Math's project

By Vinay KushwahaClass 10th CRoll no 41

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Mathematicians

• Srinivasa Ramanujan• Carl Ffriedrich gauss• Thales• Pythagoras

Copyright 2008 PresentationFx.com | Redistribution Prohibited | Image © 2008 Thomas Brian | This text section may be deleted for presentation.

Contents

Copyright 2008 PresentationFx.com | Redistribution Prohibited | Image © 2008 Thomas Brian | This text section may be deleted for presentation.

• Real numbers• Trigonometry • Area and Perimeter

Real numbers Real Numbers are just numbers like:1, 12.38, -0.8625, 3/4, √2, 1998

In fact: Nearly any number you can think of is a Real Number

Real Numbers include: Whole Numbers (like 1,2,3,4, etc) Rational Numbers (like 3/4, 0.125, 0.333..., 1.1, etc ) Irrational Numbers (like π, √3, etc )

Real Numbers can also be positive, negative or zero. So ... what is NOT a Real Number? √-1 (the square root of minus 1) is not a Real Number, it Is anImaginary Number Infinity is not a Real Number

And there are also some special numbers that mathematicians play with that are not Real Numbers

Why are they called "Real" Numbers?

Because they are not Imaginary Numbers.

The Real Numbers did not have a name before Imaginary Numbers were thought of. They got called "Real" because they were not Imaginary. That is the actual answer!

The Real Number Line• The Real Number Line is like an actual geometric line.• A point is chosen on the line to be the "origin", points

to the right will be positive, and points to the left will be negative.

• A distance is chosen to be "1", and the whole numbers can then be marked off: {1,2,3,...), and also in the negative direction: {-1,-2,-3, ...}

Any point on the line is a Real Number:

•The numbers could be rational (like 20/9)•Or irrational (like π)But you won't find Infinity, or an Imaginary Number.

Real does not mean they are in the real worldThey are not called "Real" because they show the value of something real.

In mathematics we like our numbers pure, if we write 0.5 we mean exactly half, but in the real world half may not be exact (try cutting an  apple exactly in half).

RECTANGLE SQUARE PARALLELOGRAM TRIANGLE

AREA- SIDE x SIDE E.g. - SIDE = 5 cm Area of square= S x S sq unit = (5 x 5)cm sq =25 cm sq

Perimeter = 4x side E.g. Side=60 cm Perimeter = 4x side =(4 x 60)cm =240 cm

Calculating the Area of a Triangle

Triangles with different length sides may have the same area. The area of a triangle is dependent only on the length of its base and on its perpendicular height. How can we calculate the area of a triangle from this information?

I. Calculating the area of a triangle• A. Base and height• We can choose any side of a triangle as the base. For convenience, the

same word (base) is used to denote the length of this side. Once a base has been chosen there is only one perpendicular height relative to this base; this is the straight line perpendicular to the base, passing through the opposite apex.

B. FormulaLet A be the area of a triangle of base b and perpendicular height h: .

To apply this formula A, b, and h must be expressed in corresponding units; for example: b and h in cm and A in cm2.

• C. ExampleWe choose [AB] as the base of triangle ABC, shown in figure 3.

• We apply the area formula with b = 4 cm and h = 3.5 cm.• ,

:

We apply the area formula with b = 4 cm and h = 3.5 cm.

therefore the area of triangle ABC is 7 cm2.

D. Special caseIn the case of a right-angled triangle we can choose one side of the right angle as

the base. So, the associated perpendicular height is simply the length of the other side of the right angle.

In figure 4 the area A of triangle IJK with right angle I is given by the formula: .

We apply the area formula with b = 4 cm and h = 3.5 cm.

,

II. Supplements• A. Demonstrating the formulaFigure 5 uses an example to explain the formula for calculating the

area.

