Methods for solving Equations

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Nov 30, 2013 (3 years and 9 months ago)

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Methods

for
solving

Equations

by: Rita Barrocas 11ºB

Scipione

del Ferro


Scipione

del Ferro was the first mathematician to find a
solution to the depressed cubic equation which is the
equation of the form:

ax
3

+
bx

= c


He was born on 6
th

of February of 1465 in Bologna,
Italy. His father was a papermaker which was a rising
profession due the invention of the printing with
moveable types. He attended University of Bologna
which had been founded in the 11
th

century.



by: Rita Barrocas 11ºB


In 1496, del Ferro became a lecturer in mathematics at the University
of Bologna. It’s not known about any book that del Ferro had
wroten
,
and it is
belived

that it is because he was very reserved about his
dicoveries

and only share them with close friends and selected students.
He kept his discoveries in a notebook. This notebook was later the prove
that del Ferro had indeed discovered the solution to the depressed cubic
equation.


Del Ferro died on November 5, 1526. When he died, the notebook was
given to is son
-
in
-
law Hannibal Nave. The discovery of this notebook
was what influenced
Cardano

to release his famous math book
Ars

Magna
.


Cardano

had promised
Tartaglia

that he would not reveal the method
but he felt that del Ferro's earlier discovery of the method excused him
from this promise
.


by: Rita Barrocas 11ºB

Nicolò

Tartalia



On the 16
th

Century in the
Renassaince

in
italy
, Bologna
University was known by the intense mathematics competitions.


In 1535 There was such a competition that the improbable figure
of
Nicolò

Tartaglia

discovered a
matematical

finding that was
consideres

impossible that had
preplexed

the best
matematicians

of China, India and the
islamic

world.


Niccolò Fontana became known as
Tartaglia

, that means “the
stamerer
”,
for a
communication defect that
he suffered due to an
damage
he received in a
battle during the invasion of the French
army.


He
was a poor engineer known for designing fortifications, a
evaluator
of topography
and
a bookkeeper in the Republic of
Venice.


by: Rita Barrocas 11ºB


But he was also a self
-
taught, but wildly ambitious,
mathematician.


He was distinguished by
producing
,
the first Italian
translations of works by Archimedes and Euclid from
untouched
Greek
texts,
as well as an acclaimed compilation
of mathematics of his own.


by: Rita Barrocas 11ºB


Tartaglia's

greatest contribution to
mathematical history,
occurred in 1535 when
he won the
Bologna
University
mathematics competition by demonstrating a general algebraic
formula
to solve
cubic equations
, equations
with terms
including
x
3
,
something
that was an impossibility in that time,
requiring
an
understanding of the square roots of negative
numbers. In the competition, he beat
Scipione

del
Ferro, who
had coincidentally produced his own partial solution to the
cubic equation
wich

never had happened before. However
Scipione's

solution
perhaps
predated
Tartaglia’s
, it was much
more limited, and
Tartaglia

is usually credited with the first
general solution. In the highly competitive and cut
-
throat
environment of 16th Century Italy,
Tartaglia

even hided his
solution in the form of a poem in an attempt to make it more
difficult for other mathematicians to steal it.


by: Rita Barrocas 11ºB


Tartaglia’s

definitive method was, however, leaked to
Gerolamo

Cardano

,
a
rather eccentric and confrontational mathematician, doctor and Renaissance
man, and author throughout his lifetime of some 131 books.
Cardano

published it himself in his 1545 book "
Ars

Magna" (despite having
promised
Tartaglia

that
he would not), along
with the work of his own
brilliant student
Lodovico

Ferrari. Ferrari, on seeing
Tartaglia's

cubic
solution, had realized that he could use a similar method to solve
quartic

equations
.



Tartaglia
,
Cardano

and Ferrari between them demonstrated the first uses
of
what is now known
as complex numbers, combinations of real and
imaginary numbers of the type
a

+
bi
, where
i

is the imaginary unit √
-
1. It
fell to another Bologna resident, Rafael
Bombelli
, to explain, at the end of
the


An in the end of the
1560's
, Rafael
Bombelli

that was also from bologna
was the one
tha

determinated

exactly
what imaginary numbers really were
and how they could be used.


by: Rita Barrocas 11ºB


Tartaglia

died ruined and unknown, despite having
produced the cubic equation’s solution, the first translation
of Euclid’s “Elements” in a modern European language,
formulated
Tartaglia's

Formula for the volume of a
tetrahedron, devised a method to obtain binomial
coefficients called
Tartaglia's

Triangle, that was then,
developed to the
pascal’s

triangle and become the first to
apply mathematics to the investigation of the paths of
cannonballs. Even today, the solution to cubic equations is
not known as
Tartgalia’s

Formula but as
Cardano’s
.

by: Rita Barrocas 11ºB

Horner's rule


The Horner’s rule, which has this name due to after
William George Horner, is an algorithm for the efficient
evaluation of polynomials in monomial form. It is a manual
process by which one may approximate the roots of a
polynomial equation and it is also viewed as a fast algorithm
to divide a polynomial by a linear polynomial with
Ruffini's

rule.


by: Rita Barrocas 11ºB


Given the polynomial:




where are real numbers, we need to evaluate the
polynomial with a value of
x
, for example
x
0
.


