MADE BY: SANKETH.C.PATIL

9TH ‘B’

MATHS PROJECT

Jain Public School

Davangere

International

Mathematicians

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

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

INTRODUCTION The word ‘Geometry’ comes from Greek

words ‘geo’ meaning the ‘earth’ and ‘metrein’ meaning to ‘measure’. Geometry appears to have originated from the need for measuring land.

Nearly 5000 years ago geometry originated in Egypt as an art of earth measurement. Egyptian geometry was the statements of results.

The knowledge of geometry passed from Egyptians to the Greeks and many Greek mathematicians worked on geometry. The Greeks developed geometry in a systematic manner..

Euclid was the first Greek Mathematician who initiated a new way of thinking the study of geometry

He introduced the method of proving a geometrical result by deductive reasoning based upon previously proved result and some self evident specific assumptions called AXIOMS

The geometry of plane figure is known as ‘Euclidean Geometry’. Euclid is known as the father of geometry.

His work is found in Thirteen books called ‘The Elements’.

EUCLID’S DEFINITONS

Some of the definitions made by Euclid in volume I of ‘The Elements’ that we take for granted today are as follows :-

A point is that which has no part

A line is breadth less length

The ends of a line are points

A straight line is that which has length only

Continued…..

The edges of a surface are lines

A plane surface is a surface which lies evenly with the straight lines on itself

Axioms or postulates are the assumptions which are obvious universal truths. They are not proved.

Theorems are statements which are proved, using definitions, axioms, previously proved statements and deductive reasoning.

EUCLID’S AXIOMs

SOME OF EUCLID’S AXIOMS WERE :-

Things which are equal to the same thing are equal to one another.

i.e. if a=c and b=c then a=b. Here a,b, and c are same kind of things.

If equals are added to equals, the wholes are equal.

Example :-

In fig :- 01 the line EF falls on two lines AB and CD such that the angle m + angle n < 180° on the right side of EF, then the line eventually intersect on the right side of EF

fig :- o1

Carl GaussCarl Friedrich Gauss, the "Prince of Mathematics," exhibited his calculative powers when he corrected his father's arithmetic before the age of three.His genius 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 Disquisitiones Arithmeticae, 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.

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.Euler was also a major figure in number theory, proving that the sum of the reciprocals of primes less than x is approx. (ln ln 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.

Pythagoras, 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). 

Pythagoras

Pythagoras

Born: 569 B.C. (Samos, Ionian)

Died: 475 B.C.

Era: Ancient Philosophy

Region: Western Philosophy

Parents: Mnesarchus & Pythais

Pythagoras

Contribution:

−Astronomy−Music−Religion−Mathematics−Philosophy

Pythagoras

Philosophers that influenced Pythagoras:

−Pherekydes−Thales−Anaximander

Pythagoras

Life routine:

− Wake up & have a nice breakfast− Study− Go to a reincarnation meeting− Study− Test triangles− Sleep – but not for long

Pythagoras

Places involved in Pythagoras life:

SamosEgyptBabylonCreteCroton, Italy

 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 theory

Fermat'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 numbers

Fermat 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.

Fermat

PRIME NUMBERS :

A prime number is divisible only by 1 and itself.

For example: {2, 3, 5, 7, 11, 13, 17, …}

1 could also be considered prime, but it’s not very useful.

PRIME FACTORIZATION :

• To factor a number n is to write it as a product of other numbers.

• n = a * b * c

•Example : 100 = 5 * 5 * 2 * 2

•Prime factorization of a number n is writing it as a product of prime numbers.

•Example : 143 = 11 * 13

RELATIVELY PRIME NUMBERS :

The Greatest Common Divisor (GCD) of two relatively prime numbers can be determined by comparing their prime factorizations and selecting the least powers.

GREATEST COMMON DIVISOR (GCD)

•Example : 125 = 53 and 200 = 23 * 52

•GCD(125, 200) = 20 * 52 = 25

•If the two numbers are relatively prime the GCD will be 1.

•Consider the following: 10(1, 2, 5, 10) and 21(1, 3, 7, 21)

•Thus GCD(10, 21) = 1

•It then follows, that a prime number is also relatively prime to any other number other than itself and 1.

FERMAT’S THEOREM

If ‘p’ is prime and ‘a’ is an integer not divisible by ‘p’ then

• ap-1 1 (mod p).

And for every integer ‘a’

• ap a (mod p).

This theorem is also known as Fermat’s Little Theorem

Used in public key (RSA) and primality testing.

FERMAT’S THEOREM PROOF :

Consider a set of positive integers less than ‘p’ : {1,2,3,…..,(p-1)} and multiply each element by ‘a’ and ‘modulo p’ , to get the set

X = {a mod p, 2a mod p,…, (p-1)a mod p}

No elements of X is zero and equal, since p doesn’t divide a.

Multiplying the numbers in both sets (p and X) and taking the result mod p yields

EULER’S THEOREM :

Statement

• For every ‘a’ and ‘n’ that are relatively prime:

•

na n mod1

Quotes1. “Do not say a little in many words but a great deal in a few.”

2. “Friends are as companions on a journey, who ought to aid each other to persevere in the road to a happier life.”

3. “Above the cloud with its shadow is the star with its light. Above all things reverence thyself.”

4. “Strength of mind rests in sobriety; for this keeps your reason unclouded by passion.”

5. “There is geometry in the humming of the strings, there is music in the spacing of the spheres.”

6. “Concern should drive us into action and not into a depression. No man is free who cannot control himself.”

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