Transcript (or however tried to discover) an arithmetic method for solving

ARITHMETIC METHOD FOR
SOLVING EQUATIONS
And applications
Heron, Diophantus, Brahmagupta
and Al-Khwarizmi are
remembered together because
they all found (or however tried
to discover) an arithmetic
method for solving equations. In
fact they only came up with
formulas formed by sequential
numerical calculus and they
didn’t use incognita.
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WHEN: it’s not so clear but he
probably was born on the 10 AD
and he died around the 70 AD;
WHERE: he lived in Egypt and
spent lots of his time in the library
of the university of Alexandria;
WHAT: he found some important
formulas, like the formula of the
area of a triangle, the one for the
area of a orbitary quadrilateral
and the one for the area of a
cyclic quadrilateral; he invented
the first steam engine and some
instruments for daily life;
HOW: his most famous book is “
Metrica”, but he wrote many
others; he also taught maths,
mechanics and physical science;
Nothing is really known about Hero’s life in Egypt.
As a college student Hero loved to be in the library of the
University of Alexandria, because he particularly enjoyed
the series of gardens and the thousands books.
Hero was strongly influenced by the writings of Ctesibius of
Alexandria. He may have been a student of Ctesibius.
When he became older he taught at the University of
Alexandria. Hero taught mathematics, mechanics, and
physical science. He wrote many books and he used them
as text for his students, and manuals for technicians.
1.
2.
He found the formula of the area of a triangle. This
formula may be Archimedes’s but its presentation is
credited to Heron. The area can be computed if you
know the length of one side of the triangle (a, b, c)
with “s” as the semi perimeter:
The second formula that Hero found is useful for
determining the area of an cyclic quadrilateral (which
means a quadrilateral inscribed in a circle):
At that time it was used as a toy.
A hollow ball was supported on two
brackets on the lid of a basin of
boiling water.
One bracket was hollow and
conducted steam.
The steam escaped from two bent
pipes on the top, creating a force
that made it spin around.
The movement of the ball was used to
make puppets dance.
Although it was very simple, Hero's
aeolipile illustrated Sir Isaac
Newton's third law of motion which
tells that for every action there is
an equal and opposite reaction.
Hero's steam engine has been the first
step for the development of the jet
engine.
To open the doors, the priest lit a fire on
the altar, heating the air within and
expanding it.
This expansion in volume forced water
out of the sphere and into the
bucket, which moved downwards
under the extra weight. This bucket
was connected to a rope coiled
around a spindle and, as the bucket
moved downwards, this spindle
revolved, making the doors open.
Once the fire died down, the air
contracted and, to avoid leaving a
vacuum, the water siphoned from
the bucket back into the sphere,
causing the bucket to rise with the
aid of a counterweight. As a result,
the doors swung shut.
While it’s not sure that he actually built
this particular device, one can
imagine the wonder of a
congregation seeing this machine in
action; they would surely have
believed that it was magic!
The Dioptra was a practical invention of Heron that
became fundamental for building sprawling
cites and erecting temples and monuments. It
also became a mainstay of the Greek
astronomers, allowing them to judge the position
and elevation of celestial bodies.
Maybe Heron hasn’t been his creator and maybe
the Dioptra have existed before, but he wrote
an extensive treatise about the use of this
device. The device consisted of a circular table
fixed to a sturdy stand, and this was calibrated
and inscribed with angles.
The surveyor levelled the device, using small water
levels for accuracy, and used the disc to
measure the angle between two distant objects
with the aid of a rotating bar fitted with sights.
Using triangulation, the surveyor could also measure
the distance between two objects, especially if
the terrain as too difficult for Heron’s other
surveying device, the Odometer.
Cleverly, the rotating disc could be tilted, allowing
surveyors and astronomers to make vertical
measurements.
Heron’s organ included
a small wind-wheel,
which powered a
piston and forced air
through the organ
pipes, creating
sounds and tweets,
‘like the sound of a
flute’.
This device is believed
to be the first
example of wind
powering a machine.
One of Heron’s lasting
contribute to science is the
syringe, a device he used to
control the delivery of air or
fluid with precision.
This device, as modern syringes,
used suction to keep air or
liquid and, when the plunger
was depressed, it forced the
liquid out at a controllable
rate.
This device is obviously our
syringes’ ancestor.
That fountain seemed to power
itself, and used some very
sophisticated pneumatic and
hydraulic principles.
The fountain contained two
reservoirs, one of which was
filled with water.
