pythagoras essay in 500 words

Pythagorean Theorem: History, Formula, and Proof Essay

Introduction.

Pythagoras Theory is a relatively simple theory used continuously in Standard Grade Mathematics and beyond. It is also used in Physics. It is used not only to simply solve the missing side of a right-angled triangle but also more extensively to solve Reasoning and Application problems and also can be used to solve many higher mathematics problems in trigonometry and in many topics throughout the mathematics syllabus. It is so basic that I’m sure anyone who studied it at school will remember it long after other theorems have been forgotten.

Early evidence of the theory can be traced back as far as 2000 BC with the ancient Egyptians. Pythagoras had traveled to Egypt and this may have influenced some of his beliefs. There is some evidence that they used a 3-4-5 triangle to form a perfect right angle. However, very little information pre-dates the Greeks so this remains a mystery along with many other ancient Egyptian stories. Hence the Theorem was credited to Pythagoras. What is more likely is that Pythagoras was the first to prove it. Pythagoras then generalized it to all right-angled triangles hence it is Pythagoras’ theorem.

Pythagoras was born on the island of Samos in around 582 B.C and died in around 500 BC He was a Greek philosopher and mathematician. He had been instructed in the teachings of the early Ionian philosophers Thales, Anaximander, and Anaximenes. It is believed that Pythagoras was driven out of Samos in 532 BC by Polycrates, the tyrant who ruled there. He moved to a Greek colony in southern Italy named Crotona where he set up a religious and philosophical school. A movement is known as Pythagoreanism.

Little is really known about Pythagoras’ actual work. His school practiced in secrecy thus making it difficult to distinguish between his work and that of his followers. Whether someone from the school or Pythagoras himself found the proof can’t be truly known. Hence the philosophy of Pythagoras is known through the work of his disciples. The school did make several contributions to mathematics. Pythagoras believed that all relations were based on number relations. He believed that numbers were the very essence of things and their motto was ‘all is number’ . This was based on various observations in music, astronomy, and mathematics. They were interested in the principles of mathematics, the concept of number, triangles, and the abstract idea of proof.

Proof of Pythagoras Theorem

Pythagoreans carried out investigations into odd and even numbers and of prime and square numbers. They used this to back up their belief that whole numbers and their ratios could account for all geometrical proportions. By far the greatest and most well-known theorem is the hypotenuse theorem or Pythagoras’ theorem. It allows us to find the third side of a right-angled triangle given the other two sides. The formal definition is ‘ the square of the hypotenuse of a right-angle triangle is equal to the sum of the square of the other two sides. It should be noted in Pythagoras’ terms he wouldn’t have meant square as the number multiplied by itself but as a geometrical square. The square on the smaller sides could be broke up and reassembled to make a square at the hypotenuse.

a² + b² = c²

¨ It only works for right-angled triangles

¨ We must have the lengths of two other sides

Here we can see that c is the hypotenuse and a and b are the other 2 sides.

Let a = 4, b = 3 and c =5, as shown above. The theorem claims that the area of the two smaller squares will be equal to the square of the larger one.

4² + 3² = 5²

16 + 9 = 25 as require

Draw a perpendicular from C to line AB.

c/a = a/e; c/b = b/d
ce = a 2 ; cd = b 2
ce + cd = a 2 + b 2
c(e + d) = a 2 + b 2
c 2 = a 2 + b 2

Pythagoras may have used the above methods but as his movement practiced in secrecy this is still the only estimation. Also, it cannot be fully understood if he himself or one of his disciples found this.

Euclid found the next method. Really it is not too different from the first method. Euclid’s method therefore only really supports the first proof. Euclid was first however to prove that the theorem was reversible. Therefore a²+ b² = c² can prove a triangle to be right-angled. He also generalizes Pythagoras’ theorem and proves that the area of a semi-circle along the hypotenuse of a right-angled triangle is equal to the sum of the areas of semi-circles on the other two sides. Euclid also used the following proof on his book II of elements:

A large square of side a+b is divided into two smaller squares of sides a and b respectively, and two equal rectangles with sides a and b; each of these two rectangles can be split into two equal right triangles by drawing the diagonal c. The four triangles can be arranged within another square of side a+b as shown in the figures.

The area of the square can be shown in two different ways:

As the sum of the area of the two rectangles and the squares:
As the sum of the areas of a square and the four triangles:

Now, setting the two right-hand-side expressions in these equations equal gives

Therefore, the square on c is equal to the sum of the squares on a and b”. The following are a couple more proofs, which greatly improve the pupil’s understanding of Pythagoras. Even if they only understand one proof it is still worthwhile and increases their knowledge of Pythagoras.