The area of the parallelogram is . The area of each triangle is half of the area of the parallelogram, i.e.:

B. Calculating perpendicular heightABC is a triangle with a right angle at A, where AB = 4 cm, BC = 5 cm and

AC = 3 cm. We want to calculate the height AH.The area of triangle ABC is

This area is also equal to . Therefore ; and .The height AH is therefore 2.4 cm. This area is also equal to . Therefore ; and .

Calculating the Area of a ParallelogramThe area of a rectangle of length l and width w is given by the formula:

A = l × w. Which similar formula can we use to calculate the area of a parallelogram and what are the direct applications of this formula?

• I. The formulaWe can choose two opposite sides of a parallelogram as bases. For convenience, the

same word (base) is used to denote the length of both these sides.Having chosen these bases, we can consider the perpendicular height relative to the

bases.For example, let us assume the base of the parallelogram shown in figure 1 is 10 cm and

in this case its perpendicular height is 4 cm.The area, A, of a parallelogram with base b and perpendicular height h is given by the

formula: A = b × h.

To apply this formula, b and h must be expressed in the same units; A will then be expressed in a corresponding unit. For example, if b and h are expressed in cm, then A will be expressed in cm².

Note: The area of a parallelogram is equivalent to that of a rectangle with

dimensions of b and h.

II. Application examplesA. Example 1Let us calculate the area of the parallelogram in figure 1 by using the above

rules: Example 1: b = 10 cm; h = 4 cm; 10 cm × 4 cm = 40 cm2; the area of the parallelogram is 40 cm2; Example 2: b = 5 cm; h = 8 cm; 5 cm × 8 cm = 40 cm2; the area of the parallelogram is 40 cm2.

Of course, the result is the same.B. Example 2In this example, we use the area of a parallelogram to calculate the length of

one of its sides. In figure 3, ABCD is a parallelogram where AD = 4 cm, DH = 1.6 cm and CK = 2 cm. Calculate AB.

We can now calculate the area of the parallelogram using two methods: by letting b = AD = 4 cm and h = CK = 2 cm: 4 cm × 2 cm = 8 cm2; therefore A = 8 cm2;

by letting b = AB = s cm and h = DH = 1.6 cm: s cm × 1.6 cm = 1.6s cm2; therefore A = 1.6s cm2.

The two methods must give the same result. Therefore 1.6s = 8 cm2. Therefore s = 8 cm2 ÷ 1.6 cm = 5 cm. We have found that AB = 5 cm.

There is perhaps nothing which so occupies themiddle position of mathematics as trigonometry.

– J.F. Herbart (1890)8.1 IntroductionYou have already studied about triangles, and in particular, right triangles, in yourearlier classes. Let us take some examples from our surroundings where right trianglescan be imagined to be formed. For instance :• 1. Suppose the students of a school arevisiting Qutub Minar. Now, if a studentis looking at the top of the Minar, a righttriangle can be imagined to be made,as shown in Fig 8.1. Can the studentfind out the height of the Minar, withoutactually measuring it?

2. Suppose a girl is sitting on the balconyof her house located on the bank of ariver. She is looking down at a flowerpot placed on a stair of a temple situatednearby on the other bank of the river.A right triangle is imagined to be madein this situation as shown in Fig.8.2. Ifyou know the height at which theperson is sitting, can you find the widthof the river?.

Trigonometry (from Greek trigōnon "triangle" + metron "measure"[) is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines thetrigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. The field evolved during the third century BC as a branch of geometry used extensively for astronomical studies.It is also the foundation of the practical art of surveying.

Trigonometry basics are often taught in school either as a separate course or as part of a precalculus course. The trigonometric functions are pervasive in parts of pure mathematics and applied mathematics such as Fourier analysis and the wave equation, which are in turn essential to many branches of science and technology. Spherical trigonometry studies triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. It is fundamental to astronomyand navigation. Trigonometry on surfaces of negative curvature is part of Hyperbolic geometry.

