*Pythagoras*

**Introduction**

Pythagoras was born on the island of Samos, Greece in
569 BC, and died about 500 BC in Metapontum, Lucania. He traveled extensively
in Egypt, learning math, astronomy and music. Pythagoras was also a healer,
a wrestler, and was politically active. Pythagoras was a Greek Mathematician
born in 569 B.C. who studied math, music, and astronomy.

**His Life**

Pythagoras left Samos in disgust for its ruler Polycrates.
He settled in Cretona, a Greek colony in southern Italy. There he founded a
movement with religious, political and philosophical goals. To facilitate his
movement, he created a school where his followers lived and worked. He had many
devoted followers who were called Pythagoreans. They had to adhere to certain
strict rules. Obedience, silence, abstinence from food, simplicity in dress
and possessions, and the habit of frequent self examination were required of
the Pythagoreans. They also believed in immortality and transmigration of souls.
Pythagoras created a strict order where his followers worked with Pythagoras
to make new discoveries and theories.

**His Work**

Pythagoras did much more than just discover what is now
referred to as the Pythagorean Theorem. Pythagoras and his followers contributed
to music, astronomy and mathematics. Pythagoras believed in secrecy and communalism,
so distinguishing his work from the work of his followers is almost impossible.
Some of their discoveries were right, and some were proven wrong in time.

Pythagoras worked with his followers in secret, so discerning the work of Pythagoras
and the work of his followers is almost impossible.

Among the many mathematical investigations of the Pythagoreans
were the study of odd, even, prime and square numbers. This helped them develop
a basic understanding of mathematics and geometry to build their Pythagorean
theorem. The Pythagorean theorem states that the square of the hypotenuse of
a right triangle is equal to the sum of the squares of the other two sides.
Though it was known to Babylonians 1000 years before, Pythagoras was the first
to decisively prove it. Pythagoras was the first to prove the square of the
hypotenuse is the sum of the sides squared, or Pythagorean theorem.

Pythagorean theorm:a2 + b2 = c2 (2=square)

**Conclusion**

Pythagoras had a great impact on mathematics, music, and
astronomy. His theories are still used today, in mathematics. Pythagoras is
one of the great thinkers of his time.

*Fibonacci*

**Introduction**

Fibonacci was known for many things. He was best known
for the Fibonacci Numbers, which is a number sequence that he had discovered
while solving a problem about rabbits. There is a lot more that we will talk
about him and his discoveries which are now coming up.

**Biography**

From 529 until 1500 A.D. there were no big improvements
in european mathematics. Except for Fibonacci, who was a great 13th century
mathematician. He was born in Pisa, Italy, and was the son of a pisan merchant.Fibonacci
was best known as Leonardo of Pisa.

His father was also a customs officer for the North African city of Bugia. Since Fibonacci was the son of a merchant, he was able go travel freely all over the Byzantine Empire. Merchants at the time were immuned, so they were allowed to move about freely. This allowed him to visit many of the area's centers of trade. While he was there, he was able to learn both the mathematics of the scholars and the calculating schemes in popular use, at the time.

**Accomplishments**

He published a book called Liber Abaci. In 1202 he published
the first of his four books, Liber Abaci, it used the Hindu-Arabic numbering
system, introducing the Indian numbers 1 through 9 and zero to a European audience.
In his book you can tell that he was influenced by the Muslim and Arab culture,
because he wrote many things from right to left. In his book he wrote the numerals
in descending order and his fractions came before the numeral like ?4 instead
of 4 ?

This sequence soon became known as the Fibonacci Sequence

**Fibonacci's Rabbits**

Originally this was a problem that he investigated (in
the year 1202) to see about how fast rabbits would breed in ideal circumstances.
This is assumming that none of the rabbits die, and that each pair of rabbits
does not give birth to more than one pair of rabbits each month. That is how
he discovered the sequence.

Below is one of the ways of calculating Fibonacci
Numbers.

Fibonacci Numbers, represented here by Fi, can be defined as follows.

Let F0=0 and F1=1. For all other Fi, let Fn+1 = Fn-1 + Fn.

**Effects**

His introduction of the Arabian Numerals gave the Europeans
an easier way to do calculations. Since using Roman Numerals were harder to
use because of so many letters that represent numbers in different ways.

