History of Math Essay

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History of Math



Mathematics, study of relationships among quantities, magnitudes, and properties and of
logical operations by which unknown quantities, magnitudes, and properties may be deduced.
In the past, mathematics was regarded as the science of quantity, whether of magnitudes,
as in geometry, or of numbers, as in arithmetic, or of the generalization of these two
fields, as in algebra. Toward the middle of the 19th century, however, mathematics came to
be regarded increasingly as the science of relations, or as the science that draws
necessary conclusions. This latter view encompasses mathematical or symbolic logic, the
science of using symbols to provide an exact theory of logical deduction and inference
based on definitions, axioms, postulates, and rules for combining and transforming
primitive elements into more complex relations and theorems.

This brief survey of the history of mathematics traces the evolution of mathematical ideas
and concepts, beginning in prehistory. Indeed, mathematics is nearly as old as humanity
itself; evidence of a sense of geometry and interest in geometric pattern has been found
in the designs of prehistoric pottery and textiles and in cave paintings. Primitive
counting systems were almost certainly based on using the fingers of one or both hands, as
evidenced by the predominance of the numbers 5 and 10 as the bases for most number systems
today.

Ancient Mathematics
The earliest records of advanced, organized mathematics date back to the ancient
Mesopotamian country of Babylonia and to Egypt of the 3rd millennium BC. There mathematics
was dominated by arithmetic, with an emphasis on measurement and calculation in geometry
and with no trace of later mathematical concepts such as axioms or proofs.

The earliest Egyptian texts, composed about 1800 BC, reveal a decimal numeration system
with separate symbols for the successive powers of 10 (1, 10, 100, and so forth), just as
in the system used by the Romans. Numbers were represented by writing down the symbol for
1, 10, 100, and so on as many times as the unit was in a given number. For example, the
symbol for 1 was written five times to represent the number 5, the symbol for 10 was
written six times to represent the number 60, and the symbol for 100 was written three
times to represent the number 300. Together, these symbols represented the number 365.
Addition was done by totaling separately the units-10s, 100s, and so forth-in the numbers
to be added. Multiplication was based on successive doublings, and division was based on
the inverse of this process.

The Egyptians used sums of unit fractions (a), supplemented by the fraction B, to express
all other fractions. For example, the fraction E was the sum of the fractions 3 and *.
Using this system, the Egyptians were able to solve all problems of arithmetic that
involved fractions, as well as some elementary problems in algebra. In geometry, the
Egyptians calculated the correct areas of triangles, rectangles, and trapezoids and the
volumes of figures such as bricks, cylinders, and pyramids. To find the area of a circle,
the Egyptians used the square on U of the diameter of the circle, a value of about
3.16-close to the value of the ratio known as pi, which is about 3.14.

The Babylonian system of numeration was quite different from the Egyptian system. In the
Babylonian system-which, when using clay tablets, consisted of various wedge-shaped
marks-a single wedge indicated 1 and an arrowlike wedge stood for 10 (see table). Numbers
up through 59 were formed from these symbols through an additive process, as in Egyptian
mathematics. The number 60, however, was represented by the same symbol as 1, and from
this point on a positional symbol was used. That is, the value of one of the first 59
numerals depended henceforth on its position in the total numeral. For example, a numeral
consisting of a symbol for 2 followed by one for 27 and ending in one for 10 stood for 2 ×
602 27 × 60 10. This principle was extended to the representation of fractions as
well, so that the above sequence of numbers could equally well represent 2 × 60 27 10
× (†), or 2 27 × (†) 10 × (†-2). With this sexagesimal system (base
60), as it is called, the Babylonians had as convenient a numerical system as the 10-based
system.

The Babylonians in time developed a sophisticated mathematics by which they could find the
positive roots of any quadratic equation (Equation). They could even find the roots of
certain cubic equations. The Babylonians had a variety of tables, including tables for
multiplication and division, tables of squares, and tables of compound interest. They
could solve complicated problems using the Pythagorean theorem; one of their tables
contains integer solutions to the Pythagorean equation, a2 b2 = c2, arranged so that
c2/a2 decreases steadily from 2 to about J. The Babylonians were able to sum arithmetic
and some geometric progressions, as well as sequences of squares. They also arrived at a
good approximation for ¸. In geometry, they calculated the areas of rectangles, triangles,
and trapezoids, as well as the volumes of simple shapes such as bricks and cylinders.
However, the Babylonians did not arrive at the correct formula for the volume of a
pyramid.

