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Reading time: ~25 min Reveal all steps

We have now reached an important point in geometry - being able to state and understand one of the most famous theorems in all of mathematics: Pythagoras' Theorem. It is named after the ancient Greek mathematician Pythagoras of Samos.

In any right-angled triangle, the square of the length of the Pythagoras' Theorem
hypotenuse (the side that lies opposite the right angle) is equal to the sum of the squares of the other two sides. In other words,

a b c 2 2 = 2 +

The converse is also true: if the three sides in a triangle satisfy a 2 right-angled acute obtuse. +b 2=c 2, then it must be

Right angles are everywhere, and that's why Pythagoras' Theorem is so useful.

Here you can see a The bottom of the ladder is 6m long ladder leaning on a wall. 1m away from the wall. How far does it reach up the wall?

Notice that there is a right-angled triangle formed by the ladder, the wall and the ground. Using Pythagoras' theorem, we get

Whenever you've got a right-angled triangle and know two of its sides, Pythagoras can help you find the third one.

Proving Pythagoras' Theorem

Pythagoras' theorem was known to ancient Babylonians, Mesopotamians, Indians and Chinese - but Pythagoras may have been the first to find a formal, mathematical proof.

There are actually many different ways to prove Pythagoras' theorem. Here you can see three different examples that each use a different strategy:

Rearrangement

Have a look at the figure on the right. The square has side length a+b, and contains four right-angled triangles, as well as a smaller square of area a−b a+b .

Now let's rearrange the triangles in the square. The result still contains the four right-angles triangles, as well as two squares of size and .

Comparing the area of the red area before and after the rearrangement, we see that

a 2+b 2=c 2.

This is the original proof that Pythagoras came up with.

Algebra

Here we have the same figure as before, but this time we'll use algebra rather than rearrangement to prove Pythagoras' theorem.

The large square has side length a a 2+b 2 . +b and area

It consists of four triangles, each with an area of 1 2 a b 1 2 a+b , and one square of area a×b .

If we combine all of that information, we have

a+b 2

=4×1 2 a b+c 2

a 2+2 a b+b 2

=2 a b+c 2

a 2+b 2

=c 2

And, once again, we get Pythagoras' theorem.

Similar Triangles

Here you can see another right-angled triangle. If we draw one of the altitudes, it splits the triangle into two smaller triangle. It also divides the hypotenuse c into two smaller parts which we'll call x and y. Continue

Let's separate out the two smaller triangles, so that it's clearer to see how they are related... Continue

Both smaller triangles share one angle with the original triangle. They also all have one right angle. By the AA condition, all three triangles must be similar congruent right-angled.

Now we can use the equations we already know about similar polygons:

Continue

But remember that c=x + y . Therefore

c=a 2 c+b 2 c

c 2=a 2+b 2

Once more, we've proven Pythagoras' theorem!

Much about Pythagoras' life is unknown, and no original copies of his work have survived. He founded a religious cult, the Pythagoreans, that practiced a kind of "number worship". They believed that all numbers have their own character, and followed a variety of other bizarre customs.

The Pythagoreans are credited with many mathematical discoveries, including finding the first irrational number, 2 . Irrational numbers cannot be expressed as a simple fraction - a concept the Pythagoreans found deeply troubling and (unsuccessfully) tried to cover up!

Calculating Distances

One of the most important application of Pythagoras' Theorem is for calculating distances.

On the right you can see two points in a coordinate system. We could measure their distance using a ruler, but that is not particularly accurate. Instead, let's try using Pythagoras. Continue

We can easily count the horizontal distance along the x-axis, and the vertical distance along the y-axis. If we draw those two lines, we get a right-angled triangle.

