Institute of Advanced Mathematics

Fibonacci
Life

Fibonacci was born around 1170 to Guglielmo Bonacci, a wealthy Italian merchant and, by some accounts, the consul for Pisa. Guglielmo directed a trading post in Bugia, a port in the Almohad dynasty's sultanate in North Africa. Fibonacci travelled with him as a young boy, and it was in Bugia (now Béjaïa, Algeria) that he learned about the Hindu–Arabic numeral system.[2]

Fibonacci travelled extensively around the Mediterranean coast, meeting with many merchants and learning of their systems of doing arithmetic. He soon realised the many advantages of the "Hindu-Arabic" system. In 1202 he completed the Liber Abaci (Book of Abacus or Book of Calculation) which popularized Hindu–Arabic numerals in Europe.[2]

Fibonacci became a guest of Emperor Frederick II, who enjoyed mathematics and science. In 1240 the Republic of Pisa honored Fibonacci (referred to as Leonardo Bigollo)[8] by granting him a salary.

The date of Fibonacci's death is not known, but it has been estimated to be between 1240[9] and 1250,[10] most likely in Pisa.
Liber Abaci (1202)
A page of Fibonacci's Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Roman numerals and the value in Hindu-Arabic numerals.
Main article: Liber Abaci

In the Liber Abaci (1202), Fibonacci introduced the so-called modus Indorum (method of the Indians), today known as Arabic numerals.[11][12] The book advocated numeration with the digits 0–9 and place value. The book showed the practical importance of the new numeral system by applying it to commercial bookkeeping, conversion of weights and measures, the calculation of interest, money-changing, and other applications. The book was well received throughout educated Europe and had a profound impact on European thought.
Fibonacci sequence
Main article: Fibonacci number
19th century statue of Fibonacci in Camposanto, Pisa.

Liber Abaci also posed, and solved, a problem involving the growth of a population of rabbits based on idealized assumptions. The solution, generation by generation, was a sequence of numbers later known as Fibonacci numbers. Although Fibonacci's Liber Abaci contains the earliest known description of the sequence outside of India, the sequence had been noted by Indian mathematicians as early as the sixth century.[13][14][15][16]

In the Fibonacci sequence of numbers, each number is the sum of the previous two numbers. Fibonacci began the sequence not with 0, 1, 1, 2, as modern mathematicians do but with 1,1, 2, etc. He carried the calculation up to the thirteenth place (fourteenth in modern counting), that is 233, though another manuscript carries it to the next place: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377.[17][18] Fibonacci did not speak about the golden ratio as the limit of the ratio of consecutive numbers in this sequence.
Legacy

In the 19th century, a statue of Fibonacci was constructed and erected in Pisa. Today it is located in the western gallery of the Camposanto, historical cemetery on the Piazza dei Miracoli.[19]

There are many mathematical concepts named after Fibonacci because of a connection to the Fibonacci numbers. Examples include the Brahmagupta–Fibonacci identity, the Fibonacci search technique, and the Pisano period. Beyond mathematics, namesakes of Fibonacci include the asteroid 6765 Fibonacci and the art rock band The Fibonaccis.

complex number
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation i2 = −1. In this expression, a is the real part and b is the imaginary part of the complex number.

René Descartes (1596-1650)

Descartes was educated in the Jesuit preparatory school of La Flèche and the University of Poitiers, taking a degree in law. He then spent two years in Paris where, outwardly living the life of a frivolous young gentleman, he began a serious study of mathematics. To see more of the world Descartes joined several armies as an unpaid volunteer; the brief intervals of tranquility during nine years of service provided him time to develop his mathematical and philosophic ideas. In 1628, Descartes decided to settle in Holland, where he remained for the next twenty years. There he wrote his great philosophic treatise on the scientific method, the Discours de la méthode (1637). (The still-quoted sentence, "I think, therefore I am," comes from the Discours.) In 1649, after much hesitation, Descartes accepted the invitation of the 22-year-old Queen Christina to come to Sweden as her private tutor. After only four months of winter tutoring sessions, always held at 5:00 in the morning in the ice-cold library, Descartes died of pneumonia.

The last of the three appendices to Descartes’s Discours was a 106-page essay entitled La géométrie. It provides the first printed account of what is now called analytic or coordinate geometry. The work exerted great influence after being published in a Latin translation along with explanatory notes. The Géométrie introduced many innovations in mathematical notation, most of which are still in use. With Descartes, small letters near the beginning of the alphabet indicate constants and those near the end stand for variables. He initiated the use of numerical superscripts to denote powers of a quantity, while occasionally writing aa for the second power, a2. The familiar symbols +, -, and are also encountered in Descartes’s writing.

Descartes "algebrized" the study of geometry by shifting the focus from curves to their equations, allowing the tools of algebra, rather than diagrams, to be applied to the solution of various geometric problems. The Géométrie also treated one of the most important problems of the day, that of finding tangents to curves, by describing a procedure for constructing the normal to a curve at any point (the tangent is perpendicular to the normal). Another part of the work deals with matters in the theory of equations: Descartes states that x - a is a factor of a polynomial if and only if a is a root. He also notes that the maximum number of roots is equal to the degree of the polynomial.

Beauty of NUMBERS
75 95
x 75 x 95
5625 9025

did you think how you can write it at once……
here is the method..
if any number ends from five and when we want to find the square of it…
first multiply 5 by 5 , that is 25, write it at the end..
add 1 to the first number and multiply it by the first number.. Write it at first..( ex- if the fist number is 7. multiply it by 8 )

try this..
85 x 85 =(8x9)(5x5)
=7225