The first proof involves a rectangle divided up into three triangles, each of which contains a right angle. This proof can be seen through the use of computer technology, or with something as simple as a 3×5 index card cut up into right triangles.

It can be seen that triangles 2 (in green) and 1 (in red) will completely overlap triangle 3 (in blue). Now, we can give proof of the Pythagorean Theorem using these same triangles.

Compare triangles 1 and 3.

Angles E and D, respectively, are the right angles in these triangles. By comparing their similarities, we have

and from Figure 3, BC = AD. So,

By cross-multiplication, we get:

Compare triangles 2 and 3:

By comparing the similarities of triangles 2 and 3 we get:

From Figure 1, AB = CD. By substitution,

Cross-multiplication gives:

Finally, by adding equations 1 and 2, we get:

From triangle 3,

AC = AE + EC

The next proof of the Pythagorean Theorem that will be presented is one in which a trapezoid will be used.

By the construction that was used to form this trapezoid, all 6 of the triangles contained in this trapezoid are right triangles. Thus,

Area of Trapezoid = The Sum of the areas of the 6 Triangles

And by using the respective formulas for area, we get:

The proofs led to the Pythagorean problem, which is one of the earliest problems in the history of numbers. It was to find all the right-angled triangles that fitted the Pythagorean x² + y²= z². The three integers would be known as a Pythagorean Triple. E.g. 3, 4, 5

7, 24, 25, etc

Today the Pythagorean theorem plays a significant part in many fields of mathematics. For example, it is the basics of Trigonometry, and in its arithmetic form, it unites Geometry and Algebra. Pythagoras’ Theorem is introduced to pupils in the middle of their high school career and becomes more important as they develop further in mathematics. Teaching the topic can be enhanced through computer technology and with proof so that the pupils can understand the concept behind the theory and not just the algebraic formula. This helps them see the importance rather than just substituting in numbers. This theorem and its proof were basic progress in the field of mathematics. It became a foundation of ancient geometry and had more control over theory and more practical appliances than any other. Today his theorem is used largely in building, architecture, carpenter, navigation, astronomy, and many other fields of work that involve mathematical calculations.

Each of these fields uses his theorem to try and choose either the hypotenuse or the two other sides in a right-angled triangle. Builders use Pythagoras’ theorem to calculate the magnitude of different aspects of their constructions. This allows them to work out the exact necessities of building resources needed. Architects use their theorem to work out a blueprint for the builders to use. His theory may be used to work out accurate lengths of roofs, also the structure of a house just to name a few. Carpenter uses his theorem to take the size of sides of their timber formation eg: corner fittings. Navigators and astronomers use his theorem to calculate distances between planets, satellites, and stars.

Works Cited

Maor, Eli, The Pythagorean Theorem: A 4,000-Year History . Princeton, New Jersey: Princeton University Press, 2007.

Chicago (A-D)
Chicago (N-B)

IvyPanda. (2022, May 12). Pythagorean Theorem: History, Formula, and Proof. https://ivypanda.com/essays/pythagoras-theorem-definition/

"Pythagorean Theorem: History, Formula, and Proof." IvyPanda , 12 May 2022, ivypanda.com/essays/pythagoras-theorem-definition/.

IvyPanda . (2022) 'Pythagorean Theorem: History, Formula, and Proof'. 12 May.

IvyPanda . 2022. "Pythagorean Theorem: History, Formula, and Proof." May 12, 2022. https://ivypanda.com/essays/pythagoras-theorem-definition/.

1. IvyPanda . "Pythagorean Theorem: History, Formula, and Proof." May 12, 2022. https://ivypanda.com/essays/pythagoras-theorem-definition/.

Bibliography

IvyPanda . "Pythagorean Theorem: History, Formula, and Proof." May 12, 2022. https://ivypanda.com/essays/pythagoras-theorem-definition/.

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MacTutor

Pythagoras of samos.

... was transported by the followers of Cambyses as a prisoner of war. Whilst he was there he gladly associated with the Magoi ... and was instructed in their sacred rites and learnt about a very mystical worship of the gods. He also reached the acme of perfection in arithmetic and music and the other mathematical sciences taught by the Babylonians...

