Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
The theorem can be written as an equation relating the lengths of the sides, and the hypotenuse, sometimes called the Pythagorean equation:
The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.
When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points.
The theorem can be generalized in various ways: to higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and to objects that are not triangles at all but -dimensional solids.
History
There is debate whether the Pythagorean theorem was discovered once, or many times in many places, and the date of first discovery is uncertain, as is the date of the first proof. The history of the theorem can be divided into four parts: knowledge of Pythagorean triples, knowledge of the relationship among the sides of a right triangle, knowledge of the relationships among adjacent angles, and proofs of the theorem within some deductive system.Written 1800BC, the Egyptian Middle Kingdom Berlin Papyrus 6619 includes a problem involving two squares whose areas sum to a third square, whose solution is the Pythagorean triple 6:8:10, but the problem does not mention a triangle. According to Plutarch, the ancient Egyptians did know about the 3:4:5 right triangle, identifying its sides with Osiris, Isis, and Horus respectively.
Historians of Mesopotamian mathematics have concluded that the Pythagorean rule was in widespread use during the Old Babylonian period, over a thousand years before Pythagoras was born. The Mesopotamian tablet Plimpton 322, written near Larsa also 1800BC, contains entries that can be interpreted as the sides and diagonals of 15 different Pythagorean triples. Another tablet from a similar time, YBC 7289, calculates the diagonal of a square or, equivalently, of an isosceles right triangle.
In India, the Baudhayana Shulba Sutra, the dates of which are given variously as between the 8th and 5th century BC, contains a list of Pythagorean triples and a statement of the Pythagorean theorem, both in the special case of the isosceles right triangle and in the general case, as does the Apastamba Shulba Sutra.
Byzantine Neoplatonic philosopher and mathematician Proclus, writing in the fifth century AD, states two arithmetic rules, "one of them attributed to Plato, the other to Pythagoras", for generating special Pythagorean triples. The rule attributed to Pythagoras starts from an odd number and produces a triple with leg and hypotenuse differing by one unit; the rule attributed to Plato starts from an even number and produces a triple with leg and hypotenuse differing by two units. According to Thomas L. Heath, no specific attribution of the theorem to Pythagoras exists in the surviving Greek literature from the five centuries after Pythagoras lived. However, when authors such as Plutarch and Cicero attributed the theorem to Pythagoras, they did so in a way which suggests that the attribution was widely known and undoubted. Classicist Kurt von Fritz wrote, "Whether this formula is rightly attributed to Pythagoras personally ... one can safely assume that it belongs to the very oldest period of Pythagorean mathematics." Around 300 BC, in Euclid's Elements, the oldest extant axiomatic proof of the theorem is presented, along with Euclid's formula for generating all primitive Pythagorean triples.
With contents known much earlier, but in surviving texts dating from roughly the 1st century BC, the Chinese text Zhoubi Suanjing, gives a reasoning for the Pythagorean theorem for the triangle — in China it is called the "Gougu theorem". During the Han Dynasty, Pythagorean triples appear in The Nine Chapters on the Mathematical Art, together with a mention of right triangles. Some believe the theorem arose first in China in the 11th century BC, where it is alternatively known as the "Shang Gao theorem", named after the Duke of Zhou's astronomer and mathematician, whose reasoning composed most of what was in the Zhoubi Suanjing.
Proofs using constructed squares
Rearrangement proofs
In one rearrangement proof, two squares are used whose sides have a measure of and which contain four right triangles whose sides are, and, with the hypotenuse being. In the square on the right side, the triangles are placed such that the corners of the square correspond to the corners of the right angle in the triangles, forming a square in the center whose sides are length. Each outer square has an area of as well as, with representing the total area of the four triangles. Within the big square on the left side, the four triangles are moved to form two similar rectangles with sides of length and. These rectangles in their new position have now delineated two new squares, one having side length is formed in the bottom-left corner, and another square of side length formed in the top-right corner. In this new position, this left side now has a square of area as well as. Since both squares have the area of it follows that the other measure of the square area also equal each other such that . With the area of the four triangles removed from both side of the equation what remains is.In another proof rectangles in the second box can also be placed such that both have one corner that correspond to consecutive corners of the square. In this way they also form two boxes, this time in consecutive corners, with areas and which will again lead to a second square of with the area.
English mathematician Sir Thomas Heath gives this proof in his commentary on Proposition I.47 in Euclid's Elements, and mentions the proposals of German mathematicians Carl Anton Bretschneider and Hermann Hankel that Pythagoras may have known this proof. Heath himself favors a different proposal for a Pythagorean proof, but acknowledges from the outset of his discussion "that the Greek literature which we possess belonging to the first five centuries after Pythagoras contains no statement specifying this or any other particular great geometric discovery to him." Recent scholarship has cast increasing doubt on any sort of role for Pythagoras as a creator of mathematics, although debate about this continues.
Algebraic proofs
The theorem can be proved algebraically using four copies of the same triangle arranged symmetrically around a square with side, as shown in the lower part of the diagram. This results in a larger square, with side and area. The four triangles and the square side must have the same area as the larger square,giving
A similar proof uses four copies of a right triangle with sides, and, arranged inside a square with side as in the top half of the diagram. The triangles are similar with area, while the small square has side and area. The area of the large square is therefore
But this is a square with side and area, so
Other proofs of the theorem
This theorem may have more known proofs than any other ; the book The Pythagorean Proposition contains 370 proofs.Proof using similar triangles
This proof is based on the proportionality of the sides of three similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.Let represent a right triangle, with the right angle located at, as shown on the figure. Draw the altitude from point, and call its intersection with the side. Point divides the length of the hypotenuse into parts and. The new triangle,, is similar to triangle, because they both have a right angle, and they share the angle at, meaning that the third angle will be the same in both triangles as well, marked as in the figure. By a similar reasoning, the triangle is also similar to. The proof of similarity of the triangles requires the triangle postulate: The sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the triangles leads to the equality of ratios of corresponding sides:
The first result equates the cosines of the angles, whereas the second result equates their sines.
These ratios can be written as
Summing these two equalities results in
which, after simplification, demonstrates the Pythagorean theorem:
The role of this proof in history is the subject of much speculation. The underlying question is why Euclid did not use this proof, but invented another. One conjecture is that the proof by similar triangles involved a theory of proportions, a topic not discussed until later in the Elements, and that the theory of proportions needed further development at that time.