Bust of Pythagoras at theVatican Museum.

Pythagoras of Samos was a famous Greek mathematician and philosopher, born between 580 and 572 BC, and died between 500 and 490 BC. He is known best for the proof of the important Pythagorean theorem, which is about right triangles. He started a group of mathematicians, called the Pythagoreans, who worshiped numbers and lived like monks. He was an influence for Plato. He had a great impact on mathematics, theory of music and astronomy. His theories are still used in mathematics today. He was one of the greatest thinkers of his time.

Pythagoras was born in Samos, a little island off the western coast of Asia Minor. There is not much information about his life. He was said to have had a good childhood. Growing up with two or three brothers, he was well educated. He did not agree with the government and their schooling, so he set up his own cult (little society) of followers under his rule. His followers did not have any personal possessions, and they were all vegetarians. Pythagoras taught them all, and they had to obey strict rules.

Graphical demonstration of thePythagorean theorem

Some say he was the first person to use the term philosophy. Since he worked very closely with his group, the Pythagoreans, it is sometimes hard to tell his works from those of his followers. Religion was important to the Pythagoreans. They swore their oaths by “1+2+3+4” (which equals 10). They also believed that the soul is immortal and goes through a cycle of rebirths until it can become pure. They believed that these souls were in both animal and plant life. Pythagoras himself claimed to remember having lived four different lives. He also told of hearing the voice of a dead friend in the howl of a dog being beaten, and was then attacked by an angry mob.[source?] Pythagoras’ most important belief was that the physical world was mathematical and that numbers were the real reality.

Pythagorean Theorem

See also: Pythagorean trigonometric identity

The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c). In mathematics, the Pythagorean theorem is a relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. 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. The Pythagorean theorem is named after the Greek mathematician Pythagoras (ca. 570 BC—ca. 495 BC), who by tradition is credited with its proof,[2][3] although it is often argued that knowledge of the theorem predates him. There is evidence thatBabylonian mathematicians understood the formula, although there is little surviving evidence that they used it in a mathematical framework.[4][5] Also, Mesopotamian, Indian and Chinese mathematicians have all been known for independently discovering the result, some even providing proofs of special cases.

Verification of Theorem area

2.6 Proof of Pythagorean Theorem (Indian)

The area of the inner square if Figure 4 is C ×C or C2, where the area of the outer square is, (A+B)2 = A2 +B2 + 2AB. On the other hand one may ﬁnd the area of the outer square as follows: The area of the outer square = The area of inner square + The sum of the areas of the four right triangles around the inner square, therefore

A2 +B2 + 2AB = C2 + 41

2AB, or A2 +B2 = C2.

Pythagorean Triplets

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written(a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integerk. A primitive Pythagorean triple is one in which a, b and c are co prime. A right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle. a2 + b2 = c2

Example: The smallest Pythagorean Triple is 3, 4 and 5.

Let’s check it:

32 + 42 = 52

Calculating this becomes:

9 + 16 = 25

And that is true

Triangles

And when you make a triangle with sides a, b and c it will be a right angled triangle (see Pythagoras’ Theorem for more details):

Note:

c is the longest side of the triangle, called the “hypotenuse” a and b are the other two sides

Example: The Pythagorean Triple of 3, 4 and 5 makes a Right Angled Triangle:

Here are some more examples:

5, 12, 13

9, 40, 41

52 + 122 = 132

92 + 402 = 412

25 + 144 = 169

(try it yourself)

And each triangle has a right angle!

List of the First Few

Here is a list of the first few Pythagorean Triples (not including “scaled up” versions mentioned below): (3,4,5)

(5,12,13)

(7,24,25)

(8,15,17)

(9,40,41)

(11,60,61)

(12,35,37)

(13,84,85)

(15,112,113)

(16,63,65)

(17,144,145)

(19,180,181)

(20,21,29)

(20,99,101)

(21,220,221)

(23,264,265)

(24,143,145)

(25,312,313)

(27,364,365)

(28,45,53)

(28,195,197)

(29,420,421)

(31,480,481)

(32,255,257)

(33,56,65)

(33,544,545)

(35,612,613)

(36,77,85)

(36,323,325)

(37,684,685)

… infinitely many more …

Scale Them Up

The simplest way to create further Pythagorean Triples is to scale up a set of triples. Example: scale 3,4,5 by 2 gives 6,8,10

Which also fits the formula a2 + b2 = c2:

62 + 82 = 102

36 + 64 = 100

Applications of Pythagoras Theorem

In this segment we will consider some real life applications to Pythagorean Theorem: The Pythagorean Theorem is a starting place for trigonometry, which leads to methods, for example, for calculating length of a lake. Height of a Building, length of a bridge. Here are some examples Example 2.3 To ﬁnd the length of a lake, we pointed two ﬂags at both ends of the lake, say A and B. Then a person walks to another point C such that the angle ABC is 90.

Then we measure the distance from A to C to be 150m, and the distance from B to C to be 90m. Find the length of the lake. Example 2.4 The following idea is taken from [6]. What is the smallest number of matches needed to form simultaneously, on a plane, two diﬀerent (non-congruent) Pythagorean triangles? The matches represent units of length and must not be broken or split in any way.

Example 2.5 A television screen measures approximately 15 in. high and 19 in. wide. A television is advertised by giving the approximate length of the diagonal of its screen. How should this television be advertised?

Example 2.6 In the right ﬁgure, AD = 3, BC = 5 and CD = 8. The angle ADC and BCD are right angle. The point P is on the line CD. Find the minimum value of AP +BP. Figure 8: Minimum value of AP +BP.

Statement of Pythagoras Theorem

The famous theorem by Pythagoras deﬁnes the relationship between the three sides of a right triangle. Pythagorean Theorem says that in a right triangle, the sum of the squares of the two right-angle sides will always be the same as the square of the hypotenuse (the long side). In symbols: A2 +B2 = C2