It also caused some of the people many years after he made the discovery to create a society in his name. The Fibonacci Society was founded in 1962, and a journal, The Fibonacci Quarterly, first appeared in 1963, and was dedicated to unraveling its secrets. There were a lot of secrets to be found.

**Fibonacci Forgeries**

Many people have also tried to find sequences like Fibonacci's,
but what they found were forgeries. The sequence may start off like being the
Fibonacci Sequence, but then it starts giving other numbers that are not of
the original sequence.

*Blaise Pascal*

**Background**

Blaise Pascal, the only son of Etienne Pascal, was born on June 19, 1623 in
what was Clermont (now Clermont-Ferrand), Auvergne, France. In 1632, the Pascals
left Clermont for Paris, where Blaise's father took it upon himself to educate
the family. Thus, Pascal was not allowed to study mathematics until the age
of 15, and all math texts were removed from the house. Despite all this, Blaise's
curiosity grew and he began to work on geometry himself at the age of 12. After
discovering that the sum of the angles of a triangle is two right angles, his
father relented and gave him a copy of a Euclidian geometry textbook.

**An Early Achiever**

Blaise Pascal made many discoveries between the ages of fourteen and twenty-four.
At fourteen, he attended his father's geometry meetings, and at 16, he composed
an essay on conic sections, which was published in 1640.

Between the ages of 18 and 22, he invented a
digital calculator, called a Pascaline, to assist his father in collecting taxes.

The second calculator ever to have been invented, the Pascaline resembled a
mechanical calculator of the 1940's. Also, by the age of 24, Pascal had proved
to his satisfaction that a vacuum could exist. He and Descartes argued about
this, wherein Descartes later remarked that Pascal ...has too much vacuum in
his head.

Two years later, Descartes would eat his own words.

**The Famous Triangle**

Although Pascal was not the first to study the arithmetical triangle now known
as Pascal's Triangle, he did make more contributions to it than anyone ever
had, right around 1653.

The triangle is constructed in such a way that
each number is the sum of the numbers above it and to the left of it (see fig.
2) .

The enclosed area represents the triangle. The numbers on each line, now called figurate numbers, are named in sequence, with those in the first row titled numbers of the first order, and so on. The formula for the triangle is as follows:

(m+n-2)! / (m-1)(n-1)

Where m is equal to the column and n is equal to the row.

fig. 2, Pascal's Triangle, in
table form

**Working With Fermat**

As a mathematician, Pascal is probably best known for his work with Fermat in
1654, in which he laid down the principles of the Theory of Probabilities. The
problem they were working asked how many times one must throw a pair of dice
in order to get a pair of sixes, and also how to divide the stakes if a game
of dice is incomplete. The two mathematicians solved the problem for a two player
game, but could not master the methods for a game of three or more players.

**The Last Few Years**

Throughout the period of correspondence with Fermat, Pascal was unwell. Despite
sickness, he continued work in mathematics, up until he nearly lost his life
in an accident during the October of 1654. He was unharmed, but the experience
deeply affected him psychologically. After undergoing another religious experience,
he pledged his life to Christianity.

This, along with other events, inspired him
to write two religious works, the Provincial Letters and the Pensees, in which
he states:

If God does not exist, one will lose nothing by believing in him, while if he
does exist, one will lose everything by not believing.

This quote is known as "Pascal's Wager". Blaise Pascal's last work was on the cycloid, or the curve traced by a point on the circumference of a rolling circle. He died a painful death at the age of 39 in 1662 after a cancerous growth in his stomach spread to the brain.

**Summary: Important Points**

Blaise Pascal taught himself geometry at the age of 12.

Pascal's works include: Pensees, Essay on Conic
Sections, New Experiments Concerning Vacuums, Treatise on the Equilibrium of
Liquids, The Generation of Conic Sections, Treatise on the Arithmetical Triangle,
Provincial Letters, and Letters to Carcavi.
Pascal proved Descartes wrong on more than two occasions.

Pascal was a pioneer in combining religion with logic. Even when deathly sick, Pascal continued to work on mathematic problems. Pascal kept a strong faith in God even after seeing death in his family and facing his own death. This project was prepared and presented by students Jonathan Ringler and Ryan Bourgo in 1998.