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# The History of the Pythagorean Theorem - Assignment Example

## The Pythagorean Theorem Essay Sample

❶Loomis, Elisha Scott

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So far, nothing has been shown in terms of an equation in relation to a right triangle before Pythagoras, but the right triangle was certainly known for its special properties, as can be seen in the figure below, which depicts the pattern of Oriental tiles in use at least one thousand years before Pythagoras:. Looking at this pattern, it is easily determined that the proof of the theorem lies within the pattern itself; yet its proof is credited to Pythagoras.

The Pythagorean Theorem is pure mathematics. It involves no irrational numbers but is found just about everywhere we look. Yet the Pythagorean Theorem is seen in the most profound occurrences of nature, architecture and music. One problem exists with the formula though…it only relates to integers, and therefore cannot fully translate itself into geometry as it is known today, although in plane geometry the theorem is undeniably valuable.

This proof is a product of Euclid and while neither simple nor obvious, it remains the most famous of proofs for the Pythagorean Theorem [ 6]. Setting the areas equal to each other and subtracting equals yields the proof of the theorem. This proof shows that the area of the square is the same no matter what final shape it takes: One final proof shows the practicality of the Theorem; it is not directly historical, yet it is still extremely useful and displays the versatility of the Pythagorean Theorem:.

Given that the Pythagoreans shunned anything other than whole numbers, an entire universe of geometric applications in terms of describing harmony and unity was lost to them. While the Pythagorean Theorem was indispensable for building and measuring perfect squares and rectangles it could not stand alone in terms of the patterns of nature without revision and the introduction of irrational numbers.

By applying the Pythagorean theorem to irrational numbers, a beautiful array of patterns emerges, as was initially discovered by Euclid. Other mathematicians have even managed to take the Theorem into three dimensions. Incredibly, the Pythagorean theorem as it was originally expressed, in whole numbers only, was limited. But using it as a platform for other geometries sparked an entire cosmology in mathematics and philosophy, demonstrating that the Theorem is fairly universal in its ability to be the springboard to other dimensions of mathematical expressions and mechanics.

Here is one example of the use of irrational numbers plugged into the Pythagorean Theorem, creating a crude spiral: By viewing this diagram, it is easy to imagine how the Pythagorean theorem has been used for building spiral staircases, applying the Theorem in three dimensions.

But it is also important to recognize how the Pythagorean theorem is hidden within innumerable other theorems, and by deduction it can be found even when it is not obvious. Euclid greatly advanced geometry, including the Pythagorean theorem, resulting in the Golden Spiral or Golden Ratio or Golden Mean , which uses the Pythagorean theorem as an integral part.

The Golden Ratio was explosive in the exploration of natural occurrences, harmonics, architecture and physics. There is such beauty in the Golden Ration and so much was done with it in ancient times that it lends a particular appreciation to the Pythagorean Theorem; take the Golden Ratio apart and there is the right triangle, ready to expand again into something greater. But it was Euclidian thinking that took the Pythagorean theorem to a whole new level by applying the Theorem to irrational numbers and circular functions.

Instead of stopping at the 3: Below is a diagram of the beginnings of the Golden Section, beginning with the Golden Rectangle and the geometric construction of the Golden Section, both based on the Pythagorean theorem:.

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With a hour delay you will have to wait for 24 hours due to heavy workload and high demand - for free. One can arrive at the Pythagorean theorem by studying how changes in a side produce a change in the hypotenuse and employing calculus.

The triangle ABC is a right triangle, as shown in the upper part of the diagram, with BC the hypotenuse. At the same time the triangle lengths are measured as shown, with the hypotenuse of length y , the side AC of length x and the side AB of length a , as seen in the lower diagram part. If x is increased by a small amount dx by extending the side AC slightly to D , then y also increases by dy.

Therefore, the ratios of their sides must be the same, that is:. This is more of an intuitive proof than a formal one: The converse of the theorem is also true: It can be proven using the law of cosines or as follows:. Construct a second triangle with sides of length a and b containing a right angle. Since both triangles' sides are the same lengths a , b and c , the triangles are congruent and must have the same angles.

Therefore, the angle between the side of lengths a and b in the original triangle is a right angle. The above proof of the converse makes use of the Pythagorean theorem itself. The converse can also be proven without assuming the Pythagorean theorem.

A corollary of the Pythagorean theorem's converse is a simple means of determining whether a triangle is right, obtuse, or acute, as follows. The following statements apply: Dijkstra has stated this proposition about acute, right, and obtuse triangles in this language:. In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths.

Some well-known examples are 3, 4, 5 and 5, 12, A primitive Pythagorean triple is one in which a , b and c are coprime the greatest common divisor of a , b and c is 1. One of the consequences of the Pythagorean theorem is that line segments whose lengths are incommensurable so the ratio of which is not a rational number can be constructed using a straightedge and compass.

