Just Who Was This Pythagoras, Anyway?
Pythagoras (569-500 B.C.E.) was born on the island of Samos in Greece, and did much traveling through Egypt, learning, among other things, mathematics. Not much more is known of his early years. Pythagoras gained his famous status by founding a group, the Brotherhood of Pythagoreans, which was devoted to the study of mathematics. The group was almost cult-like in that it had symbols, rituals and prayers. In addition, Pythagoras believed that "Number rules the universe,"and the Pythagoreans gave numerical values to many objects and ideas. These numerical values, in turn, were endowed with mystical and spiritual qualities. Legend has it that upon completion of his famous theorem, Pythagoras sacrificed 100 oxen. Although he is credited with the discovery of the famous theorem, it is not possible to tell if Pythagoras is the actual author. The Pythagoreans wrote many geometric proofs, but it is difficult to ascertain who proved what, as the group wanted to keep their findings secret. Unfortunately, this vow of secrecy prevented an important mathematical idea from being made public. The Pythagoreans had discovered irrational numbers! If we take an isosceles right triangle with legs of measure 1, the hypotenuse will measure sqrt 2. But this number cannot be expressed as a length that can be measured with a ruler divided into fractional parts, and that deeply disturbed the Pythagoreans, who believed that "All is number." They called these numbers "alogon," which means "unutterable." So shocked were the Pythagoreans by these numbers, they put to death a member who dared to mention their existence to the public. It would be 200 years later that the Greek mathematician Eudoxus developed a way to deal with these unutterable numbers.

Pythagoras of Samos
Who is Pythagoras?

Born: about 569 BC in Samos, Ionia
Died: about 475 BC
Pythagoras is often described as the first pure...

...In mathematics, the Pythagoreantheorem — 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 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).
Thetheorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:[1]
where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
The Pythagoreantheorem is named after the Greek mathematician Pythagoras (ca. 570 BC—ca. 495 BC), who by tradition is credited with its discovery and proof,[2][3] although it is often argued that knowledge of the theorem predates him. There is evidence that Babylonian mathematicians understood the formula, although there is little surviving evidence that they used it in a mathematical framework.[4][5]
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...

...The PythagoreanTheorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a Greek seaport in Southern Italy. He is credited with many contributions to mathematics although some of them may have actually been the work of his students.
ThePythagoreanTheorem 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. They were devout in their belief that any two lengths were integral multiples of some unit length. Many attempts were made to suppress the knowledge that the square root of 2 is irrational. It is even said that the man who divulged the secret was drowned at sea.
The PythagoreanTheorem is a statement about triangles containing a right angle. The PythagoreanTheorem states that:
"The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides."
Figure 1
According to...

...
PythagoreanTheorem
Pythagoras was born in Samos, Greece around 570 BCE. From there he emigrated to Croton, Italy where most of his most important ideas and theories would develop. Early on, Pythagoras founded a society of disciples where he introduced the idea eternal recurrence into Greek thought, and it was Pythagoras’ ambition to reveal the basis of divine order. This is how Pythagoras came to mathematics, and he saw math as a purifier of the soul, and saw numbers in everything. He was convinced that the divine principles of the universe can be expressed in terms of relationships of numbers.
Over 4000 years ago, the Babylonians and the Chinese already knew that a triangle with the sides of 3, 4, and 5 must be a right triangle. They used this knowledge to construct right angles. Pythagoras studied them a bit closer and found that the two shorter sides of the triangles squared and then added together, equal exactly the square of the longest side. And he proved that this doesn't only work for the special triangles, but for any right triangle. This can be written in the form a^2 + b^2 = c^2, and today this is what is known as the PythagoreanTheorem.
The PythagoreanTheorem was one of the first times in human history that people could calculate a length or distance using only outside information. The train of thought used by Pythagoras was the first time the idea...

... When using the PythagoreanTheorem, the hypotenuse or its length is often labeled with a lower case c. The legs (or their lengths) are often labeled a and b.
Either of the legs can be considered a base and the other leg would be considered the height (or altitude), because the right angle automatically makes them perpendicular. If the lengths of both the legs are known, then by setting one of these sides as the base ( b ) and the other as the height ( h ), the area of the right triangle is very easy to calculate using this formula:
(1/2)
This is intuitively logical because another congruent right triangle can be placed against it so that the hypotenuses are the same line segment, forming a rectangle with sides having length b and width h. The area of the rectangle is b × h, so either one of the congruent right triangles forming it has an area equal to half of that rectangle.
Right triangles can be neither equilateral, acute, nor obtuse triangles. Isosceles right triangles have two 45° angles as well as the 90° angle. All isosceles right triangles are similar since corresponding angles in isosceles right triangles are equal. If another triangle can be divided into two right triangles, then the area of the triangle may be able to be determined from the sum of the two constituent right triangles. Also the Pythagoreantheorem can be used for non right triangles. a2+b2=c2-2c
The side lengths of a...