... he formed a school in the city [ of Samos ] , the 'semicircle' of Pythagoras, which is known by that name even today, in which the Samians hold political meetings. They do this because they think one should discuss questions about goodness, justice and expediency in this place which was founded by the man who made all these subjects his business. Outside the city he made a cave the private site of his own philosophical teaching, spending most of the night and daytime there and doing research into the uses of mathematics...

... he tried to use his symbolic method of teaching which was similar in all respects to the lessons he had learnt in Egypt. The Samians were not very keen on this method and treated him in a rude and improper manner.

... Pythagoras was dragged into all sorts of diplomatic missions by his fellow citizens and forced to participate in public affairs. ... He knew that all the philosophers before him had ended their days on foreign soil so he decided to escape all political responsibility, alleging as his excuse, according to some sources, the contempt the Samians had for his teaching method.

(1) that at its deepest level, reality is mathematical in nature, (2) that philosophy can be used for spiritual purification, (3) that the soul can rise to union with the divine, (4) that certain symbols have a mystical significance, and (5) that all brothers of the order should observe strict loyalty and secrecy.

It is hard for us today, familiar as we are with pure mathematical abstraction and with the mental act of generalisation, to appreciate the originality of this Pythagorean contribution.

The Pythagorean ... having been brought up in the study of mathematics, thought that things are numbers ... and that the whole cosmos is a scale and a number.

Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly. This feeling modern mathematics has deliberately eliminated, but we still find overtones of it in fiction and poetry. Ten was the very best number: it contained in itself the first four integers - one, two, three, and four [1 + 2 + 3 + 4 = 10] - and these written in dot notation formed a perfect triangle.

After [ Thales , etc. ] Pythagoras transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner: he it was who discovered the theory of irrational and the construction of the cosmic figures.

I emulate the Pythagoreans who even had a conventional phrase to express what I mean "a figure and a platform, not a figure and a sixpence", by which they implied that the geometry which is deserving of study is that which, at each new theorem, sets up a platform to ascend by, and lifts the soul on high instead of allowing it to go down among the sensible objects and so become subservient to the common needs of this mortal life.

... the following philosophical and ethical teachings: ... the dependence of the dynamics of world structure on the interaction of contraries, or pairs of opposites; the viewing of the soul as a self-moving number experiencing a form of metempsychosis, or successive reincarnation in different species until its eventual purification ( particularly through the intellectual life of the ethically rigorous Pythagoreans ) ; and the understanding ...that all existing objects were fundamentally composed of form and not of material substance. Further Pythagorean doctrine ... identified the brain as the locus of the soul; and prescribed certain secret cultic practices.

In their ethical practices, the Pythagorean were famous for their mutual friendship, unselfishness, and honesty.

Cylon, a Crotoniate and leading citizen by birth, fame and riches, but otherwise a difficult, violent, disturbing and tyrannically disposed man, eagerly desired to participate in the Pythagorean way of life. He approached Pythagoras, then an old man, but was rejected because of the character defects just described. When this happened Cylon and his friends vowed to make a strong attack on Pythagoras and his followers. Thus a powerfully aggressive zeal activated Cylon and his followers to persecute the Pythagoreans to the very last man. Because of this Pythagoras left for Metapontium and there is said to have ended his days.

... was violently suppressed. Its meeting houses were everywhere sacked and burned; mention is made in particular of "the house of Milo" in Croton, where 50 or 60 Pythagoreans were surprised and slain. Those who survived took refuge at Thebes and other places.

References ( show )