Trigonometric RatiosIn Section 8.1, you have seen some right trianglesimagined to be formed in different situations.Let us take a right triangle ABC as shownin Fig. 8.4.Here, CAB (or, in brief, angle A) is an∠acute angle. Note the position of the side BCwith respect to angle A. It faces A. We call it∠the side opposite to angle A. AC is thehypotenuse of the right triangle and the side ABis a part of A. So, we call it the side∠adjacent to angle A. Fig. 8.4

Note that the position of sides changewhen you consider angle C in place of A(see Fig. 8.5).You have studied the concept of ‘ratio’ inyour earlier classes. We now define certain ratiosinvolving the sides of a right triangle, and callthem trigonometric ratios.The trigonometric ratios of the angle Ain right triangle ABC (see Fig. 8.4) are definedas follows :

The ratios defined above are abbreviated as sin A, cos A, tan A, cosec A, sec Aand cot A respectively. Note that the ratios cosec A, sec A and cot A are respectively,the reciprocals of the ratios sin A, cos A and tan A.

Srinivasa Ramanujan

Born 22 December 1887Erode, Madras Presidency

Died 26 April 1920 (aged 32)Chetput, Madras, Madras Presidency

Residence KumbakonamNationality IndianFields MathematicsAlma mater Government Arts College

Pachaiyappa's College

Academic advisors G. H. HardyJ. E. Littlewood

Known forLandau–Ramanujan constantMock theta functionsRamanujan conjectureRamanujan primeRamanujan–Soldner constantRamanujan theta functionRamanujan's sumRogers–Ramanujan identitiesRamanujan's master theorem

Influences G. H. Hardy

Signature

Srinivasa Ramanujan FRS ( pronunciation (help·info)) (Tamil:  ஸ்ரீநிவாச ராமானுஜன்) (22 December 1887 – 26 April 1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. Living in India with no access to the larger mathematical community, which was centered in Europe at the time, Ramanujan developed his own mathematical research in isolation. As a result, he sometimes rediscovered known theorems in addition to producing new work. Ramanujan was said to be a natural genius by the English mathematician G.H. Hardy, in the same league as mathematicians like Euler and Gauss.[1]

Born in a poor Brahmin family, Ramanujan's introduction to formal mathematics began at age 10. He demonstrated a natural ability, and was given books on advanced trigonometrywritten by S. L. Loney that he mastered by the age of 12; he even discovered theorems of his own, and re-discovered Euler's identity independently.[2] He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan had conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant.Ramanujan received a scholarship to study at Government College in Kumbakonam, but lost it when he failed his non-mathematical coursework. He joined another college to pursue independent mathematical research, working as a clerk in the Accountant-General's office at the Madras Port Trust Office to support himself.[3] In 1912–1913, he sent samples of his theorems to three academics at the University of Cambridge. G. H. Hardy, recognizing the brilliance of his work, invited Ramanujan to visit and work with him at Cambridge. He became a Fellow of the Royal Society and a Fellow of Trinity College, Cambridge. Srinivasa died of illness, malnutrition, and possibly liver infection in 1920 at the age of 32.During his short lifetime, Ramanujan independently compiled nearly 3900 results (mostly identities and equations).[4] Most of his claims have now been proven correct, although a small number of these results were actually false and some were already known. [5] He stated results that were both original and highly unconventional, such as the Ramanujan prime and the Ramanujan theta function, and these have inspired a vast amount of further research.[6] However, the mathematical mainstream has been rather slow in absorbing some of his major discoveries. The Ramanujan Journal, an international publication, was launched to publish work in all areas of mathematics influenced by his work.[7]

In December 2011, in recognition of his contribution to mathematics, the Government of India declared that Ramanujan's birthday (22 December) should be celebrated every year asNational Mathematics Day, and also declared 2012 the National Mathematical Year.