Using Pythagoras,

d 2

=${b.x-a.x} 2+${b.y-a.y} 2

d 2

=${(b.x-a.x)**2 + (b.y-a.y)**2}

d

=${(b.x-a.x)**2 + (b.y-a.y)**2} =${round(distance(a,b),4)}

This method works for any two points:

If you are given two points with coordinates ( The Distance Formula
x 1, y 1) and (x 2, y 2), the distance between them is

d 2=x 2−x 1 2+y 2−y 1 2

d=x 2−x 1 2+y 2−y 1 2

Pythagorean Triples

As you moved the However there are a few examples of right-angled triangles where the lengths of vertices of the triangle in the previous step, you might have noticed that in most cases, the length of the hypothenuse all three sides happens to be whole numbers. d ended up being a decimal number fraction integer.

One famous example is the 3-4-5 triangle. Since 3 2+4 2=5 2, any triangle with sides of length 3, 4 and 5 must be right-angled.

The ancient Egyptians didn't know about Pythagoras' theorem, but they did know about the 3-4-5 triangle. When building the pyramids, they used knotted ropes of lengths 3, 4 and 5 to measure perfect right angles.

Three integers like this are called Pythagorean Triples. (3, 4, 5) is one example of a Pythagorean triple. If we multiply every number by 2, we get another Pythagorean triple: (6, 8, ).

We can think of these triples as grid points in a coordinate systems. For a valid Pythagorean triples, the distance from the origin to the grid point has to be a whole number. Using the coordinate system below, can you find any other Pythagorean triples?

Do you notice any pattern in the distribution of these points?

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Isosceles and Equilateral Triangles

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Pythagoras' Theorem - Triangles and Trigonometry - Mathigon

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Reading time: ~25 min Reveal all steps We have now reached an important point in geometry - being able to state and understand one of the most famous theorems in all of mathematics: Pythagoras' Theorem. It is named after the ancient Greek mathematician Pythagoras of Samos.

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BB.4 Converse of the Pythagorean Theorem: Is it a right triangle?

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BB.4 Converse of the Pythagorean Theorem: Is it a right triangle?

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Research and share an important or interesting fact about Square Roots

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square root, in mathematics, a factor of a number that, when multiplied by itself, gives the original number. For example, both 3 and -3 are square roots of 9. As early as the 2nd millennium bc, the Babylonians possessed effective methods for approximating square roots. See root.

William L. Hosch.

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Square root, in mathematics, a factor of a number that, when multiplied by itself, gives the original number. For example, both 3 and -3 are square roots of 9. As early as the 2nd millennium bc, the Babylonians possessed effective methods for approximating square roots. See

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Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)-or, in familiar algebraic notation, a ² + b ² = c ². Although the theorem has long been associated with Greek mathematician-philosopher Pythagoras (c. 570-500/490 ), it is actually far older. Four Babylonian tablets from circa 1900-1600 bce indicate some knowledge of the theorem, with a very accurate calculation of the square root of 2 (the length of the hypotenuse of a right triangle with the length of both legs equal to 1) and lists of special integers known as Pythagorean triples that satisfy it (e.g., 3, 4, and 5; 3 ² + 4 ² = 5 ², 9 + 16 = 25). The theorem is mentioned in the Baudhayana Sulba-sutra of India, which was written between 800 and 400 bce. Nevertheless, the theorem came to be credited to Pythagoras. It is also proposition number 47 from Book I of Euclid's Elements.

According to the Syrian historian Iamblichus (c. 250-330 ce), Pythagoras was introduced to mathematics by Thales of Miletus and his pupil Anaximander. In any case, it is known that Pythagoras traveled to Egypt about 535 bce to further his study, was captured during an invasion in 525 bce by Cambyses II of Persia and taken to Babylon, and may possibly have visited India before returning to the Mediterranean. Pythagoras soon settled in Croton (now Crotone, Italy) and set up a school, or in modern terms a monastery ( see Pythagoreanism), where all members took strict vows of secrecy, and all new mathematical results for several centuries were attributed to his name. Thus, not only is the first proof of the theorem not known, there is also some doubt that Pythagoras himself actually proved the theorem that bears his name. Some scholars suggest that the first proof was the one shown in the figure. It was probably independently discovered in several different cultures.