Greek Mathematics
The Greeks adopted elements of mathematics from both the Babylonians and the Egyptians.
The new element in Greek mathematics, however, was the invention of an abstract
mathematics founded on a logical structure of definitions, axioms, and proofs. According
to later Greek accounts, this development began in the 6th century BC with Thales of
Miletus and Pythagoras of Samos, the latter a religious leader who taught the importance
of studying numbers in order to understand the world. Some of his disciples made important
discoveries about the theory of numbers and geometry, all of which were attributed to
Pythagoras.

In the 5th century BC, some of the great geometers were the atomist philosopher Democritus
of Abdera, who discovered the correct formula for the volume of a pyramid, and Hippocrates
of Chios, who discovered that the areas of crescent-shaped figures bounded by arcs of
circles are equal to areas of certain triangles. This discovery is related to the famous
problem of squaring the circle-that is, constructing a square equal in area to a given
circle. Two other famous mathematical problems that originated during the century were
those of trisecting an angle and doubling a cube-that is, constructing a cube the volume
of which is double that of a given cube. All of these problems were solved, and in a
variety of ways, all involving the use of instruments more complicated than a straightedge
and a geometrical compass. Not until the 19th century, however, was it shown that the
three problems mentioned above could never have been solved using those instruments alone.

In the latter part of the 5th century BC, an unknown mathematician discovered that no unit
of length would measure both the side and diagonal of a square. That is, the two lengths
are incommensurable. This means that no counting numbers n and m exist whose ratio
expresses the relationship of the side to the diagonal. Since the Greeks considered only
the counting numbers (1, 2, 3, and so on) as numbers, they had no numerical way to express
this ratio of diagonal to side. (This ratio, ¸, would today be called irrational.) As a
consequence the Pythagorean theory of ratio, based on numbers, had to be abandoned and a
new, nonnumerical theory introduced. This was done by the 4th-century BC mathematician
Eudoxus of Cnidus, whose solution may be found in the Elements of Euclid. Eudoxus also
discovered a method for rigorously proving statements about areas and volumes by
successive approximations.

Euclid was a mathematician and teacher who worked at the famed Museum of Alexandria and
who also wrote on optics, astronomy, and music. The 13 books that make up his Elements
contain much of the basic mathematical knowledge discovered up to the end of the 4th
century BC on the geometry of polygons and the circle, the theory of numbers, the theory
of incommensurables, solid geometry, and the elementary theory of areas and volumes.

The century that followed Euclid was marked by mathematical brilliance, as displayed in
the works of Archimedes of Syracuse and a younger contemporary, Apollonius of Perga.
Archimedes used a method of discovery, based on theoretically weighing infinitely thin
slices of figures, to find the areas and volumes of figures arising from the conic
sections. These conic sections had been discovered by a pupil of Eudoxus named Menaechmus,
and they were the subject of a treatise by Euclid, but Archimedes' writings on them are
the earliest to survive. Archimedes also investigated centers of gravity and the stability
of various solids floating in water. Much of his work is part of the tradition that led,
in the 17th century, to the discovery of the calculus. Archimedes was killed by a Roman
soldier during the sack of Syracuse. His younger contemporary, Apollonius, produced an
eight-book treatise on the conic sections that established the names of the sections:
ellipse, parabola, and hyperbola. It also provided the basic treatment of their geometry
until the time of the French philosopher and scientist René Descartes in the 17th century.

After Euclid, Archimedes, and Apollonius, Greece produced no geometers of comparable
stature. The writings of Hero of Alexandria in the 1st century AD show how elements of
both the Babylonian and Egyptian mensurational, arithmetic traditions survived alongside
the logical edifices of the great geometers. Very much in the same tradition, but
concerned with much more difficult problems, are the books of Diophantus of Alexandria in
the 3rd century AD. They deal with finding rational solutions to kinds of problems that
lead immediately to equations in several unknowns. Such equations are now called
Diophantine equations (see Diophantine Analysis).