Attention from mathematicians

He met deputy collector V. Ramaswamy Aiyer, who had recently founded the Indian Mathematical Society.[38] Ramanujan, wishing for a job at the revenue department where Ramaswamy Aiyer worked, showed him his mathematics notebooks. As Ramaswamy

Aiyer later recalled:I was struck by the extraordinary mathematical results contained in it [the notebooks]. I

had no mind to smother his genius by an appointment in the lowest rungs of the revenue department.[39]

Ramaswamy Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras.[38] Some of these friends looked at his work and gave him letters of

introduction to R. Ramachandra Rao, the district collector forNellore and the secretary of the Indian Mathematical Society.[40][41][42] Ramachandra Rao was impressed by

Ramanujan's research but doubted that it was actually his own work. Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable Bombay

 mathematician, in which Saldhana expressed a lack of understanding for his work but concluded that he was not a phony.[43] Ramanujan's friend, C. V. Rajagopalachari, persisted with Ramachandra Rao and tried to quell any doubts over Ramanujan's

academic integrity. Rao agreed to give him another chance, and he listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of 

divergent series, which Rao said ultimately "converted" him to a belief in Ramanujan's mathematical brilliance.[43] When Rao asked him what he wanted, Ramanujan replied that he needed some work and financial support. Rao consented and sent him to Madras. He

continued his mathematical research with Rao's financial aid taking care of his daily needs. Ramanujan, with the help of Ramaswamy Aiyer, had his work published in

the Journal of Indian Mathematical Society.

One of the first problems he posed in the journal was:He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end, Ramanujan supplied the solution to the problem himself. On page 105 of his first notebook, he formulated an equation that could be used to solve the infinitely nested radicals problem.Using this equation, the answer to the question posed in the Journal was simply 3.[45] Ramanujan wrote his first formal paper for the Journal on the properties of Bernoulli numbers. One property he discovered was that the denominators (sequence A027642 in OEIS) of the fractions of Bernoulli numbers were always divisible by six. He also devised a method of calculating Bn based on previous Bernoulli numbers. One of these methods went as follows:It will be observed that if n is even but not equal to zero,(i) Bn is a fraction and the numerator of  in its lowest terms is a prime number,(ii) the denominator of Bn contains each of the factors 2 and 3 once and only once,(iii)  is an integer and  consequently is an odd integer.In his 17-page paper, "Some Properties of Bernoulli's Numbers", Ramanujan gave three proofs, two corollaries and three conjectures.[46] Ramanujan's writing initially had many flaws. As Journal editor M. T. Narayana Iyengar noted:Mr. Ramanujan's methods were so terse and novel and his presentation so lacking in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him. [47]

Ramanujan later wrote another paper and also continued to provide problems in the Journal.[48] In early 1912, he got a temporary job in the Madras Accountant General's office, with a salary of 20 rupees per month. He lasted for only a few weeks.[49] Toward the end of that assignment he applied for a position under the Chief Accountant of the Madras Port Trust. In a letter dated 9 February 1912, Ramanujan wrote:Sir,I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject. I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me. [50]

Attached to his application was a recommendation from E. W. Middlemast, a mathematics professor at the Presidency College, who wrote that Ramanujan was "a young man of quite exceptional capacity in Mathematics".[51] Three weeks after he had applied, on 1 March, Ramanujan learned that he had been accepted as a Class III, Grade IV accounting clerk, making 30 rupees per month. [52] At his office, Ramanujan easily and quickly completed the work he was given, so he spent his spare time doing mathematical research. Ramanujan's boss, Sir Francis Spring, and S. Narayana Iyer, a colleague who was also treasurer of the Indian Mathematical Society, encouraged Ramanujan in his mathematical pursuits.