Book I of the Elements ends with Euclid's famous "windmill" proof of the Pythagorean theorem. ( See Sidebar: Euclid's Windmill.) Later in Book VI of the Elements, Euclid delivers an even easier demonstration using the proposition that the areas of similar triangles are proportionate to the squares of their corresponding sides. Apparently, Euclid invented the windmill proof so that he could place the Pythagorean theorem as the capstone to Book I. He had not yet demonstrated (as he would in Book V) that line lengths can be manipulated in proportions as if they were commensurable numbers (integers or ratios of integers). The problem he faced is explained in the Sidebar: Incommensurables.

A great many different proofs and extensions of the Pythagorean theorem have been invented. Taking extensions first, Euclid himself showed in a theorem praised in antiquity that any symmetrical regular figures drawn on the sides of a right triangle satisfy the Pythagorean relationship: the figure drawn on the hypotenuse has an area equal to the sum of the areas of the figures drawn on the legs. The semicircles that define Hippocrates of Chios's lunes are examples of such an extension. ( See Sidebar: Quadrature of the Lune.)

In the Nine Chapters on the Mathematical Procedures (or Nine Chapters), compiled in the 1st century ce in China, several problems are given, along with their solutions, that involve finding the length of one of the sides of a right triangle when given the other two sides. In the Commentary of Liu Hui, from the 3rd century, Liu Hui offered a proof of the Pythagorean theorem that called for cutting up the squares on the legs of the right triangle and rearranging them ("tangram style") to correspond to the square on the hypotenuse. Although his original drawing does not survive, the next figure shows a possible reconstruction.

The Pythagorean theorem has fascinated people for nearly 4,000 years; there are now more than 300 different proofs, including ones by the Greek mathematician Pappus of Alexandria (flourished c. 320 ce), the Arab mathematician-physician Thābit ibn Qurrah (c. 836-901), the Italian artist-inventor Leonardo da Vinci (1452-1519), and even U.S. Pres. James Garfield (1831-81).

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Pythagorean theorem | Definition & History

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Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)-or, in familiar algebraic notation, a2 + b2 = c2. Although the theorem has long been associated with Greek mathematician-philosopher Pythagoras (c.

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Pythagoras (l.c. 571- c. 497 BCE) was a Greek philosopher whose teachings emphasized the immortality and transmigration of the soul (reincarnation), virtuous, humane behavior toward all living things, and the concept of "number" as truth in that mathematics not only cleared the mind but allowed for an objective comprehension of reality.

He is best known in the modern day for the Pythagorean Theorem, a mathematical formula which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides. This formula has been applied to measuring distance and space as, for example, in planning and executing the construction of a building. Although attributed to Pythagoras by ancient writers, modern scholars cite evidence from Babylonian texts, written some time before Pythagoras, which discuss the same formula or, at least, one very similar.

Almost nothing is known of Pythagoras' life even though later writers (such as Diogenes Laertius, l. c.180-240 CE) attempted to put together biographies based on stories and fragments from earlier works. Laertius' biography of Pythagoras is the most complete but, unfortunately, the author never cites the sources he drew from and so it is impossible to corroborate many of his claims.

The difficulty in any discussion of Pythagoras is to separate the actual man & his teachings from the mythology which surrounded him.

Pythagoras' influence on later philosophers, and the development of Greek philosophy generally, was enormous. Plato (l. c. 428/427-348/347 BCE) references Pythagoras in a number of his works and Pythagorean thought, as understood and relayed by other ancient writers, is the underlying form of Plato's philosophy. Plato's famous student Aristotle (l. 384-322 BCE) also incorporated Pythagorean teachings into his own thought and Aristotle's works would influence philosophers, poets, and theologians (among many others) from his time through the Middle Ages (c. 476-1500 CE) and into the modern day. Although Pythagoras remains a mysterious figure in antiquity, therefore, he also stands as one of the most significant in the development of philosophical and religious thought.

