This document provides brief biographies of famous mathematicians including Euclid, Carl Gauss, Leonhard Euler, Pythagoras, Aryabhata, and Fermat. It discusses their major contributions such as Euclid establishing the principles of geometry, Gauss' work in number theory and algebra, Euler making advances in many fields of mathematics through his notations and methods, Pythagoras discovering relationships in music and numbers, Aryabhata developing India's place value system and early work in trigonometry and algebra, and Fermat's foundational work in number theory and discoveries in analytic geometry and calculus.

1. Famous Mathematicians

2. INDEX1. Euclid2. Carl Gauss3. Leonhard Euler4. Pythagoras5. Aryabhata6. Fermat

3. EuclidEuclid also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry“. His Elements is one of the most influential works in the history of mathematics. Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor. Euclid may have been a student of Aristotle. He founded the school of mathematics at the great university of Alexandria. He was the first to prove that there are infinitely many prime numbers; he stated and proved the unique factorization theorem; and he devised Euclid's algorithm for computing gcd. He introduced the Mersenne primes and observed that (M2+M)/2 is always perfect (in the sense of Pythagoras) if M is Mersenne. Among several books attributed to Euclid are The Division of the Scale, The Optics, The Cartoptrics. Several of his masterpieces have been lost, including works on conic sections and other advanced geometric topics. Apparently Desargues' Homology Theorem was proved in one of these lost works; this is the fundamental theorem which initiated the study of projective geometry

4. Carl GaussCarl Friedrich Gauss, the "Prince of Mathematics," exhibited his calculative powers when he corrected his father's arithmetic before the age of three.Hisgenius was confirmed at the age of nineteen when he proved that the regular n-gon was constructible, for odd n, if and only if n is the product of distinct prime Fermat numbers. At age 24 he published DisquisitionesArithmeticae, probably the greatest book of pure mathematics ever. Gauss may be the greatest theorem prover ever. Several important theorems and lemmas bear his name; he was first to produce a complete proof of Euclid's Fundamental Theorem of Arithmetic and first to produce a rigorous proof of the Fundamental Theorem of Algebra. Gauss himself used "Fundamental Theorem" to refer to Euler's Law of Quadratic Reciprocity; Gauss was first to provide a proof for this, and provided eight distinct proofs for it over the years. Gauss proved the n=3 case of Fermat's Last Theorem for a class of complex integers; though more general, the proof was simpler than the real integer proof, a discovery which revolutionized algebra. Other work by Gauss led to fundamental theorems in statistics, vector analysis, function theory, and generalizations of the Fundamental Theorem of Calculus.

5. Euler may be the most influential mathematician who ever lived he ranks #77 on Michael Hart's famous list of the Most Influential Persons in History. His notations and methods in many areas are in use to this day. Just as Archimedes extended Euclid's geometry to marvelous heights, so Euler took marvelous advantage of the analysis of Newton and Leibniz. He gave the world modern trigonometry.He invented graph theory.Eulerwas also a major figure in number theory, proving that the sum of the reciprocals of primes less than x is approx. (lnln x). Euler was also first to prove several interesting theorems of geometry, including facts about the 9-point Feuerbach circle; relationships among a triangle's altitudes, medians, and circumscribing and inscribing circles; and an expression for a tetrahedron's area in terms of its sides. Euler was first to explore topology, proving theorems about the Euler characteristic. he settled an arithmetic dispute involving 50 decimal places of a long convergent series. Four of the most important constant symbols in mathematics (π, e, i = √-1, and γ = 0.57721566...) were all introduced or popularized by Euler.

6. PythagorasPythagoras, who is sometimes called the "First Philosopher," studied under Anaximander, Egyptians, Babylonians, and the mystic Pherekydes. he became the most influential of early Greek mathematicians. Pythagoras discovered that harmonious intervals in music are based on simple rational numbers. This led to a fascination with integers and mystic numerology; he is sometimes called the "Father of Numbers" and once said "Number rules the universe. The Pythagorean Theorem was known long before Pythagoras, but he is often credited with the first proof. He also discovered the simple parametric form of Pythagorean triplets (xx-yy, 2xy, xx+yy).

7. Place value system and zeroThe place-value system, first seen in the 3rd century Bakhshali Manuscript, was clearly in place in his work. While he did not use a symbol for zero, the French mathematician Georges If rah explains that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficientsAryabhataApproximation of πAryabhata worked on the approximation for pi (π), and may have come to the conclusion that π is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:TrigonometryIn Ganitapada 6, Aryabhata gives the area of a triangle as "for a triangle, the result of a perpendicular with the half-side is the area."AlgebraIn Aryabhatiya Aryabhata provided elegant results for the summation of series of squares and cubes:

8. Fermat Fermat practically founded Number Theory, and also played key roles in the discoveries of Analytic Geometry and Calculus. He was also an excellent geometer and discovered probability theoryFermat's most famous discoveries in number theory include the ubiquitously-used Fermat's Little Theorem; the n = 4 case of his conjectured Fermat's Last Theorem the fact that every natural number is the sum of three triangle numbersFermat developed a system of analytic geometry which both preceded and surpassed that of Déscartes; he developed methods of differential and integral calculus which Newton acknowledged as an inspiration.

9. Quotes“Do not say a little in many words but a great deal in a few.” “Friends are as companions on a journey, who ought to aid each other to persevere in the road to a happier life.”“Above the cloud with its shadow is the star with its light. Above all things reverence thyself.”“Strength of mind rests in sobriety; for this keeps your reason unclouded by passion.”“There is geometry in the humming of the strings, there is music in the spacing of the spheres.”“Concern should drive us into action and not into a depression. No man is free who cannot control himself.”

10. Made By:-TejasavKhattarClass:- IX-D