Thales of Miletus (Θαλῆς ὁ Μιλήσιος)

Thales

Born ca. 620–625 BC

Died ca. 547–546 BCSchool Ionian, Milesian

school, Naturalism

Main interests Ethics, Metaphysics, Mathematics, Astronomy

Notable ideas Water is the physis, Thales' theorem, intercept theorem

Influenced by•Babylonian astronomy & Ancient Egyptian mathematics and religion

Influenced•Pythagoras, Anaximander, Anaximenes

Thales

Thales of Miletus ( /ˈθeɪliːz/; Greek: Θαλῆς, Thalēs; c. 624 BC – c. 546 BC) was a pre-Socratic Greek philosopher from Miletus in Asia Minor, and one of the Seven Sages of Greece. Many, most notably Aristotle, regard him as the first philosopher in the Greek tradition.[1] According to Bertrand Russell, "Western philosophy begins with Thales."[2] Thales attempted to explain natural phenomena without reference to mythology and was tremendously influential in this respect. Almost all of the other pre-Socratic philosophers follow him in attempting to provide an explanation of ultimate substance, change, and the existence of the world—without reference to mythology. Those philosophers were also influential, and eventually Thales' rejection of mythological explanations became an essential idea for the scientific revolution. He was also the first to define general principles and set forth hypotheses, and as a result has been dubbed the "Father of Science", though it is argued that Democritusis actually more deserving of this title.[3][4]

In mathematics, Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. As a result, he has been hailed as the first true mathematician and is the first known individual to whom a mathematical discovery has been attributed. Also, Thales was the first person known to have studied electricity.

TheoriesThe Greeks often invoked idiosyncratic explanations of natural phenomena by reference to the will of anthropomorphic gods and heroes. Thales, however, aimed to explain natural phenomena via a rational explanation that referenced natural processes themselves. For example, Thales attempted to explain earthquakes by hypothesizing that the Earth floats on water, and that earthquakes occur when the Earth is rocked by waves, rather than assuming that earthquakes were the result of supernatural processes. Thales was a Hylozoist (those who think matter is alive). [13] It is unclear whether the interpretation that he treated matter as being alive might have been mistaken for his thinking the properties of nature arise directly from material processes, more consistent with modern ideas of how properties arise as emergent characteristics ofcomplex systems involved in the processes of evolution and developmental change.Thales, according to Aristotle, asked what was the nature (Greek Arche) of the object so that it would behave in its characteristic way. Physis (φύσις) comes from phyein (φύειν), "to grow", related to our word "be".[14] (G)natura is the way a thing is "born",[15] again with the stamp of what it is in itself.Aristotle[16] characterizes most of the philosophers "at first" (πρῶτον) as thinking that the "principles in the form of matter were the only principles of all things", where "principle" is arche, "matter" is hyle ("wood" or "matter", "material") and "form" is eidos.Arche is translated as "principle", but the two words do not have precisely the same meaning. A principle of something is merely prior (related to pro-) to it either chronologically or logically. An arche (from ἄρχειν, "to rule") dominates an object in some way. If the arche is taken to be an origin, then specific causality is implied; that is, B is supposed to be characteristically B just because it comes from A, which dominates it.The archai that Aristotle had in mind in his well-known passage on the first Greek scientists are not necessarily chronologically prior to their objects, but are constituents of it. For example, in pluralism objects are composed of earth, air, fire and water, but those elements do not disappear with the production of the object. They remain as archai within it, as do the atoms of the atomists.What Aristotle is really saying is that the first philosophers were trying to define the substance(s) of which all material objects are composed. As a matter of fact, that is exactly what modern scientists are attempting to accomplish in nuclear physics, which is a second reason why Thales is described as the first western scientist.

