Earliest Evidence of Pythagoras’ Theorem

October 24, 2012 · Posted in Firsts in History, Math · Comments Off on Earliest Evidence of Pythagoras’ Theorem 

To answer the question of what is the first evidenced knowledge of the familiar equation, a^2 + b^2 = c^2, named after the Greek philosopher Pythagoras (569-500 B.C.E.), depends on who you ask.

Credit for this geometrical proof has been attributed to, of course, namesake Pythagoras, but also to the ancient Babylonians via the tablet Plimpton 322, the ancient Chinese from the Zhou Bi Suan Jing (c. 100 B.C.E.- c. 100 C.E.), the Indian mathematician Bhaskara, and to Euclid who included a variation in his text The Elements.

Though the jury may be out on the rightful owner of being the first, it is evident that the ancients understood the theorem before Pythagoras got around to writing his proof.

The Chinese mathematics text,

 

chinise_proof

Visual proof of Pythagoras Theorem from Chinise Math text called Zhou Bi Suan Jing

 

The Zhou Bi Suan Jing (周髀算经) is a book called The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven. It is one of the oldest of Chinese mathmatical texts.

Compilation of the contents took place first in the Zhou Dynasty (1046 BCE—256 BCE), and continued into the Western Han Dynasty (202 BCE – 220 CE). Its contents include 246 problems, along with the corresponding answers and arithmetic algorithms. Found within this collection is a recorded proof of the Pythagorean Theorem.

chinese proof

Another example is found in the Plimpton 322. The Babylonian tablet was written sometime around 1800 BCE in ancient Iraq (also called Mesopotamia). It is housed at Columbia and has a table consisting of fifteen columns of Pythagorean triples. Triples are a set of three positive integers a, b and c, where a2 = b2 + c2

plimpton_322

The Babylonian Tablet showing the pythagorean theorem

 

Sources: Zhou Bi Suan Jing , http://www.maa.org/news/monthly105-120.pdf

 

 

First African-American Grand Master Of Chess

August 19, 2012 · Posted in Black History, Math · Comments Off on First African-American Grand Master Of Chess 

There are only about a thousand grand masters of chess in the world and only one of them is African-American: Maurice Ashley.

He wasn’t even good enough to make his high school chess team. But he studied hard and became a master when he was 20, then, 14 years later– a grand master– a ranking just short of world champion.

He’s 45 now and Maurice Ashley has made chess his life. He travels the world bringing chess to kids who might not otherwise be aware of it, often playing…and winning! against an entire room of young hopefuls lined up before him at their chessboards.

Some of the upstarts he may have to keep an eye on: three young African-American New Yorkers who recently became masters before their 13th birthdays!!

Source

Pythagoras for Kids

July 21, 2011 · Posted in Math · Comment 

 

Pythagoras

http://www.historyforkids.org/learn/greeks/science/math/pythagoras.htm

Pythagoras lived in the 500s BC, and was one of the first Greek mathematical thinkers. He spent most of his life in the Greek colonies in Sicily and southern Italy. He had a group of followers (like the later disciples of Jesus) who followed him around and taught other people what he had taught them. The Pythagoreans were known for their pure lives (they didn’t eatbeans, for example, because they thought beans were not pure enough). They wore their hair long, and wore only simple clothing, and went barefoot. Both men and women were Pythagoreans.

Pythagoreans were interested in philosophy, but especially in music and mathematics, two ways of making order out of chaos. Music is noise that makes sense, and mathematics is rules for how the world works.

Pythagoras himself is best known for proving that the Pythagorean Theorem was true. The Sumerians, two thousand years earlier, already knew that it was generally true, and they used it in their measurements, but Pythagoras is said to have proved that it would always be true. We don’t really know whether it was Pythagoras that proved it, because there’s no evidence for it until the time ofEuclid, but that’s the tradition. Some people think that the proof must have been written around the time of Euclid, instead.

Here is the proof:

The Pythagorean Theorem says that in a right triangle, the sum of the squares of the two right-angle sides will always be the same as the square of the hypotenuse (the long side). A2 + B2 = C2. Try it yourself: if Side A is 4 inches long, and Side B is 3 inches long, then 4×4=16, and 3×3=9, and 9+16=25, and so Side C will be 5 inches long. Try it with other size triangles and see if this is still true (you can use a calculator, or your computer, to figure out the square roots).

But how can you know that this is always true, every single time, no matter what size the triangle is?

Take a straight line and divide it into two pieces, and call one piece a and the other piece b, like this:

Now make a square with this line on each side, like this:

and draw in the lines where A meets B on each side to make four smaller shapes. So now you have one square with area AxA (the big yellow one) and one square with area BxB (the little green one) and two rectangles with area AxB (the light blue ones). So the area of the whole square is (A+B) x (A+B) or the area is (AxA) + 2(AxB) + (BxB).

Or you might say that
(A+B)2 = A2 + 2AB + B2

Now draw diagonal lines across the blue rectangles, making four smaller blue triangles. Call those lines C. Do you see that you have made four blue right triangles, whose sides are A, B, and C?

Now imagine that you take these triangles and rearrange them (or if you print it out you can cut them up with scissors and really rearrange them) around the edges of the square like this:

The little triangles take up part of the square. The area of all four triangles together is the same as the two blue rectangles you made them from, so that is 2AB.

The area of the pink square in the middle is CxC or C2.

And the area of the whole big square is, as we have already seen,
A2 + 2AB + B2
So A2 + 2AB + B2 = 2AB + C2
We can subtract 2AB from both sides, so
that gives (ta da!)

A2 + B2 = C2

Here’s an animated short video showing another way to prove the Pythagorean Theorem.