As water was poured into the
upper tray, it flowed down to
the first reservoir, where it
compressed the air.
This compressed air was forced into
the second reservoir, where it
forced the water out and
created a powerful jet.
This device operated until the
bottom reservoir became filled
with water, when it had to be
reset.
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WHEN: he was born around
200-214 and he died
between 284 and 298;
WHERE: he studied at the
University of Alexandria, in
Egypt;
WHAT: he recognized
fractions as numbers and
he improved algebraic
notation;
HOW: his major contribute is
a collection of 13 books
called Arithmetica, he
wrote books with a series of
about 150 problems.
We only know a little about his life, but we can
count how many years he lived thanks for an
algebraic puzzle rhyme: “Here lies Diophantus:
God gave him his boyhood one-sixth of his life,
One-twelfth more as youth while whiskers grew
rife; And then yet one-seventh eve marriage
begun; In five years there came a bouncing new
son. Alas, the dear child of master and sage Met
fate at just half his dad's final age. Four years yet
his studies gave solace from grief; Then leaving
scenes earthly he, too found relief."
Did you solve the puzzle? The answer is 84 years old.
This is his major work and the most
prominent work on algebra in Greek
mathematics. It is a collection of problems
giving numerical solutions of determinate
and indeterminate equations. Nowadays we
only have six of the original thirteen books
which belonged to Arithmetica. Though
there are some who believe that four Arab
books discovered in 1968 are also by
Diophantus. Some Diophantine problems
from Arithmetica have been found in Arabic
sources.
Diophantus himself refers to a work which
consists of a collection of lemmas called “The
Porisms” (or Porismata), but this book is
completely lost. Many scholars and researchers
believe that The Porisms may have actually
been a part included inside Arithmetica or
indeed may have been the rest of Arithmetica.
However we know three lemmas contained in
The Porisms bacause Diophantus refered to
them in the Arithmetica. One of them is that
the difference of the cubes of two rational
numbers is equal to the sum of the cubes of two
other rational numbers.
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WHEN: he probably was born
in 598 and he died in the 670;
WHERE: he lived in India;
WHAT: he found solutions for
linear equations, he used
firstly the zero, he found the
formula for the area of a
cyclic quadrilateral, he wrote
the theorem of rational
triangles, he gave important
ideas of astronomy;
HOW: his major book is
“Brahmasphutasiddhanta”,
but he also wrote other three
books about astronomy and
arithmetic.
He mostly studied mathematics and astronomy and his works are
really important for the modern studies.
MATHEMATICS:
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He was the first to use zero as a number;
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He gave the solution of the general linear equation;
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He gave rules for dealing with five types of combinations of
fractions;
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He gave an approximate and an exact formula for the area of
a cyclic quadrilateral;
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He gave a theorem on rational triangles;
ASTRONOMY:
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He rebutted the idea that the Moon is farther from the Earth
than the Sun;
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He found methods for calculating the position of heavenly
bodies over time and for calculating solar and lunar eclipses;
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he observed that the Earth and heaven were spherical and
that the Earth is moving.
For a cyclic quadrilateral with sides of length a, b, c, and d, the
area is given by
where s is the semiperimeter
Given:
Draw chord AC. Extend AB and CD so
they meet at point P:
Angle ADC and Angle ABC subtend the same chord AC from
the two arcs of the circle. Therefore they are
supplementary. Angle ADP is supplementary to Angle ADC.
So
Triangle PBC and Triangle PDA are similar. The ratio of similarity
is
Area of Triangle PDA = * Area of Triangle PBC
Area ABCD = Area of Triangle PBC - Area of Triangle PDA
Perhaps Brahmagupta's most important innovation
was his treatment of the number zero.
Brahmagupta was the first who studied the behavior
of zero in common arithmetical equations, relating
zero to positive and negative numbers (which he
called fortunes and debts).
He correctly stated that multiplying any number by
zero yields a result of zero, but erred, as did many
other ancient mathematicians, in attempting to
define division by zero.
Nevertheless, Brahmagupta is sometimes referred to
as the "Father of Zero”.
He was the first one who described addition, subtration,
multiplication and division.
1.
2.
ADDITION: “[The sum] of two positives is positives, of two
negatives negative; of a positive and a negative [the
sum] is their difference; if they are equal it is zero. The
sum of a negative and zero is negative, [that] of a
positive and zero positive, [and that] of two zeros zero”.
SUBTRACTION: “A negative minus zero is negative, a
positive [minus zero] positive; zero [minus zero] is zero.
