The Pythagorean Theorem is Pythagoras' most famous mathematical contribution. According to legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen. The later discovery that the square root of 2 is irrational and therefore, cannot be expressed as a ratio of two integers, greatly troubled Pythagoras and his followers · Topics: Pythagorean theorem, Triangle, Hypotenuse Pages: 3 ( words) Published: March 9, In mathematics, 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-angled triangle, the area of the square whose side is the hypotenuse (the side 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. Thus, from the theorem we have the following relationship: Area of red + Area of yellow = Area of purple There
The Pythagorean Theorem
We use cookies to give you the best experience possible. A limited time offer! Get a custom sample essay written according to your requirements urgent 3h delivery guaranteed. By and large, most anyone who has taken high school algebra has encountered this theorem without every really understanding it.
In words, the Pythagorean Theorem states that the square on the hypotenuse of a right triangle has an area equal to the combined areas of the squares on the other two sides. The Egyptians, Babylonians and Chinese knew of the right triangle long before Pythagoras came along, and the Egyptians had devised a right triangle of rope with twelve evenly spaced knots for measurements. This paper will explore several aspects of the Pythagorean Theorem, from its history to how it found in art, how it relates to music and how it occurs in everyday life, whether people know it or not.
The History Of the Pythagorean Theorem It is not conclusively known that Pythagoras can be solely credited with his famous theorem, but it is known that he was the first to prove why a right triangle behaves the way it does. Born in Samos now in Turkey in about B. Various scenarios of his early life put him in Egypt, Babylonia, and Italy, where he founded his school that survived for a few hundred years, pythagorean theorem essay.
Pythagoras was not interested in solving mathematical problems; he focused on pure mathematics and geometric relationships. Pythagoras and his Brotherhood the Pythagoreans traveled fairly extensively, and it is known that they visited and studied in Egypt.
The Pythagoreans were a mystical group, and Pythagoras himself identified numbers with mysticism. It is possible that Pythagoras saw the knotted triangle in Egypt and went about assigning a numerical value to the geometry of the triangle.
Nothing of his was written down, though, and while the theorem is attributed to him it could well have been one of his brethren or students that invented the theorem; since pythagorean theorem essay in the Pythagorean school was community property, and since his name was the name of the school, we have come to know the famous theorem as it is today.
How do we know that they Pythagorean Theorem was known before Pythagoras? So far, nothing has been shown in terms of an equation in relation to a right triangle before Pythagoras, pythagorean theorem essay, 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 pythagorean theorem essay, and therefore cannot fully translate itself into geometry as it is known pythagorean theorem essay, although in plane geometry the theorem is undeniably valuable.
Or can it? 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: [8]. 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 pythagorean theorem essay 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, pythagorean theorem essay, 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, pythagorean theorem essay, 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: Applying the Pythagorean Theorem to an isosceles triangle with sides 1,1, we can begin to add new triangles, each with a hypotenuse that increases by using the first hypotenuse as the leg of the new triangle:[10]. 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, pythagorean theorem essay.
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 Meanwhich 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 pythagorean theorem essay 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 right triangles, pythagorean theorem essay, square roots were introduced that were not whole numbers.
Below is a pythagorean theorem essay 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:. Bulaevsky, Jacobo. Head, pythagorean theorem essay, Angie. University Of Georgia Department of Mathematics Education. Lindauer, Heather. Martin Gardner. Chapter Regula, Timothy. Weisstein, Eric W, pythagorean theorem essay. Pythagorean theorem essay University, Sanders, Sorry, but copying text is forbidden on this website.
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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.
This proof shows that the area of the square is the same no matter what final shape it takes: [8] 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: III. Evolution Of the Pythagorean Theorem Given that the Pythagoreans shunned anything other than whole numbers, pythagorean theorem essay, an entire universe of geometric applications in terms of describing harmony and unity was lost to them.
Here is one example of the use of irrational numbers plugged into the Pythagorean Pythagorean theorem essay, creating a crude spiral: Applying the Pythagorean Theorem to an isosceles triangle with sides 1,1, we can begin to add new triangles, each with a hypotenuse that increases by using the first hypotenuse as the leg of the new triangle:[10] Here is the second triangle pythagorean theorem essay Continuing, we see the building blocks for the Golden 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, pythagorean theorem essay.
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: Bibliography Bulaevsky, Jacobo.
pythagorean theorem essay Head, Angie. html Lindauer, Heather, pythagorean theorem essay. htm Martin Gardner. htm Regula, pythagorean theorem essay, Timothy.
html Weisstein, Eric W. html [5] U of Minnesota Geometry Center. html [7] Angie Head,University of Georgia Dept. Of Mathematics. html [10] Cathleen V. html [12] ibid, pythagorean theorem essay. Computer Science. Scientific Method. We can write a custom essay According to Your Specific Requirements Order an essay. All rights reserved. send By clicking pythagorean theorem essay, you agree to our terms of service and privacy policy. Copying is only available for logged-in users.
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· The pythagorean theorem says that the area of a square on the hypotenuse is equal to the sum of the areas of the squares on the legs. If the lengths of the legs are A and B, and the length of the hypotenuse is C, then, a^2 +b^2=c2. There are many different proofs of The Pythagorean Theorem is one of the earliest known theorems to ancient civilizations. It was named after Pythagoras, a Greek mathematician and philosopher. The theorem bears his name although we have evidence that the Babylonians knew this relationship some years earlier Summary In mathematics, the Pythagorean Theorem is a relation in Euclidean geometry among the three sides of a right triangle. The theorem is named after the Greek mathematician Pythagoras who lived in the 6th century B.C. "In any right angle triangle, the area of
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