...PythagoreanTheorem
Diana Lorance
MAT126
Dan Urbanski
March 3, 2013
PythagoreanTheorem
In this paper we are going to look at a problem that can be seen in the “Projects” section on page 620 of the Math in our World text. The problem discusses Pythagorean triples and asks if you can find more Pythagorean triples than the two that are listed which are (3,4, and 5) and (5,12, and 13) (Bluman, 2012). The Pythagoreantheorem states that for any right triangle, the sum of the squares of the length of the sides of the triangle is equal to the square of the length of the side opposite of the right angle (hypotenuse) and can be shown as a² + b² = c² (Bluman, 2012). We will be using a formula to find five more Pythagorean Triples and then verify each of them in the PythagoreanTheorem equation.
The formula that I have decided to illustrate is (2m)² + (m2 - 1)² = (m2 + 1)² where m is any natural number, this formula is attributed to Plato (c. 380 B. C.) (Edenfield, 1997). A natural number is any number starting from one that is not a fraction or negative (MathIsFun, 2011). The triples will be the square roots of each part of the equation. We will test this formula with the natural numbers of 5, 8, and 10. When we use 5 the formula looks like this: (2x5)² + (5² - 1)² = (5² + 1)², 10² + (25 – 1)² = (25 + 1)², 100 + 24² =...

... The assignment for the week is on page 371 number 98. We will be using PythagoreanTheorem, quadratic, zero factor, and compound equation, to solve this equation. We will explain step by step to solve how many paces to reach Castle Rock for Ahmed and Vanessa had to accomplish to meet there goal. Ahmed has half of a treasure map, which indicates that the treasure is buried in the desert 2x + 6 paces from Castle Rock. Vanessa has the other half of the map. Her half indicates that to find the treasure, one must get to Castle Rock, walk x paces to the north, and then walk 2x + 4 paces to the east. If they share their information, then they can find x and save a lot of digging. What is x (Dugoploski, 2012)?
With the information we know that Vanessa must walk North x paces, then 2x + 4 paces to the East. We do not know which direction Ahmed must go, however, we assume that they will end up in the same location. Using a piece of paper, I drew the triangle and it is right triangle. Now I can use the PythagoreanTheorem to help solve for x.
The PythagoreanTheorem is a^2 + b^2 = c^2. Letting a = x, b= 2x+4, and c = 2x + 6.
a^2 + b^2 = c^2 PythagoreanTheorem
x^2 + (2x+4)^2 = (2x+6)^2 Putting the binomials into the PythagoreanTheorem.
x^2 + 4x^2 + 16x +16 = 4x^2 + 24x + 36 Binomials squared. This is 4x^2 on both sides of...

...PYTHAGOREANTHEOREM
More than 4000 years ago, the Babyloneans and the Chinese already knew that a triangle with the sides of 3, 4 and 5 must be a right triangle. They used this knowledge to construct right angles. By dividing a string into twelve equal pieces and then laying it into a triangle so that one side is three, the second side four and the last side five sections long, they could easily construct a right angle.
A Greek scholar named Pythagoras, who lived around 500 BC, was also fascinated by triangles with these special side ratios. He studied them a bit closer and found that the two shorter sides of the triangles squared and then added together, equal exactly the square of the longest side. And he proved that this doesn't only work for the special triangles, but for any right triangle. Today we would write it somehow like this: a2 + b2= c2. In the time of Pythagoras they didn't use letters yet to replace variables. (They weren't introduced until the 16th century by Vieta.) Instead they wrote down everything in words, like this: if you have a right triangle, the squares of the two sides adjacent to the right angle will always be equal to the square of the longest side.
We can't be sure if Pythagoras really was the first person to have found this relationship between the sides of right triangles, since no texts written by him were found. In fact, we can't even prove the guy lived. But the theorem a2 + b2= c2 got his...

...Rock. Vanessa has the other half of the map. Her half indicates that to find the treasure, one must get to Castle Rock, walk x paces to the north, and then walk 2x + 4 paces to the east. If they share their information, then they can find x and save a lot of digging. What is x? (Dugoploski)
With that information we know that Vanessa must walk North x paces, then 2x + 4 paces to the East. We do not know which direction Ahmed must go, but assume that they will end up in the same location. Using a piece of paper I drew the triangle and it is a right triangle. Now I can use the PythagoreanTheorem to help solve for x.
The PythagoreanTheorem is a^2+b^2=c^2. Letting a = x, b = 2x+4, and c = 2x+6
a^2+b^2=c^2 PythagoreanTheorem
x^2+〖(2x+4)〗^2=〖(2x+6)〗^2 Putting the binomials into the PythagoreanTheorem.
x^2+〖4x〗^2+16x+16=〖4x〗^2+24x+36 Binomials squared.
x^2+16x+16=24x+36 Subtract 〖4x〗^2 out
x^2+16x-24x+16-36=0 Subtracted 24x+36 to the other side
x^2-8x-20=0 Now I have a quadratic equation with a zero
factor.
(x-10)(x+2)=0 Coefficient of x^2 is 1. Put a x inside two sets
of parenthesis. Use...

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