K von Fritz, Biography in Dictionary of Scientific Biography ( New York 1970 - 1990) . See THIS LINK .
Biography in Encyclopaedia Britannica. http://www.britannica.com/biography/Pythagoras
R S Brumbaugh, The philosophers of Greece ( Albany, N.Y., 1981) .
M Cerchez, Pythagoras ( Romanian ) ( Bucharest, 1986) .
Diogenes Laertius, Lives of eminent philosophers ( New York, 1925) .
P Gorman, Pythagoras, a life (1979) .
T L Heath, A history of Greek mathematics 1 ( Oxford, 1931) .
Iamblichus, Life of Pythagoras ( translated into English by T Taylor ) ( London, 1818) .
I Levy, La légende de Pythagore de Grèce en Ralestine ( Paris, 1927) .
L E Navia, Pythagoras : An annotated bibliography ( New York, 1990) .
D J O'Meara, Pythagoras revived : Mathematics and philosophy in late antiquity ( New York, 1990) .
Porphyry, Vita Pythagorae ( Leipzig, 1886) ,
Porphyry, Life of Pythagoras in M Hadas and M Smith, Heroes and Gods ( London, 1965) ..
E S Stamatis, Pythagoras of Samos ( Greek ) ( Athens, 1981) .
B L van der Waerden, Science Awakening ( New York, 1954) .
C J de Vogel, Pythagoras and Early Pythagoreanism (1966) .
H Wussing, Pythagoras, in H Wussing and W Arnold, Biographien bedeutender Mathematiker ( Berlin, 1983) .
L Ya Zhmud', Pythagoras and his school ( Russian ) , From the History of the World Culture 'Nauka' ( Leningrad, 1990) .
C Byrne, The left-handed Pythagoras, Math. Intelligencer 12 (3) (1990) , 52 - 53 .
H S M Coxeter, Polytopes, kaleidoscopes, Pythagoras and the future, C. R. Math. Rep. Acad. Sci. Canada 7 (2) (1985) , 107 - 114 .
W K C Guthrie, A History of Greek Philosophy I (1962) , 146 - 340 .
F Lleras, The theorem of Pythagoras ( Spanish ) , Mat. Ense nanza Univ. 19 (1981) , 3 - 12 .
B Russell, History of Western Philosophy ( London, 1961) , 49 - 56 .
G Tarr, Pythagoras and his theorem, Nepali Math. Sci. Rep. 4 (1) (1979) , 35 - 45 .
B L van der Waerden, Die Arithmetik der Pythagoreer, Math. Annalen 120 (1947 - 49) , 127 - 153 , 676 - 700 .
L Zhmud, Pythagoras as a Mathematician, Historia Mathematica 16 (1989) , 249 - 268 .
L Ya Zhmud', Pythagoras as a mathematician ( Russian ) , Istor.-Mat. Issled. 32 - 33 (1990) , 300 - 325 .

Additional Resources ( show )

Other pages about Pythagoras:

See Pythagoras on a timeline
Pythagoras's theorem
Pythagorus in competition with Boethius in Margaista Philosophica (1504)
An entry in The Mathematical Gazetteer of the British Isles
Astronomy: The Structure of the Solar System
Heinz Klaus Strick biography
Miller's postage stamps

Other websites about Pythagoras:

Dictionary of Scientific Biography
Encyclopaedia Britannica
G Don Allen
Internet Encyclopedia of Philosophy
Google books
Sci Hi blog

Honours ( show )

Honours awarded to Pythagoras

Lunar features Crater Pythagoras
Popular biographies list Number 2

Cross-references ( show )

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History Topics: Perfect numbers
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History Topics: Pythagoras's theorem in Babylonian mathematics
History Topics: The Golden ratio
History Topics: The Indian Sulbasutras
History Topics: The Ten Mathematical Classics
History Topics: The history of cartography
History Topics: The real numbers: Pythagoras to Stevin
Student Projects: Indian Mathematics - Redressing the balance: Chapter 10
Student Projects: Indian Mathematics - Redressing the balance: Chapter 19
Student Projects: Indian Mathematics - Redressing the balance: Chapter 5
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Other: 1936 ICM - Oslo
Other: 2009 Most popular biographies
Other: Earliest Known Uses of Some of the Words of Mathematics (A)
Other: Earliest Known Uses of Some of the Words of Mathematics (H)
Other: Earliest Known Uses of Some of the Words of Mathematics (M)
Other: Earliest Known Uses of Some of the Words of Mathematics (P)
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Pythagoras: Life, work and achievements

Although famous throughout the world, Pythagoras’ life is shrouded in mystery.

Pythagoras, a pioneer of early Greek philosophy, mathematics and natural science

Pythagoras Theory

In his footsteps, additional resources, bibliography.

Born in Samos in around 570 B.C, Pythagoras is commonly said to be the first pure mathematician who proposed that everything is a number.

Although he is most famous for his mathematical theorem, Pythagoras also made extraordinary developments in astronomy and geometry. He also developed a theory of music while and founded a philosophical and religious school in Croton, Italy. It was here he taught that "the whole cosmos is a scale and a number", according to the University of St Andrews .

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While playing on his lyre, which was an ancient Greek stringed instrument, Pythagoras discovered that the vibrating strings created a beautiful sound when the ratios of the lengths of the wires were whole numbers, and that this was also true of other instruments. He combined this discovery with his understanding of the planets, conceiving the theory that when the planets were in harmony, it created beautiful music that man was incapable of hearing.