When a positive is to be subtracted from a negative or a
negative from a positive, then it is to be added”.
1.
2.
MULTIPLICATION: “The product of a negative and a
positive is negative, of two negatives positive, and of
positives positive; the product of zero and a negative, of
zero and a positive, or of two zeros is zero”.
DIVISION: “A positive divided by a positive or a negative
divided by a negative is positive; a zero divided by a
zero is zero; a positive divided by a negative is negative;
a negative divided by a positive is [also] negative.
A negative or a positive divided by zero has that [zero] as
its divisor, or zero divided by a negative or a positive
[has that negative or positive as its divisor]. The square
of a negative or of a positive is positive; [the square] of
zero is zero. That of which [the square] is the square is
[its] square-root”.
“Imaging two triangles within [a cyclic quadrilateral] with unequal
sides, the two diagonals are the two bases. Their two segments
are separately the upper and lower segments [formed] at the
intersection of the diagonals.
The two [lower segments] of the two diagonals are two sides in a
triangle; the base [of the quadrilateral is the base of the
triangle].
Its perpendicular is the lower portion of the [central]
perpendicular; the upper portion of the [central]
perpendicular is half of the sum of the [sides] perpendiculars
diminished by the lower [portion of the central
perpendicular]”.
Brahmagupta changed the
idea that the Moon is
farther from the Earth than
the Sun. He explained this
showing the illumination of
the Moon by the Sun. He
told that since the Moon is
closer to the Earth than the
Sun, the degree of the
illuminated part of the
Moon depends on the
relative positions of the Sun
and the Moon, and this can
be computed from the size
of the angle between the
two bodies.
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WHEN: he probably was born around
780 AD and he died around 850 AD;
WHERE: he lived in Baghdad and he
studied at “the House of Wisdom”;
WHAT: he worked on the first and
second degree (linear and quadratic
terms), he founded the mathematical
concept of the algorithm, he worked
on geography and astronomy;
HOW: he wrote two important books:
“The book of calculations by
completion and balancing” and “The
work of Al-Khwarizmi” (which was
named after him). His most important
contribute is his skill of getting easy
something really difficult.
Al-Khwarizmi received this name because in his book appeared for
the first time the word “algebra”.
His book is divided in two parts: in the first one he discussed about
linear and quadratic terms, in the second one he focused in the
aspect of business and the applications involved.
Al-Khwarizmi found six standard forms of equations:
1. squares equal to roots (10x²= 20x)
2. squares equal to numbers (10x² = 25)
3. roots equal to numbers (10x = 20)
4. squares and roots equal to numbers (x² + 10x = 39)
5. squares and numbers equal to roots (x² + 39 = 10x)
6. roots and numbers equal to squares (10x + 39 = x²)
This is the method found by Al-Khwarizmi
for solving a quadratic equation.
He used geometry in order to explain the
equation x²+10x=39.
He starts with a square of side x, which
therefore represents x².
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To the square we must add 10x and this
is done by adding four rectangles each
of breadth 10/4 and length x to the
square.
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Now the figure has area x² + 10x which
is equal to 39. We now complete the
square by adding the four little squares
each of area 5/2 x 5/2 = 25/4.
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Hence the outside square has area
4x25/4+39 = 25 + 39 = 64.
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The side of the square is therefore 8. But
the side is of length 5/2 + x + 5/2 so x + 5
= 8, giving x = 3.
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He did all his works rhetorically or spoken. In fact notation didn’t appear
until the 16th century.
He didn’t use negative numbers, so we can imagine he read Euclid’s
“Element” which explain the non-use of negative.
He was the first one who used the number zero: this is because he
integrated Greek and Hindu knowledge.
He reviewed Ptolemy’s ideas on geography and he checked them in
detail. With other geographers he also produced the first map of the
known world changing it into a globe.
WEBSITES USED:
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www.bsu.edu/web/cvjones/AlgBridge/father.htm
www.mathsisgoodforyou.com/people/alkhwarizmi.h
tm
www.math.twsu.edu/history/men/diophantus.html
www.nndb.com/people/744/000104432/
www.alexandrias.tripod.com/hero.htm
www.brahmagupta.net
www-groups.dcs.stand.ac.uk/.../Mathematicians/Heron.html
www.experiment-resources.com
en.wikipedia.org/wiki/Brahmagupta
Made by:
Bandini Francesca
“C.Colombo” classical high school
Genoa, Italy