The Pythagorean theorem may be the best-known equation in mathematics. Its origins reach back to the beginnings of civilization, and today every student continues to study it. What most nonmathematicians don't understand or appreciate is why this simply stated theorem has fascinated countless generations.
I first met Alfred S. Posamentier in 2001, when he was the Dean of Education at City College in New York. The energy level of his intellectual engagement with my work is part of an experience I will never forget.

In his entertaining and informative book, The Pythagorean Theorem: The Story of Its Power and Beauty, Posamentier, a veteran math educator, makes the importance of the Pythagorean theorem delightfully clear.

The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power. Scores of references to it can be found in literature, plays, musicals, songs, stamps and cartoons. It can be introduced to students during the middle school years and becomes increasingly important during high school. It is not enough to merely state the algebraic formula for the Pythagorean Theorem.

According to University of Georgia graduate student Stephanie Morris:

Students need to see the geometric connections as well. The teaching and learning of the Pythagorean Theorem can be enriched and enhanced through the use of dot paper, geoboards, paper folding, and computer technology, as well as many other instructional materials.
The Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:  a^2 + b^2 = c^2!,  where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.

The theorem is named after the Greek mathematician Pythagoras, who by tradition (only) is credited with its discovery and proof. The theorem has numerous proofs, possibly the most of any mathematical theorem. These are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids.

Posamentier begins his book with a brief history of Pythagoras and the early use of his theorem by the ancient Egyptians, Babylonians, Indians, and Chinese, who used it intuitively long before Pythagoras's name was attached to it.

He then shows the many ingenious ways in which the theorem has been proved visually using highly imaginative diagrams. Some of these go back to ancient mathematicians; others are comparatively recent proofs, including one by the twentieth president of the United States, James A. Garfield.

After demonstrating some curious applications of the theorem, Posamentier then explores the Pythagorean triples, pointing out the many hidden surprises of the three numbers that can represent the sides of the right triangle (e.g, 3, 4, 5 and 5, 12, 13). And many will truly amaze the reader.

Next, Posamentier turns to the "Pythagorean means" (the arithmetic, geometric, and harmonic means). By comparing their magnitudes in a variety of ways, he gives the reader a true appreciation for these mathematical concepts.

The final two chapters view the Pythagorean theorem from an artistic point of view--namely, how Pythagoras's work manifests itself in music and how the Pythagorean theorem can influence fractals.

This lucid presentation and gift for conveying the significance of this key equation to those with little math background will inform, entertain, and inspire the reader, once again demonstrating the power and beauty of mathematics!

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Comment Preferences

• Informative diary I use the Pythagorean Theory(6+ / 0-)

much in my trade. It is very much in play in the calculation of ohm's law and is used in figuring out the bends in conduit. I am an Electrician.

• Tipped and rec'd(3+ / 0-)
Recommended by:
nancat357, palantir, Annalize5

"Corruptio Optimi Pessima" (Corruption of the best is the worst)

• Thanks Mr. Bear!(3+ / 0-)
Recommended by:
palantir, Lujane, nancat357

I did pretty well in math, and I love all the funny ways that numbers work together.

I'm lucky.

• Oops, I may have assumed wrong.(3+ / 0-)
Recommended by:
palantir, Lujane, nancat357

Should I have said Ms. Bear?

I'm lucky.

[ Parent ]

• words and numbers(5+ / 0-)

I love stuff like this. I'm an English teacher but every week or so something catches my attention from math or science that I just got to talk over for the proverbial mini-lesson. Last week it was the N D Tyson "most astounding fact about the universe."  Also the Vihart fibonacci stuff on youtube I liked too much and students watched with eyes glued to the screen. This guy was generalizing a bit much about teachers and math education for me (I also do a GED class and adults worry as much about their writing skills as their math in my experience) but that Kaprekar number was too cool and will be on the board before the end of the week.

• I gotta' question: the way you wrote the equation(3+ / 0-)
Recommended by:
sviscusi, palantir, Monsieur Georges

... is:  a^2 + b^2 = c^2!

Normally I'm used to seeing the exclamation point used to denote "logical NOT," as in  "A != B," meaning "A does NOT equal B."

Clearly "logical NOT" isn't the correct interpretation of how you used the exclamation point.

What does it mean in the equation the way you wrote it?

"Minus two votes for the Democrat" equals "plus one vote for the Republican." Arithmetic doesn't care about your feelings.

• I think it means(4+ / 0-)
Recommended by:

Holy cow, isn't that cool!^2

• oh, ok:-) (4+ / 0-)

Well, it is cool.  One of the really nifty things about the physical universe is that so much of it can be described in math, and we can reasonably expect the rest of it to eventually be described in math, even where the math is probabilistic (e.g. QM and whatever body of theory will eventually exceed the scope of QM as it has done for Einsteinian relativity, which in turn did the same for Newtonian mechanics).

"Minus two votes for the Democrat" equals "plus one vote for the Republican." Arithmetic doesn't care about your feelings.

[ Parent ]

• Well, I had a different thought(2+ / 0-)
Recommended by:

about the exclamation point.  I think of it as "factorial."  But I figured it had to do with excitement.  ;-)

I'm lucky.

[ Parent ]

• When I used to teach the(4+ / 0-)

Pythagorean Thm in HS geometry I always tried to encourage my students to dig further into the history of Pythagoras with a few historical/legendary tidbits about the man:

1. Pythagoras didn't "discover" the Pythagorean thm, he learned it while he studied for years in the great temples of Egypt.  Egyptian farmers were using the PT for centuries to restore the boundaries of their farms each year after the Nile overflowed its banks.

2. When Pythagorus returned to Greece he headed up a religious cult which believed that everything in the universe could be defined in terms of number. They were stymied, however, when they couldnt find a unique number to represent the hypotenuse of a right triangle with legs of 1 and 1. They continued to work on it unsuccessfully and the legend is that they drowned the member the cult who first disclosed publically that there was no unique number to define the square root of 2.

• I've got some video proofs, if you're interested(1+ / 0-)
Recommended by:
Lujane
• Useful, but beauty is in the eye of the beholder(0+ / 0-)

As a builder, I thought the most useful class I ever took was trigonometry. Ive used the Pythgorean Theorem so often in my life i could not name all the times.
but beautiful?  My Granddaughters are beautiful, the P Theorem is just useful IMHO.
I like other useful things like guns and tools but theyre not beautiful

Happy just to be alive

• Calling ET(3+ / 0-)
Recommended by:
Monsieur Georges, Lujane, Annalize5

I seem to recall reading once about a scientist who proposed drawing a huge diagram of the Pythagorean Theorem out in Siberia or some vacant piece of real estate, large enough so that extraterrestrial life, if it existed, would be able to see it through their telescope and know that Earth posessed life intelligent enough to grasp basic mathematical concepts.

It was never carried out, though.

"All the World's a Stage and Everyone's a Critic." -- Mervyn Alquist

• A lot of engineering and physics math(2+ / 0-)
Recommended by:
Annalize5, Monsieur Georges

is simply constant use and re-use of the Pythagorean theorem. The use of vectors in rectangular co-ordinates reduces a lot of problem solution to either finding a resultant hypotenuse knowing the x and y vectors (or some corresponding set of axes, like north and east), or else breaking down a vector in some arbitrary direction into x and y components.

So if you want to analyze the forces in a structure, like a beam or truss, or find the phase angle of an electrical signal in a circuit, or compute the trajectory of an artillery shell in terms of height and distance or the course of a jet given its velocity, heading and the wind direction, you end up going back to Pythagoras.

It's never too late to have a happy childhood - Tom Robbins