Pythagoras concluded that mathematics and music were interconnected and that knowledge of one area led to an understanding of the other, according to the University of Connecticut . He also believed that music had healing properties and would often play his lyre for the sick and dying.

Little is known about the life of Pythagoras and, as a result, many bizarre myths have sprung up around the man.

It was claimed amongst other things that he had taken part in the Olympics and was awarded laurels for pugilism, or boxing, when he was a young man. It was also said that he had fought in the Trojan Wars during a previous life.

This last myth reflects Pythagoras' genuine belief in metempsychosis, which argues that all souls are everlasting and, when the physical body dies, it simply floats away and finds a new body to live in, according to Stanford University . Later reports stated that he had been able to clearly recall four previous lives.

His fascination with astronomy ,as with many ancient Greeks, combined with his deep understanding of numbers led Pythagoras to confirm that the Earth was in fact a sphere and, through patient study, he discovered that the Evening Star and the Morning Star were the same planet, Venus .

Pythagoras' Theory states that in a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides, according to Encyclopedia Britannica.

In other words, when a triangle has a right angle and squares are made of each of the three sides, then the biggest square has the same area as the other two squares combined. The equation can be used to work out the length of a third side if only two measurements are given.

The Babylonians discovered this mathematical phenomena circa 1900 – 1600 BC but Pythagoras may have been first to prove it, according to New Scientist .

Although Pythagoras’ Theory is still taught in every classroom today, no one would recognise his original school of thought as it combined his mathematical teachings with philosophy and religion. His followers, the Pythagoreans

created a secret commune, filled with strange rules and regulations, according to Encyclopaedia Britannica .

Much of his written work was stored in the Great Library of Alexandria . Far from being the master mathematician that we think of today, Pythagoras was known for his belief in reincarnation, religious rituals and almost magical abilities, according to Stanford University. For example, it was said that he could be in two places at the same time. Today, these mystical elements have been almost forgotten and he is now looked upon as a founding father of science and mathematics .

Greek philosopher Plato created the world's first university, known as the Platonic Academy, in ancient Athens. Although different from a modern day university, the Academy was a place where people could meet and share their academic beliefs. Plato based a large proportion of his teachings on the thoughts of Pythagoras and his Pythagorean disciples, according to Stanford University.

Like Pythagoras, Aristotle was interested in the concept of a soul, according to the University of Washington . He wrote "On the Soul", which set out to examine the psychology of mankind, the principles of which are still referred to by psychologists today. Aristotle combined metaphysics with scientific investigation just as Pythagoras had achieved with metaphysics and the Number Theory. He was also inspired by Pythagoras’ interest in astronomy, ultimately developing the physical model of the heavens.

To find out more about Pythagoras, check out “ Pythagoras: His Life, Teaching, and Influence ”, by Christoph Riedweg and “ P ythagoras: His Lives And The Legacy Of A Rational Universe ”, by Kitty Ferguson.

Mickaël Launay & Stephen S. Wilson, " It All Adds Up: The Story of People and Mathematics ", William Collins, 2019.
NRICH, " All is Number ", University of Cambridge, 2017.
Michael Marshall, " Babylonians calculated with triangles centuries before Pythagoras ", New Scientist, August 2021.
Stanford Encyclopedia of Philosophy, " Pythagoras ", University of Stanford, October 2018.
Holger Thesleff, " Pythagoreanism ", Encyclopedia Britannica, May 2020.
Encyclopedia Britannica, " Pythagorean theorem ", May 2020.
University of Connecticut, " 3.7 Music of the Spheres and the Lessons of Pythagoras ", accessed in March 2022.
Silvano Leonessi, " The Pythagorean Philosophy of Numbers ", Rosicrucian Digest, Volume 1, 2009.
J. J. O'Connor & E. F. Robertson, " Pythagoras of Samos ", University of St Andrews, January 1999.
Brent Swancer, " The Great Pythagoras and his Mystical Cult ", Mysterious Universe, January 2021.
Dimtry Sudakov, " Pythagoras and his theory of reincarnation ", Pravda.ru, May 2013.

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Jo is a freelance journalist, academic lawyer and lecturer specialising in criminal law and forensics. Jo has written for several magazines, including Real Crime and All About History. She is also the author of a number of true crime books and horror anthologies, such as "Murderous East Anglia" and "Strangers".

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History Of Pythagoras

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Topic: Famous Person , Pythagoras

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