Pythagoras's theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root operation. The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer. In each right triangle, Pythagoras's theorem establishes the length of the hypotenuse in terms of this unit.

For more detail, see Quadratic irrational. Incommensurable lengths conflicted with the Pythagorean school's concept of numbers as only whole numbers. The Pythagorean school dealt with proportions by comparison of integer multiples of a common subunit.

So the three quantities, r , x and y are related by the Pythagorean equation,. Note that r is defined to be a positive number or zero but x and y can be negative as well as positive. Geometrically r is the distance of the z from zero or the origin O in the complex plane.

This can be generalised to find the distance between two points, z 1 and z 2 say. The required distance is given by. The distance formula in Cartesian coordinates is derived from the Pythagorean theorem. If Cartesian coordinates are not used, for example, if polar coordinates are used in two dimensions or, in more general terms, if curvilinear coordinates are used, the formulas expressing the Euclidean distance are more complicated than the Pythagorean theorem, but can be derived from it.

A typical example where the straight-line distance between two points is converted to curvilinear coordinates can be found in the applications of Legendre polynomials in physics. The formulas can be discovered by using Pythagoras's theorem with the equations relating the curvilinear coordinates to Cartesian coordinates.

Performing the squares and combining terms, the Pythagorean formula for distance in Cartesian coordinates produces the separation in polar coordinates as:.

This formula is the law of cosines , sometimes called the generalized Pythagorean theorem. This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity. The Pythagorean theorem relates the cross product and dot product in a similar way: The relationship follows from these definitions and the Pythagorean trigonometric identity.

This can also be used to define the cross product. By rearranging the following equation is obtained. This can be considered as a condition on the cross product and so part of its definition, for example in seven dimensions. A generalization of the Pythagorean theorem extending beyond the areas of squares on the three sides to similar figures was known by Hippocrates of Chios in the 5th century BC, [45] and was included by Euclid in his Elements: If one erects similar figures see Euclidean geometry with corresponding sides on the sides of a right triangle, then the sum of the areas of the ones on the two smaller sides equals the area of the one on the larger side.

This extension assumes that the sides of the original triangle are the corresponding sides of the three congruent figures so the common ratios of sides between the similar figures are a: The basic idea behind this generalization is that the area of a plane figure is proportional to the square of any linear dimension, and in particular is proportional to the square of the length of any side.

Thus, if similar figures with areas A , B and C are erected on sides with corresponding lengths a , b and c then:. For example, the starting center triangle can be replicated and used as a triangle C on its hypotenuse, and two similar right triangles A and B constructed on the other two sides, formed by dividing the central triangle by its altitude.

See Einstein's proof by dissection without rearrangement. The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines: Clearing fractions and adding these two relations:. Pappus's area theorem is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the three sides in place of squares squares are a special case, of course.

The upper figure shows that for a scalene triangle, the area of the parallelogram on the longest side is the sum of the areas of the parallelograms on the other two sides, provided the parallelogram on the long side is constructed as indicated the dimensions labeled with arrows are the same, and determine the sides of the bottom parallelogram.

The lower figure shows the elements of the proof. Focus on the left side of the figure. The left green parallelogram has the same area as the left, blue portion of the bottom parallelogram because both have the same base b and height h. However, the left green parallelogram also has the same area as the left green parallelogram of the upper figure, because they have the same base the upper left side of the triangle and the same height normal to that side of the triangle.

Repeating the argument for the right side of the figure, the bottom parallelogram has the same area as the sum of the two green parallelograms. In terms of solid geometry, Pythagoras's theorem can be applied to three dimensions as follows. Consider a rectangular solid as shown in the figure. The length of diagonal BD is found from Pythagoras's theorem as:. Using horizontal diagonal BD and the vertical edge AB , the length of diagonal AD then is found by a second application of Pythagoras's theorem as:.

This one-step formulation may be viewed as a generalization of Pythagoras's theorem to higher dimensions. However, this result is really just the repeated application of the original Pythagoras's theorem to a succession of right triangles in a sequence of orthogonal planes.

A substantial generalization of the Pythagorean theorem to three dimensions is de Gua's theorem , named for Jean Paul de Gua de Malves: If a tetrahedron has a right angle corner like a corner of a cube , then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces.

This result can be generalized as in the " n -dimensional Pythagorean theorem": Then the square of the volume of the hypotenuse of S is the sum of the squares of the volumes of the n legs. This statement is illustrated in three dimensions by the tetrahedron in the figure. The "hypotenuse" is the base of the tetrahedron at the back of the figure, and the "legs" are the three sides emanating from the vertex in the foreground. As the depth of the base from the vertex increases, the area of the "legs" increases, while that of the base is fixed.

The theorem suggests that when this depth is at the value creating a right vertex, the generalization of Pythagoras's theorem applies. In a different wording: The Pythagorean theorem can be generalized to inner product spaces , [56] which are generalizations of the familiar 2-dimensional and 3-dimensional Euclidean spaces.

