 3rd millennium BC, incorporate geometric ideas such as circles, ellipses, and Pythagorean
triples in their design.
The earliest known mathematics in ancient India dates from 3000-2600 BC in the Indus
Valley Civilization (Harappan civilization) of North India and Pakistan. This civilization
developed a system of uniform weights and measures that used the decimal system, a
surprisingly advanced brick technology which utilized 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 included 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, 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. Archeological evidence has led some
to suspect that this civilization used a base 8 numeral system and had a value of π, the ratio of
the length of the circumference of the circle to its diameter.
.
Ancient Near East (c. 1800-500 BC)
Mesopotamia
Babylonian mathematics refers to any mathematics of the people of Mesopotamia (modern
Iraq) from the days of the early Sumerians until the beginning of the Hellenistic period.
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). 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 were 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
Egyptian mathematics refers to mathematics written in the Egyptian language. From the
Hellenistic period, Greek replaced Egyptian as the written language of Egyptian scholars, and
from this point Egyptian mathematics merged with Greek and Babylonian mathematics to
give rise to Hellenistic mathematics. Mathematical study in Egypt later continued under the
Arab Empire as part of Islamic mathematics, when Arabic became the written language of
Egyptian scholars.
The oldest mathematical text discovered so far is the Moscow papyrus, which is an Egyptian
Middle Kingdom papyrus dated c. 2000—1800 BC. Like many ancient mathematical texts, it
consists of what are today called "word problems" or "story problems", which were
apparently intended as entertainment.
The Rhind papyrus (c. 1650 BC) 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, including composite and prime numbers; arithmetic, geometric and
harmonic means; and simplistic understandings of the perfect number theory (namely, that of
the number 6). It also shows how to solve first order linear equations as well as arithmetic
and geometric series.
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 π 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 cotangent.
Finally, the Berlin papyrus (c. 1300 BC) shows that ancient Egyptians could solve a secondorder algebraic equation .
Hypatia was the daughter of Theon of Alexandria who was a teacher of mathematics with the
Museum of Alexandria in Egypt.
Hypatia studied with her father, and with many others including Plutarch the Younger. She
herself taught at the Neoplatonist school of philosophy. She became the salaried director of
this school in 400. She probably wrote on mathematics, astronomy and philosophy, including
about the motions of the planets, about number theory and about conic sections.
Hypatia corresponded with and hosted scholars from others cities. Synesius, Bishop of
Ptolemais, was one of her correspondents and he visited her frequently. Hypatia was a popular
lecturer, drawing students from many parts of the empire.
Hypatia dressed in the clothing of a scholar or teacher, rather than in women's clothing. She
moved about freely, driving her own chariot, contrary to the norm for women's public
behavior. She exerted considerable political influence in the city.
Christian bishop, Cyril, a future saint probably objected to Hypatia on a number of counts:
She represented heretical teachings, including experimental science and pagan religion. She
was an associate of Orestes. And she was a woman who didn't know her place. Cyril's
preaching against Hypatia is said to have been what incited a mob led by fanatical Christian
monks in 415 to attack Hypatia as she drove her chariot through Alexandria. They dragged
her from her chariot and, according to accounts from that time, stripped her, killed her,
stripped her flesh from her bones, scattered her body parts through the streets, and burned
some remaining parts of her body in the library of Caesareum.
Hypatia's students fled to Athens, where the study of mathematics flourished after that. The
Neoplatonic school she headed continued in Alexandria until the Arabs invaded in 642.
When the library of Alexandria was burned by the Arab conquerors, used as fuel for baths, the
works of Hypatia were destroyed. We know her writings today through the works of others
who quoted her -- even if unfavorably -- and a few letters written to her by contemporaries.
Ancient Indian mathematics (c. 900 BC — AD 200)
Brahmi numerals in the first century CE
Vedic mathematics began in the early Iron Age, with the Shatapatha Brahmana (c. 9th
century BC), which approximates the value of π 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.
Pāṇini (c. 5th century BC) formulated the grammar rules for Sanskrit. Pingala (roughly 3rd1st centuries BC) in his treatise of prosody uses a device corresponding to a binary numeral
system. His discussion of the combinatorics of meters, corresponds to the binomial theorem.
Pingala's work also contains the basic ideas of Fibonacci numbers (called mātrāmeru).
Between 400 BC and AD 200, Jaina 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, quadratic equations, sequences and
progressions, permutations and combinations, squaring and extracting square roots, and finite
and infinite powers. The Bakhshali 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.
Here's how to use VERTICALLY AND CROSSWISE for multiplying numbers close to 100.

Suppose you want to multiply 88 by 98.
Not easy,you might think. But with
VERTICALLY AND CROSSWISE you can give
the answer immediately, using the same method
as above.
Both 88 and 98 are close to 100.
88 is 12 below 100 and 98 is 2 below 100.
You can imagine the sum set out like this:
As before the 86 comes from
subtracting crosswise: 88 - 2 = 86
(or 98 - 12 = 86: you can subtract
either way, you will always get
the same answer).
And the 24 in the answer is
just 12 x 2: you multiply vertically.
So 88 x 98 = 8624
This is so easy it is just mental arithmetic
A quick way to square numbers that end in 5 using the formula BY ONE MORE THAN THE
ONE BEFORE.