Life & Works

What is known of Pythagoras comes from later writers piecing together fragments of his life from contemporaries and students. It is known that Pythagoras was born on the island of Samos, off Asia Minor, where his ancestors had settled after leaving Phlius, a city in the northwest Peloponnese, after the civil war there in 380 BCE. He received a quality education as his father, Mnesarchus, was a wealthy merchant. He may have studied in Babylon and in Egypt and possibly had the best Greek tutors of the time. All of this is speculative, however, as the information comes from later writers who accepted, uncritically, what others wrote about him. If there was an authoritative biography of Pythagoras, or original works by the man himself, they are long lost. Scholar Forrest E. Baird comments:

Pythagoras was associated with so many legends that few scholars dare to say much about his life, his personality, or even his teachings, without adding that we cannot be sure our information is accurate. That there was a man named Pythagoras who founded the sect called the Pythagoreans, we need not doubt; among the witnesses to his historicity was his younger contemporary Heraclitus, who thought ill of him. Nevertheless, it is notoriously difficult to distinguish between the teachings of Pythagoras himself and those of his followers, the Pythagoreans. (14)

The historicity of Pythagoras has never been questioned. Heraclitus (l. c. 500 BCE), as Baird notes, considered Pythagoras highly overrated and another contemporary, the visionary Xenophanes of Colophon (l. c. 570-c. 478 BCE), mocked Pythagoras for his belief in reincarnation. The difficulty in any discussion of Pythagoras is in trying to separate the actual man and his teachings from the mythology which surrounded him even in his own lifetime.

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Pythagorean Beliefs

As noted, none of Pythagoras' writings - if he wrote anything - have survived and, owing to the secrecy he demanded of his students, the specifics of his teachings were carefully kept. The philosopher Porphyry (l. c. 234 - c. 305 CE), who wrote a later biography of Pythagoras, noted:

What he taught his disciples no one can say for certain, for they maintained a remarkable silence. All the same, the following became generally known. First, he said that the soul is immortal; second, that it migrates into other kinds of animals; third, that the same events are repeated in cycles, nothing being new in the strict sense; and finally, that all things with souls should be regarded as akin. Pythagoras seems to have been the first to introduce these beliefs to Greece. (Robinson, 58)

The Greek historian Herodotus (l.c. 484 - c. 425/413 BCE) alludes to Pythagoras (though famously refuses to name him) in his Histories:

Moreover, the Egyptians are the first to have maintained the doctrine that the soul of man is immortal and that, when the body perishes, it enters into another animal that is being born at the same time, and when it has been the complete round of the creatures of the dry land and of the sea and of the air it enters again into the body of a man at birth; and its cycle is completed in three thousand years. There are some Greeks who have adopted this doctrine, some in former times and some in later, as if it were their own invention; their names I know but refrain from writing down. (Book II.123)

Like the Pythagorean Theorem, Pythagoras' concept of the transmigration of souls may also have been borrowed. Scholar George G. M. James, in his work Stolen Legacy: The Egyptian Origins of Western Philosophy, points out that all of the great Pre-Socratic philosophers either studied in Egypt or in the Egyptian Mystery Schools of Asia Minor (James, 9). Thales (l.c. 585 BCE), considered the first Western philosopher, studied in Babylon and two other of the most significant Pre-Socratics - Anaximander (l. c. 610-c.546 BCE) and Anaximenes (l. c. 546 BCE) - both traveled extensively and had access to the Mystery Schools which focused on Egyptian religious thought.

Whatever the reason, the secrecy of Pythagoras' teachings added greatly to his mystique & reputation.

It is more than likely, then, that Pythagoras' thought was actually Egyptian spirituality transplanted to Greece. Pythagoras' famous secrecy may have been intended to keep this fact from circulating too widely and discrediting him as an original thinker. He is said to have been quite charismatic and a powerful public speaker and it would have undermined his authority if his philosophy was revealed as simply re-packaged Egyptian belief.