For example, a function may be considered as a vector with infinitely many components in an inner product space, as in functional analysis. In an inner product space, the concept of perpendicularity is replaced by the concept of orthogonality: The inner product is a generalization of the dot product of vectors.

The dot product is called the standard inner product or the Euclidean inner product. However, other inner products are possible.

The concept of length is replaced by the concept of the norm v of a vector v , defined as: In an inner-product space, the Pythagorean theorem states that for any two orthogonal vectors v and w we have. This form of the Pythagorean theorem is a consequence of the properties of the inner product:. A further generalization of the Pythagorean theorem in an inner product space to non-orthogonal vectors is the parallelogram law: Any norm that satisfies this equality is ipso facto a norm corresponding to an inner product.

The Pythagorean identity can be extended to sums of more than two orthogonal vectors. Another generalization of the Pythagorean theorem applies to Lebesgue-measurable sets of objects in any number of dimensions. Specifically, the square of the measure of an m -dimensional set of objects in one or more parallel m -dimensional flats in n -dimensional Euclidean space is equal to the sum of the squares of the measures of the orthogonal projections of the object s onto all m -dimensional coordinate subspaces.

The Pythagorean theorem is derived from the axioms of Euclidean geometry , and in fact, the Pythagorean theorem given above does not hold in a non-Euclidean geometry. Here two cases of non-Euclidean geometry are considered— spherical geometry and hyperbolic plane geometry ; in each case, as in the Euclidean case for non-right triangles, the result replacing the Pythagorean theorem follows from the appropriate law of cosines.

The sides are then related as follows: This equation can be derived as a special case of the spherical law of cosines that applies to all spherical triangles:. By expressing the Maclaurin series for the cosine function as an asymptotic expansion with the remainder term in big O notation ,. Substituting the asymptotic expansion for each of the cosines into the spherical relation for a right triangle yields. The constants a 4 , b 4 , and c 4 have been absorbed into the big O remainder terms since they are independent of the radius R.

This formula is a special form of the hyperbolic law of cosines that applies to all hyperbolic triangles: On an infinitesimal level, in three dimensional space, Pythagoras's theorem describes the distance between two infinitesimally separated points as:. Such a space is called a Euclidean space. However, in Riemannian geometry , a generalization of this expression useful for general coordinates not just Cartesian and general spaces not just Euclidean takes the form: Sometimes, by abuse of language, the same term is applied to the set of coefficients g ij.

It may be a function of position, and often describes curved space. A simple example is Euclidean flat space expressed in curvilinear coordinates. For example, in polar coordinates:. 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.

According to Joran Friberg, a historian of mathematics, evidence indicates that the Pythagorean theorem was well-known to the mathematicians of the First Babylonian Dynasty 20th to 16th centuries BC , which would have been over a thousand years before Pythagoras was born, thus an example of Stigler's law of eponymy. Bartel Leendert van der Waerden — conjectured that Pythagorean triples were discovered algebraically by the Babylonians. In India , the Baudhayana Sulba Sutra , the dates of which are given variously as between the 8th and 5th century BC, [75] contains a list of Pythagorean triples discovered algebraically, a statement of the Pythagorean theorem, and a geometrical proof of the Pythagorean theorem for an isosceles right triangle.

Projecting a sphere to a plane. Point Line segment ray Length. Volume Cube cuboid Cylinder Pyramid Sphere. Hyperbolic triangle and Gaussian curvature. Sally; Paul Sally American Mathematical Society Bookstore. The Story of Its Power and Beauty , p. School of Mathematics and Statistics. Retrieved 25 January In this article we examine four Babylonian tablets which all have some connection with Pythagoras's theorem. Certainly the Babylonians were familiar with Pythagoras's theorem.

The comparatively late writers who attribute it to him add the story that he sacrificed an ox to celebrate his discovery.

The exact sciences in antiquity Republication of Brown University Press 2nd ed. For a different view, see Dick Teresi The Ancient Roots of Modern Science.

That notion is pretty much laid to rest, however, by Eleanor Robson New Light on Plimpton ". The American Mathematical Monthly. Mathematical Association of America. The generally accepted view today is that the Babylonians had no awareness of trigonometric functions. See also Abdulrahman A. The Moment of Proof: Mathematical Epiphanies , pp.

A 4,year History , p. God created the integers: Running Press Book Publishers. This proof first appeared after a computer program was set to check Euclidean proofs. Retrieved 19 December Retrieved 27 February

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The Pythagorean theorem states that: "The area of the square built on the hypotenuse of a right triangle is equal to the sum of the squares on the remaining two sides." According to the Pythagorean Theorem, the sum of the areas of the red and yellow squares is equal to the area of the purple square.

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One of the topics that almost every high school geometry student learns about is the Pythagorean Theorem. When asked what the Pythagorean Theorem is, students will often state that a2+b2=c2 where a, b, and c are sides of a right triangle.