Applied Mathematics in Greece
Paralleling the studies described in pure mathematics were studies made in optics,
mechanics, and astronomy. Many of the greatest mathematical writers, such as Euclid and
Archimedes, also wrote on astronomical topics. Shortly after the time of Apollonius, Greek
astronomers adopted the Babylonian system for recording fractions and, at about the same
time, composed tables of chords in a circle. For a circle of some fixed radius, such
tables give the length of the chords subtending a sequence of arcs increasing by some
fixed amount. They are equivalent to a modern sine table, and their composition marks the
beginnings of trigonometry. In the earliest such tables-those of Hipparchus in about 150
BC-the arcs increased by steps of 71°, from 0° to 180°. By the time of the astronomer
Ptolemy in the 2nd century AD, however, Greek mastery of numerical procedures had
progressed to the point where Ptolemy was able to include in his Almagest a table of
chords in a circle for steps of 3°, which, although expressed sexagesimally, is accurate
to about five decimal places.

In the meantime, methods were developed for solving problems involving plane triangles,
and a theorem-named after the astronomer Menelaus of Alexandria-was established for
finding the lengths of certain arcs on a sphere when other arcs are known. These advances
gave Greek astronomers what they needed to solve the problems of spherical astronomy and
to develop an astronomical system that held sway until the time of the German astronomer
Johannes Kepler.

Medieval and Renaissance Mathematics
Following the time of Ptolemy, a tradition of study of the mathematical masterpieces of
the preceding centuries was established in various centers of Greek learning. The
preservation of such works as have survived to modern times began with this tradition. It
was continued in the Islamic world, where original developments based on these
masterpieces first appeared.

Islamic and Indian Mathematics
After a century of expansion in which the religion of Islam spread from its beginnings in
the Arabian Peninsula to dominate an area extending from Spain to the borders of China,
Muslims began to acquire the results of the “foreign sciences.” At centers such as the
House of Wisdom in Baghdad, supported by the ruling caliphs and wealthy individuals,
translators produced Arabic versions of Greek and Indian mathematical works.

By the year 900 AD the acquisition was complete, and Muslim scholars began to build on
what they had acquired. Thus mathematicians extended the Hindu decimal positional system
of arithmetic from whole numbers to include decimal fractions, and the 12th-century
Persian mathematician Omar Khayyam generalized Hindu methods for extracting square and
cube roots to include fourth, fifth, and higher roots. In algebra, al-Karaji completed the
algebra of polynomials of Muhammad al-Khwarizmi. Al-Karaji included polynomials with an
infinite number of terms. (Al-Khwarizmi's name, incidentally, is the source of the word
algorithm, and the title of one of his books is the source of the word algebra.) Geometers
such as Ibrahim ibn Sinan continued Archimedes' investigations of areas and volumes, and
Kamal al-Din and others applied the theory of conic sections to solve optical problems.
Using the Hindu sine function and Menelaus's theorem, mathematicians from Habas al-Hasib
to Nasir ad-Din at-Tusi created the mathematical disciplines of plane and spherical
trigonometry. These did not become mathematical disciplines in the West, however, until
the publication of De Triangulis Omnimodibus by the German astronomer Regiomontanus.

Finally, a number of Muslim mathematicians made important discoveries in the theory of
numbers, while others explained a variety of numerical methods for solving equations. The
Latin West acquired much of this learning during the 12th century, the great century of
translation. Together with translations of the Greek classics, these Muslim works were
responsible for the growth of mathematics in the West during the late Middle Ages. Italian
mathematicians such as Leonardo Fibonacci and Luca Pacioli, one of the many 15th-century
writers on algebra and arithmetic for merchants, depended heavily on Arabic sources for
their knowledge.

Western Renaissance Mathematics
Although the late medieval period saw some fruitful mathematical considerations of
problems of infinity by writers such as Nicole Oresme, it was not until the early 16th
century that a truly important mathematical discovery was made in the West. The discovery,
an algebraic formula for the solution of both the cubic and quartic equations, was
published in 1545 by the Italian mathematician Gerolamo Cardano in his Ars Magna. The
discovery drew the attention of mathematicians to complex numbers and stimulated a search
for solutions to equations of degree higher than 4. It was this search, in turn, that led
to the first work on group theory (Group) at the end of the 18th century, and to the
theory of equations developed by the French mathematician Évariste Galois in the early
19th century.

The 16th century also saw the beginnings of modern algebraic symbolism (Mathematical
Symbols), as well as the remarkable work on the solution of equations by the French
mathematician François Viète. His writings influenced many mathematicians of the following
century, including Pierre de Fermat in France and Isaac Newton in England.

Mathematics Since the 16th Century
Europeans dominated in the development of mathematics after the Renaissance.
17th Century
During the 17th century, the greatest advances were made in mathematics since the time of
Archimedes and Apollonius. The century opened with the discovery of logarithms by the
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