Whether he concealed his teachings for this reason or some other cannot be ascertained. It is possible he simply felt the masses would not understand or appreciate his ideas. Whatever the reason, the secrecy added greatly to his mystique and reputation. His belief in the immortality of the soul and reincarnation led naturally to a vegetarian lifestyle with an emphasis on doing no harm to any other living thing and this asceticism, which he also demanded of his followers, elevated his reputation as a holy man even further. Diogenes Laertius describes his diet and habits:

Some say that he was satisfied with honey alone or a bit of honeycomb or bread (he did not touch wine during the day); or, for a treat, vegetables boiled or raw. Seafood he ate but rarely. His robe, which was white and spotless, and his bedclothes, which were also white, were of wool; for linen had not yet reached those parts. He was never observed to relieve himself, or to have intercourse, or to be drunk. He used to avoid laughter and all pandering to scurrilous jokes and vulgar stories. (VIII.19)

Laertius describes Pythagoras as a pescatarian, eating fish and sea food, but most other ancient authors maintain he was a strict vegetarian abstaining from the meat of any living thing which could be regarded as having a soul. He likewise abstained from sex and remained celibate to maintain spiritual power and clarity of thought. In disengaging from worldly pleasures such as sex and food, he freed himself from the distractions of the body to focus on the improvement of the soul.

This ascetism was thought by some to go too far. He and his followers were known to especially abstain from eating, or even touching, beans (one account of his death, in fact, claims that he would not enter a bean field to escape pursuers and so was killed). Laertius also mentions Xenophanes' satirical criticism of Pythagoras' belief in the transmigration of souls:

Once, they say, that he [Pythagoras] was passing by when a puppy was being whipped, and he took pity and said, `Stop! Do not beat it! For it is the soul of a friend that I recognized when I heard it giving tongue. (VIII.36)

To Xenophanes, who rejected reincarnation, Pythagoras' beliefs were as foolish as claiming one could recognize a departed friend's voice in the bark of a dog. To Pythagoras, however, vegetarianism, pacifism, and humane treatment of other living things were all part of the path to inner peace and, by extension, world peace in that humans could never live in harmony as long as they killed, ate, and were cruel to animals. Poor treatment of animals, and eating animal flesh, devalued all life by maintaining that some creatures (humans) were worth more in life than others. Pythagoras believed that all creatures were created equal and should be treated with respect.

He was considered by contemporaries and later writers as a mystic - not a mathematician as his is sometimes defined in the present day - and his school was associated with spiritual salvation and miraculous revelation. A central belief, which would significantly influence Plato, was that philosophic inquiry was vital to the salvation of the soul and apprehension of ultimate truth. An aspect of that truth was that nothing ever significantly changed and all was eternal and eternally recurring. According to the ancient writer, and student of Aristotle's, Eudemus of Rhodes (l. c. 370 - c. 300 BCE), Pythagoras believed in eternal recurrence as a logical, mathematical necessity. Eudemus writes:

If one were to believe the Pythagoreans, that events recur in an arithmetical cycle, and that I shall be talking to you again sitting just as you are now, with this pointer in my hand, and that everything else will be just as it is now, then it is plausible to suppose that the time, too, will be the same as now. (Baird, 16)

In this belief, Pythagoras prefigures the great German philosopher Fredrich Nietzsche (l. 1844-1900 CE) and his Theory of Eternal Recurrence in which Nietzsche claims that, in the absence of the "finish line" of a God who renders judgment after death, one's life will automatically reset and repeat itself in precisely the same way. Nietzsche's theory has often been interpreted as an encouragement to carefully consider how one spends one's time since one will have to relive every event, large or small, eternally; this may have also been suggested by Pythagoras' teachings.

