Pythagoras was a very significant person in the history of the world. He was a man who was not content with accepting things as they are. He needed explanations and reasons. Pythagoras was an ancient Greek mathematician and philosopher. Pythagoras was responsible for important developments in the history of mathematics, astronomy, and the theory of music.

The thing that Pythagoras is probably the most famous for is the Pythagorean Theorem. The Pythagorean Theorem is used in the field of mathematics and it states the following: the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two other sides. This means that if one makes a square (with all sides equal in length) out of a triangle with a right angle, the areas of the squares made from the two shorter sides, when added together, equal the area of the square made from the long side. Another geometrical discovery made by Pythagoras is that the diagonal of a square is not a rational multiple of its side. The latter discovery proved the existence of irrational numbers and therefore changed the entire Greek mathematical belief that whole numbers and their ratios could account for geometrical properties. He also discovered a formula to find out how many degrees there are in a polygon. Pythagoras came up with (n-2)180°= the number of degrees in a polygon, where (n) represents the number of sides in the polygon. For example, a triangle has three sides, 3-2=1, 1x180=180, which is the total sum of all the inner angles of a triangle. Along with that he found out that the sum of all the outer angles of a polygon is always equal to three hundred sixty degrees. This is true for every single polygon, regardless of the number of the sides.

Pythagoras and his disciples connected music with mathematics and found that intervals between notes can be expressed in numerical terms. Pythagoras noticed that vibrating strings produce harmonious tones when the ratios of the lengths of the strings...

...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 square root of their product. The harmonic mean of two numbers is the arithmetic mean of their reciprocals. Write a program that takes two floating-point numbers as inputs and displays these three means.
7. Write a C++ program to take a depth (in kilometers) inside the earth as input data; compute and display the temperature at that depth in degrees Celsius and Fahrenheit. The relevant formulas are:
Celsius = 10 x (depth) + 20 (Celsius temperature at depth in km)
Farhrenheit = 1.8 x (Celsius) + 32
8. The PythagoreanTheorem states that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. For example, if two sides of a right triangle have lengths 3 and 4, then the hypotenuse must have a length of 5. The integers 3, 4, and 5 together form a Pythagorean triple. There is an infinite number of such triples. Given two positive integers, m and n, where m > n, a Pythagorean triple can be generated by the following formulas:
Side1 = m2- n2
Side2 = 2mn
Hypotenuse = side12 + side22
Group 2
1. In a laboratory, the time of an experiment is measured in seconds. Write a C++ program to enter the time in seconds, convert and print out it as a number of hours, minutes and seconds. Use appropriate format for the output.
2. Write a C++ program to enter an integer number of 4 digits and print it out...

...-------------------------------------------------
PythagoreanTheorem
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 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).
The theorem 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.
These two formulations show two fundamental aspects of this theorem: it is both a statement about areas and about lengths. Tobias Dantzig refers to these as areal and metric interpretations.[2][3] Some proofs of the theorem are based on one interpretation, some upon the other. Thus, Pythagoras' theorem stands with one foot in geometry and the other in algebra, a connection made clear originally byDescartes in his work La Géométrie, and extending today into other branches of mathematics.[4]
The Pythagoreantheorem has been modified to apply outside its original domain. A number...

...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:
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.
The converse of the theorem is also true:
For any three positive numbers a, b, and c such that a2 + b2 = c2, there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides of lengths a and b.
An alternative statement is:
For any triangle with sides a, b, c, if a2 + b2 = c2, then the angle between a and b measures 90°.
This converse also appears in Euclid's Elements (Book I, Proposition 48):
"If in a triangle the square on one of the sides equals the sum of the squares on the remaining two 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...

...THE WIZARD OF OZ 2
The Wizard of Oz
Scarecrow’s Speech on PythagoreanTheorem
The Pythagoreantheorem is one of the earliest theorems known to ancient civilization. The well-known theorem is named after the Greek mathematician and philosopher, Pythagoras. In the Wizard of Oz, after the Scarecrow gets a brain, he states the Pythagoreantheorem. However, he mistakenly says it applies to an isosceles triangle when it applies to a right triangle. He not only says the wrong triangle, he also gets the equation wrong.
The Scarecrow says, “The sum of the square root of two sides of an isosceles triangle is equal to the square root of the third side.” The correct equation for the Pythagoreantheorem is, “The sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.” The isosceles triangle is a triangle with at least two equal sides; it also has two equal angles.
The Pythagoreantheorem is a statement about triangles containing a right angle. A right triangle is a triangle with a ninety-degree angle. With the Pythagoreantheorem, you take a triangle with a right angle and make a square on each of the three sides; the biggest square has the exact same area as the two other squares put together.
A square root of...

...Brief History of the PythagoreanTheorem
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...

...PythagoreanTheorem:
Some False Proofs
Even smart people make mistakes. Some mistakes are getting published and thus live for posterity to learn from. I'll list below some fallacious proofs of the Pythagoreantheorem that I came across. Some times the errors are subtle and involve circular reasoning or fact misinterpretation. On occasion, a glaring error is committed in logic and leaves one wondering how it could have avoided being noticed by the authors and editors.
Proof 1
One such error appears in the proof X of the collection by B. F. Yanney and J. A. Calderhead (Am Math Monthly, v.3, n. 6/7 (1896), 169-171.)
Suppose the theorem true. Then AB² = AC² + BC², BC² = CD² + BD², and AC² = AD² + CD². Combining the three we get
AB² = AD² + 2CD² + BD².
But CD² = AD·BD. Therefore,
AB² = AD² + 2AD·BD + BD².
From which
AB = AD + BD,
which is true. The supposition is true.
Critique
By the same token, assume 1 = 2. Then, by symmetry, 2 = 1. By Euclid's Second Common Notion, we may add the the two identities side by side: 3 = 3. Which is true, but does not make the assumption(1 = 2) even one bit less false.
As we know, falsity implies anything, truth in particular.
Proof 2
This proof is by E. S. Loomis (Am Math Monthly, v. 8, n. 11 (1901), 233.)
Let ABC be a right triangle whose sides are tangent to the circle O. Since CD = CF, BE = BF, and AE = AD = r = radius of circle, it is easily shown...