GeometryThales was known for his innovative use of geometry. His understanding was theoretical as well as practical. For example, he said:Megiston topos: hapanta gar chorei (Μέγιστον τόπος· άπαντα γαρ χωρεί)”Space is the greatest thing, as it contains all things”Topos is in Newtonian-style space, since the verb, chorei, has the connotation of yielding before things, or spreading out to make room for them, which is extension. Within this extension, things have a position. Points, lines, planes andsolids related by distances and angles follow from this presumption.Thales understood similar triangles and right triangles, and what is more, used that knowledge in practical ways. The story is told in DL (loc. cit.) that he measured the height of the pyramids by their shadows at the moment when his own shadow was equal to his height. A right triangle with two equal legs is a 45-degree right triangle, all of which are similar. The length of the pyramid’s shadow measured from the center of the pyramid at that moment must have been equal to its height.This story indicates that he was familiar with the Egyptian seked, or seqed - the ratio of the run to the rise of a slope (cotangent). The seked is at the base of problems 56, 57, 58, 59 and 60 of the Rhind papyrus - an ancient Egyptian mathematics document.In present day trigonometry, cotangents require the same units for run and rise (base and perpendicular), but the papyrus uses cubits for rise and palms for run, resulting in different (but still characteristic) numbers. Since there were 7 palms in a cubit, the seked was 7 times the cotangent.Thales' Theorem: To use an example often quoted in modern reference works, suppose the base of a pyramid is 140 cubits and the angle of rise 5.25 seked. The Egyptians expressed their fractions as the sum of fractions, but the decimals are sufficient for the example. What is the rise in cubits? The run is 70 cubits, 490 palms. X, the rise, is 490 divided by 5.25 or 931⁄3cubits. These figures sufficed for the Egyptians and Thales. We would go on to calculate the cotangent as 70 divided by 931⁄3 to get 3/4 or .75 and looking that up in a table of cotangents find that the angle of rise is a few minutes over 53 degrees.Whether the ability to use the seked, which preceded Thales by about 1000 years, means that he was the first to define trigonometry is a matter of opinion. More practically Thales used the same method to measure the distances of ships at sea, said Eudemus as reported by Proclus (“in Euclidem”). According to Kirk & Raven (reference cited below), all you need for this feat is three straight sticks pinned at one end and knowledge of your altitude. One stick goes vertically into the ground. A second is made level. With the third you sight the ship and calculate the seked from the height of the stick and its distance from the point of insertion to the line of sight.The seked is a measure of the angle. Knowledge of two angles (the seked and a right angle) and an enclosed leg (the altitude) allows you to determine by similar triangles the second leg, which is the distance. Thales probably had his own equipment rigged and recorded his own sekeds, but that is only a guess.

Thales’ Theorem is stated in another article. (Actually there are two theorems called Theorem of Thales, one having to do with a triangle inscribed in a circle and having the circle's diameter as one leg, the other theorem being also called the intercept theorem.) In addition Eudemus attributed to him the discovery that a circle is bisected by its diameter, that the base angles of an isosceles triangle are equal and that vertical angles are equal. According to a historical Note[24], when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as: all straight angles are equal, equals added to equals are equal, and equals subtracted from equals are equal. It would be hard to imagine civilization without these theorems.It is possible, of course, to question whether Thales really did discover these principles. On the other hand, it is not possible to answer such doubts definitively. The sources are all that we have, even though they sometimes contradict each other.

Pythagoras (Πυθαγόρας)

Bust of Pythagoras of Samos in the Capitoline Museums, RomeBorn c. 570 BC

Samos

Died c. 495 BC (aged around 75)Metapontum

Era Ancient philosophy

Region Western philosophy

School Pythagoreanism

Main interests Metaphysics, Music, Mathematics, Ethics, Politics

Notable ideas Musica universalis, Golden ratio[citation needed], Pythagorean tuning, Pythagorean theorem

Influenced by•Thales, Anaximander, Pherecydes

Influenced•Philolaus, Alcmaeon, Parmenides, Plato, Euclid, Empedocles,

 Hippasus, Kepler

Pythagoras

Pythagoras of Samos (Ancient Greek: Πυθαγόρας ὁ Σάμιος [Πυθαγόρης in Ionian Greek] Pythagóras ho Sámios "Pythagoras the Samian", or simply Πυθαγόρας; b. about

570 – d. about 495 BC[1][2]) was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about