Even if Pythagoras himself did not frame the concept in this way, he must have articulated it somehow for later Pythagoreans to have repeated it. The concept of the cyclical nature of life and the immortality of the soul were at the heart of Pythagorean thought and influenced many writers and thinkers of ancient Greece but none as significant as Plato.

Pythagoras & Plato

It is possible that Plato began as a student of Socrates, adhering to dialectic in establishing truth, and then only gradually moved toward embracing the idealism of Pythagoras - as some scholars have claimed - but it seems more probable that Socrates himself was aligned with Pythagorean thought. There is really no way of establishing any claim along these lines since most of what we know of Socrates comes from Plato's Dialogues which were written after Socrates' death when Plato was already of a mature philosophical mind.

However he was introduced to it, Pythagorean thought significantly influenced Plato's philosophy which included the concept of an ultimate truth not subject to opinion, of an ethical way of living in line with that truth, the soul's immortality, the necessity of salvation through philosophy, and of learning-as-recollection. Pythagorean concepts are apparent throughout Plato's work but most notably in the dialogues of the Meno and Phaedo.

In the Meno, Plato's main character Socrates shows how what one calls "learning" is actually only "remembering" lessons from a past life. He proves his claim by having a young, uneducated slave solve a geometrical problem. Plato argues that, if one dies with one's mind intact, one will 'remember' what one learned during that life when one is born into the next. What one thinks one 'learns' in this life, one is actually only 'remembering' from one's past life and what one knew in that past life was remembered from a previous one.

Plato never addresses the obvious problem with this theory: at some point, the soul must have had to actually "learn" and not just "remember". His claim that one "remembers" what one learned in the ether in between lives - not just in mortal form - does not address the concern because the soul still would need to "learn" at some point, whether in the body or out of it.

Pythagoras' assertion that "things are numbers" and that one could understand the physical world through mathematics also features in the Meno, not only through Socrates' interaction with the slave but through his argument that virtue is a singular quality inherent in all people, regardless of their age, sex, or social status, in the same way that "number" informs and defines the known world; one recognizes reality through a distinction between unity and duality.

This claim would go toward the development of Plato's famous Theory of Forms in which he describes an objective world of Truth, above the mortal realm, which underlies and informs all human truths and gives them their value of "truthfulness". Without this Realm of Forms, Plato argued, there could be no actual truth; only opinion about what one felt was true.

To Pythagoras, mathematics was the path toward enlightenment and understanding and, as he claimed, "Ten is the very nature of number" and by this 'number' he meant not only a unit of measurement but a means by which the world could be grasped and understood. He noted how people can count up to ten on their fingers and, having reached ten, revert back to the one-unit and begin again. In the same way, a soul entered a body, lived for a certain time, died, and reverted back to where it started from, only to then travel the same path again.

This concept is explored fully in Plato's Phaedo, the account of Socrates' last day in prison before his execution, which focuses on the immortality of the soul and the afterlife. Right from the start of the dialogue, Plato makes use of Pythagoras' link to Philius, in choosing Echecrates of Phlius as interlocutor and audience to Phaedo, the narrator. Further, the characters of Simmias and Cebes of Thebes - Socrates' central interlocutors in the account Phaedo relates - are both Pythagoreans. Plato's choice of Echecrates links the dialogue directly to Pythagorean thought from the first line but, through Simmias and Cebes, Pythagorean concepts are introduced and developed throughout.

Toward the end of the dialogue, after various proofs have been given by Socrates for the immortality of the soul, he concludes with this exchange with Cebes:

Tell me, [Socrates said], what is that which must be in the body to make it alive?

A soul, [Cebes] replied.

And is this always so?

Of course, [Cebes] said.

Then the soul always brings life to whatever contains her?

No doubt, [Cebes] answered.

And is there an opposite to life or not?

Yes.

What is it?

Death.

And we have already agreed that the soul cannot ever receive the opposite of what she brings?"

Yes, certainly we have, said Cebes.