752 = 5625
752 means 75 x 75.
The answer is in two parts: 56 and 25.
The last part is always 25.
The first part is the first number, 7, multiplied by the number "one more", which is 8:
so 7 x 8 = 56

Similarly 852 = 7225 because 8 x 9 = 72.
Greek and Hellenistic mathematics (c. 550 BC—AD 300)
Greek mathematics refers to mathematics written in Greek between about 600 BCE and 450
CE. 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 of the
period following Alexander the Great is sometimes called Hellenistic mathematics.
Greek 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.
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.
Thales used geometry to solve problems such as calculating the height of pyramids and the
distance of ships from the shore. Pythagoras is credited with the first proof of the Pythagorean
theorem, though the statement of the theorem has a long history. In his commentary on
Euclid, Proclus states that Pythagoras expressed the theorem that bears his name and
constructed Pythagorean triples algebraically rather than geometrically. The Academy of
Plato had the motto "let none unversed in geometry enter here".
The Pythagoreans discovered the existence of irrational numbers. Eudoxus (408 —c.355 BC)
developed the method of exhaustion, a precursor of modern integration. Aristotle (384—c.322
BC) first wrote down the laws of logic. Euclid (c. 300 BC) is the earliest example of the
format still used in mathematics today, definition, axiom, theorem, proof. He also studied
conics. His book, Elements, was known to all educated people in the West until the middle of
the 20th century. In addition to the familiar theorems of geometry, such as the Pythagorean
theorem, Elements includes a proof that the square root of two is irrational and that there are
infinitely many prime numbers. The Sieve of Eratosthenes (ca. 230 BC) was used to discover
prime numbers.
Some say the greatest of Greek mathematicians, if not of all time, was Archimedes (c.287—
212 BC) of Syracuse.
Death of Archimedes
When Syracuse was taken, Archimedes was describing mathematical figures upon the earth, and
when one of the enemy came upon him, sword in hand, and asked his name, he was so engrossed
with the desire of preserving the figures entire, that he answered only by an earnest request to the
soldier to keep off, and not break in upon his circle. The soldier, conceiving himself scorned, ran
Archimedes through the body, the purple streams gushing from which soon obscured all traces of the
problem on which he had been so intent. Thus fell this illustrious man, from the mere neglect to tell
his name; for it is due to the Roman general, Marceline, to state, that he had given special orders to
his men to respect the life and person of the philosopher
He used the method of exhaustion to calculate the area under the arc of a parabola with the
summation of an infinite series, and gave remarkably accurate approximations of Pi. He also
defined the spiral bearing his name, formulas for the volumes of surfaces of revolution and an
ingenious system for expressing very large numbers.
Greek Women Philosophers
Women were able to contribute to the "search for wisdom" during the period between
800 BC and 500 BC in Greece. Greek women had virtually no political rights in the
male dominated society. Although woman did receive some education, a woman's
place was in the house, supervising the daily running of the household ("Women's
Life"). Keeping the women inside made it almost impossible "for women of the
'respectable' class to pursue a profession" (Finnegan). Philosophy was a male
dominated profession even less likely to include women "since its practice often
involved the discussion of theories in groups or sects" (Finnegan). The minority of
women who were able to enter the field of philosophy overcame subjugation and
found great opportunities in education in order to learn to read, write, and think.
Greek women received their education either in the home or from well educated
experts. Girls were educated in their homes in areas of reading, writing, arithmetic,
spinning, weaving, embroidery, singing, dancing and playing a musical instrument
(Olsen 10). Spartan girls received a formal education more similar to the training
boys received (Roice). Respectable women were not talked about in public. It was
taboo to even mention them by name (Finnegan). Therefore, proof of female literary
production is almost non-existent. With a low percentage of literate women, the few
working Athenian women had occupations that "could hardly be described as
professional careers, but rather, as the least condemnable means of earning a wage"
(Finnegan). The women hardly came in contact with men because they were avoided
as many believed that the women belonged in the home. There were various women
who were able to pursue a chosen career and achieve fame.
In a time when it was common belief that a woman's nature was different from man's
but not of lesser value, some women were major contributors to the works of the
Pythagorean school. These select women entered the Pythagorean society on an equal
basis as men. Pythagoras, known as the 'Feminist Philosopher' because many of his
works were influenced by various women, founded the Pythagorean school.
Pythagoras was well respected and often men "gave their wives into the charge of
Pythagoras to learn his doctrines" (Finnegan). After his death in a fire in the home of
one of his daughters, his wife Theano became the director of the school. Their three
daughters Arignote, Myia, and Damo were also educated at the school and helped to
continue it's teaching.
Pythagoreanism described a harmony in the "cosmos" that exhibits order and beauty.
Numbers could explain all things in the universe. Since numbers include both odds
and evens, all things have a contradiction- light and dark, the limited and unlimited,
good and evil, male and female.
Theano was the most famous woman of the Pythagoreans. She wrote on the "number
theory" and explained it as a principle to create order that helped to distinguish one
thing from another.
Theano also wrote about the ethics a woman should adopt in daily life. It defined that
a wife's sexual activity must be restricted to pleasing her husband. She cannot have
any other lovers. If a woman has sex with her husband, she remains "pure," since "in
the context of marriage, chastity and virtue are not identified with abstinence"
(Waithe 14). But a woman can never be "pure" if she has sex with someone other
than her husband. Her view of romantic love was that it was nothing but "the natural
inclination of an empty soul" (Waithe 14). A woman's responsibility was solely to
maintain law, justice and harmony in the home. A woman who did not abide by this
contributed to the chaos and disorder of society.