Pythagoras was written down centuries after he lived, so very little reliable information is known about him. He was born on the island of Samos, and might have travelled widely

in his youth, visiting Egypt and other places seeking knowledge. Around 530 BC, he moved to Croton, a Greek colony in southern Italy, and there set up a religious sect. His followers pursued the religious rites and practices developed by Pythagoras, and studied his philosophical theories. The society took an active role in the politics of Croton, but this

eventually led to their downfall. The Pythagorean meeting-places were burned, and Pythagoras was forced to flee the city. He is said to have ended his days in Metapontum.Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. He is often revered as a greatmathematician, mystic and scientist, but he

is best known for the Pythagorean theorem which bears his name. However, because legend and obfuscation cloud his work even more than with the other pre-Socratic

philosophers, one can give account of his teachings to a little extent, and some have questioned whether he contributed much to mathematics and natural philosophy. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors. Whether or not his disciples believed that everything was related to mathematics and that numbers were the ultimate reality is unknown. It was said

that he was the first man to call himself a philosopher, or lover of wisdom, [3] and Pythagorean ideas exercised a marked influence on Plato, and through him, all

ofWestern philosophy.

Pythagorean theoremA visual proof of the Pythagorean theorem

Since the fourth century AD, Pythagoras has commonly been given credit for discovering thePythagorean theorem, a theorem in geometry that states that in a right-angled triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas

of the squares of the other two sides—that is, .a²+b²=c²While the theorem that now bears his name was known and previously utilized by

the Babylonians andIndians, he, or his students, are often said to have constructed the first proof. It must, however, be stressed that the way in which the Babylonians handled Pythagorean numbers implies that they knew that the principle was generally applicable, and knew some kind of proof,

which has not yet been found in the (still largely unpublished) cuneiform sources. [46] Because of the secretive nature of his school and the custom of its students to attribute everything to their teacher,

there is no evidence that Pythagoras himself worked on or proved this theorem. For that matter, there is no evidence that he worked on any mathematical or meta-mathematical problems. Some

attribute it as a carefully constructed myth by followers of Plato over two centuries after the death of Pythagoras, mainly to bolster the case for Platonic meta-physics, which resonate well with the ideas

they attributed to Pythagoras. This attribution has stuck down the centuries up to modern times.[47] The earliest known mention of Pythagoras's name in connection with the theorem occurred five

centuries after his death, in the writings of Cicero and Plutarch.

Carl Friedrich Gauss Carl Friedrich Gauss (1777–1855), painted byChristian Albrecht Jensen

Born30 April 1777

Braunschweig, Duchy of Brunswick-Wolfenbüttel,Holy Roman EmpireDied

23 February 1855(aged 77)Göttingen, Kingdom of Hanover

ResidenceKingdom of Hanover

NationalityGermanFields

Mathematics and PhysicsInstitutions

University of GöttingenAlma mater

University of HelmstedtDoctoral advisor

Johann Friedrich Pfaff

Other academic advisorsJohann Christian Martin Bartels

Doctoral studentsFriedrich Bessel

Christoph GudermannChristian Ludwig Gerling

Richard DedekindJohann EnckeJohann Listing

Bernhard RiemannChristian Peters

Moritz CantorOther notable students

Gotthold EisensteinGustav KirchhoffErnst Kummer

Johann DirichletAugust Ferdinand Möbius

Julius WeisbachL. C. Schnürlein

Known forSee full listInfluenced

Sophie GermainNotable awards

Copley Medal (1838)

Johann Carl Friedrich

Gauss (  /ˈɡaʊs/; German: Gauß   listen (help·info), Latin: Carolus Fridericus Gauss) (30

April 1777 – 23 February 1855) was a German mathematician and physical scientist who

contributed significantly to many fields, including number theory, statistics, analysis,differential

geometry, geodesy geophysics, electrostatics, astronomy and optics. Sometimes referred to as the Princeps

mathematicorum(Latin, "the Prince of Mathematicians" or "the foremost of mathematicians") and "greatest

mathematician since antiquity", Gauss had a remarkable influence in many fields of mathematics and science and is

ranked as one of history's most influential mathematicians. He referred to mathematics as "the queen

of sciences"