...

What do we call that which does not admit death?

The immortal, [Cebes] said.

And the soul does not admit death?

No.

Then soul is immortal?

It is.

Good, [Socrates] said. Shall we say that this is proved? What do you think?

( Phaedo, 105c-e)

The mathematical proofs Socrates offers earlier concerning even and uneven numbers finally leads to the above proof that "even" cannot admit of the "odd" in order to remain itself (even) and so life (the soul) cannot admit of death and still remain life; therefore, the soul must be immortal. This entire argument typifies Pythagorean thought as understood by ancient writers and practiced by Pythagorean sects of Plato's time.

Conclusion

The Phaedo also establishes the geography of the afterlife which would later be used by the Church in creating the concepts of Hell, Purgatory, and Heaven. The concept of purgatory first appears in Phaedo 108b-d, judgment of the dead in 113d-e, Hell in 113e-114a, and heaven in 109d-110b. Plato's argument for an ultimate, undeniable, realm of truth, from which all other truths are established, is also evident in the gospel narratives of the Bible, most notably the Gospel of John, and in the epistles of St. Paul.

Even though nothing can be said for certain regarding Pythagoras' life or original teachings, enough of his thought was developed by later disciples and admirers to have influenced the greatest Greek philosopher of antiquity. Plato's work established the discipline of philosophy and has permeated every other, to greater or lesser degrees, for the past 2,000 years. The details of Pythagoras' life may never be fully known but his influence continues to be felt, world-wide, in the present day.

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Pythagoras (l.c. 571- c. 497 BCE) was a Greek philosopher whose teachings emphasized the immortality and transmigration of the soul (reincarnation), virtuous, humane behavior toward all living things, and the concept of "number" as truth in that mathematics not only cleared the mind but allowed for an objective comprehension of reality.

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Interactivate: Pythagorean Explorer

Pythagorean Explorer: Calculate the length of one side of an automatically generated right triangle by using the Pythagorean Theorem, and then check your answers. Pythagorean Explorer is one of the Interactivate assessment explorers.

On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty enhancement, and interactive curriculum development at all levels. Student development of numerical models and simulations integrated with core curriculum provides an opportunity to gain practical experience in computational science. Additionally, the National Computational Science Institute (NCSI) provides nation-wide workshops portraying resources and instructional ideas to middle school, high school, and undergraduate instructors for use in the classroom. Resources and materials offered to these instructors are available free of charge from Shodor's website and are largely developed by Shodor student interns. Shodor's academic program efficiently guides participants from excitement to experience to expertise through computational explorations, research opportunities, and service.

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Interactivate: Pythagorean Explorer On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty enhancement, and interactive curriculum development at all levels. Student development of numerical models and simulations integrated with core curriculum provides an opportunity to gain practical experience in computational science.

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Interactive Maths - The Interactive Way to Teach Mathematics

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Use the GeoGebra Activity below to investigate the areas of the squares on the sides of right angled triangles. The two orange points N and B can be selected and dragged along the lines they are on....

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500 Lightstick-Toting Math Enthusiasts ‘Pythagorized’ NYC’s Iconic Flatiron Building In Epic Fashion

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For the Museum of Math’s first anniversary.

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Try out these challenge problems. Share your answer in the comments section under each problem. Let's see if you can come to a consensus on these answers...

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Find words with a minimum of four letters that include the center letter. (Ex. "goal" is legal, but "loco" is not because it does not use the center letter).

BONUS: Find a word that uses ALL letters at least once.

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Use any combination of numbers and operations provided to reach the target number. Each number can only be used once.

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Use any combination of numbers and operations provided to reach the target number. Each number can only be used once.

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Use any combination of numbers and operations provided to reach the target number. Each number can only be used once.

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Here is a collection of puzzles and problems from Mathigon.

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A selection of our favourite mathematical puzzles and problems. Most are simple to explain, but the solutions require clever and unconventional thinking.