To do this, we delineate a new succession of constants :





by: Rita Barrocas 11ºB


Then
b
0

is the value of
p
(
x
0
).



the polynomial can be written in the form





so, by iteratively substituting the
b
i

into the expression:



by: Rita Barrocas 11ºB

François

Viete


François
Viete

is considered for some to
be the father of
algebra
as it’s known

today
;


his innovative
look at how equations are used to solve
problems changed
the

methods
behind analysis and

that was
what allowed algebra to evolve .


François
Viete’s

way
of implicating new notation and
theory behind equations
permitted
for the
ancient

Greek
method to be
absolutely
rebuilt,
allowing others
to
investigate

p
roblems with a new view.



by: Rita Barrocas 11ºB


During the time that men have tried to discover the extent
π

, they have come up with many ideas of figuring out as many
digits as possible.


The earliest attempts at unraveling the mysteries of
π

were
really just "guess and check" figures. They included
everything from 22/7 to 211875/67441.

It was enough to
satisfy the needs during the centuries, however, the
mathematicians continued the research
.





by: Rita Barrocas 11ºB


Next, they evolved to the next and latest phase in the
calculation of pi: infinite products and sums.
This

trend

began

with

Francois

Viete’s

formula:




This type of equation allows one to calculate one term at a time,
consequently allowing one mathematician to work on one term
and another to pick up where the other one left off.


Viete's

method was extremely slow and clumsy, however it
created a base for almost all advancements in
π

that had been
done after that.


by: Rita Barrocas 11ºB

Gauss

Gauss developed the Gaussian function and it is a function
of the form:



for some real constants
a
,
b
,
c

> 0, and
e

≈ 2.718281828
(Euler’s number)

by: Rita Barrocas 11ºB




The graph of a Gaussian function has the characteristic of having a symmetric "bell
curve" shape that quickly falls off towards plus or minus infinite. The parameter
a

is
the height of the curve's culmination,
b

is the position of the centre of the culmination
and
c

controls the thickness of the shape.

Gaussian functions are usually used in statistics where they describe the normal distributions
in signal processing where they serve to define Gaussian filters, in image processing where two
-
dimensional Gaussians are used for Gaussian blurs, and in mathematics where they are used to
solve heat equations and diffusion

by: Rita Barrocas 11ºB

John
Atanasoff


John
Atanasoff

was

a
bulgarian

matematician

who

has

born

in

1903
and

he

was

the

inventor
of

the first general purpose
electronic computer
.


Atanasoff

didn’t

invented

it

for
fame

or

glory
,
he

invented

ir
for
necessity


At that time, the Monroe Calculator, a mechanical machine,
was the easiest method of performing mathematics
automatically, but
Atanasoff

wanted to improve upon this
device using various methods.



His initial research involved using combinations of IBM
tabulators and Monroe calculators.

by: Rita Barrocas 11ºB


Later, in 1936, he invented an analog calculator that could be
used to analyze surface geometry.


However, He find these solutions were still relatively
rudimentary and what he really desired was an electronic
calculator.



In late 1937, he discovered a solution that would work and, with
a $650 donation, he worked on building a prototype.



In November of 1939, he and a graduate student, Clifford
Berry, constructed the
Atanasoff
-
Berry Computer. The
computer was able to solve up to 29 linear equations
simultaneously. Its calculating power came principally from
vacuum tubes and regenerative capacitor memory


by: Rita Barrocas 11ºB


In 1940,
Atanasoff

and Berry performed research that
suggested that their machine was the first of its kind.


On January 15, 1941, was told to the world about the existence
of the device by a news article in the Des Moines Register .



In June of 1941, a scientist, called John
Mauchly
,
vwent

to
Atanasoff's

home and examined the machine during four days.



In September,
Atanasoff

took a job during the war as Chief
of the Acoustic Division at the Naval Ordnance Laboratory.




by: Rita Barrocas 11ºB


During this time,
Atanasoff

believed that the Iowa State
College was applying for a patent on the computer., however,
they were not.



In 1943,
Mauchly
, the man who had examined
Atanasoff's

computer, began working on ENIAC, a general purpose
electronic computer.



In 1945, the US Navy decided to start working on a project
known as the NOL computer, project and putted
Atanasoff

in charge . But the project was canceled since
Atanasoff's

work on acoustic research was considered more important.


by: Rita Barrocas 11ºB


In 1947,
Mauchly

and John Eckert, his partner, applied for
a patent on their ENIAC , which was only granted in 1964.
The rights to the patent were sold to Remington Rand, which
began manufacturing commercial computer systems.


On the 26
th

of May of 1967 ,the patent was contested and the
trial lasted until the 13rd, of March of 1972, with more than
30,000 evidence exhibits and 77 witnesses.


On the 19
th

of October of 1973, the United States District
Court Judge Earl Larson ruled that the patent was invalid
while the ENIAC was derived from the
Atanasoff
-
Berry
Computer.

Was proved that
Atanasoff

and Berry were the rightful
owners of the patent and the creators of the first general
purpose electronic computer, but the issue was neglected until
much later.

by: Rita Barrocas 11ºB



In 1952,
Atanasoff

founded the Ordnance Engineering
Corporation, but sold the company to
Aerojet

General
Corporation in 1956.


In 1961, he started Cybernetics Incorporated company.


In 1981,
Atanasoff

was awarded the Computer Pioneer
Medal from the IEEE.


In 1990, had been given to him the United States National
Medal of Technology from President George Bush.


On the 15
th

of June of 1995 he died from natural causes. Today,


by: Rita Barrocas 11ºB