PYTHAGORAS of SAMOS

1. PYTHAGORAS of SAMOS circa 580-500 B.C.

2. There are no know first hand accounts of Pythagoras life and work. His history is built on myth and legend which makes it difficult to discern fact and fiction. What is certain is Pythagoras studies the properties of numbers, the relationships between them and patterns they form with his work being responsible for the first golden age of mathematics.[1 pp 6-7] Pythagoras gained his mathematical skills traveling throughout the world, in particular from Egyptian and Babylonian cultures. Both of these cultures went beyond using numbers as a counting tool and could perform calculations to build sophisticated accounting systems, design and engineer complicated structures, reconstruct boundaries that washed away in the annual flooding of the Nile River. Pythagoras realized that Egyptians and Babylonians conducted calculations by blindly applying formulas handed down from generation to generation that would give the correct answer. Nobody had ever questioned them to explore the logic underlying the equations.[1pp 6-7] It is estimated that Pythagoras traveled for about twenty years of his life before returning to Samos. In returning to the island he found a new political structure in place. The once liberal island had fallen under the rule of the very conservative Polycrates, who asked Pythagoras to join his court in an effort to silence him. Pythagoras lived in a cave on a remote part of the island where he started a school with one pupil whose name was also thought to be Pythagoras. He paid this person to be his student at first because the student was very reluctant. As the student became more enthusiastic for knowledge he told the student he could no longer afford to pay him, at that point the student offered to pay for his education. He established the Semicircle of Pythagoras school for a short time but the political climate was not correct in Samos to maintain it.[1 p 8]

3. Pythagoras went to the city of Croton in southern Italy where he found a very wealthy patron by the name of Milo. Milo was more famous than Pythagoras gaining fame as an excellent athlete who had been champion of the Olympic and Pythian Games a record twelve times. Milo sets aside part of his house and provides Pythagoras with enough room to establish a school. Pythagoras founds the Pythagorean Brotherhood who were over 300 aristocrats capable of understanding his teachings. The curriculum consisted of four areas: arithmetica (number theory), harmonia, geometria and astrologia.[1p 8] The manner that Pythagoras organized his school was to divide students into two groups acousti (listeners) and mathematici. The acousti would listen to his lectures for three years behind a curtain then they could be initiated into the inner circle of the mathematici. The law prevented women from attending public meetings, but at least 24 were admitted to his school. He followed the custom of teachers of the east of passing on his lessons by word of mouth. He did not seem to write down any of his teachings. Members of his community were bound not to disclose to outsiders anything taught by Pythagoras or discovered by others in the brotherhood.[2 pp 86-88] At the age of 60 he married one of his students a women named Theno who was in her late teens or early twenties. Reports conflict about her. Some say she was Milo's daughter, some say Pythagoras and they were never married. It was certain she was a good student and mathematician who continued to develop his work in her life.[2 pp 86-88] Pythagoras and his followers believed that numbers were found throughout nature and the world in developing an almost cult like following. What set them apart from others was the philosophy that "knowledge is the greatest purification," and to them knowledge meant mathematics.[2 pp 86-88]

4. Music was a good example of how numbers were contained in nature. The Pythagoreans found that all harmonious notes were found by making the strings on an instrument (such as a lyre) in simple ratio with each other using whole numbers such as 2:1, 3:2 and 4:3.[2 pp 87-88] • Astronomy was also an area that was studied and certain numeric relationships were found. Pythagoras held that the seven known planets (including sun and moon) where carried around the earth by a crystal sphere. It was impossible for gigantic spheres to whirl endlessly without generating noise the system created a celestial harmony, which Pythagoras alone among mortal men could hear. • Pythagoras studied properties of numbers which would be familiar to mathematicians today, such as even and odd numbers, triangular numbers, perfect numbers etc. However to Pythagoras numbers had personalities which we hardly recognize as mathematics today: • Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly. This feeling modern mathematics has deliberately eliminated, but we still find overtones of it in fiction and poetry. Ten was the very best number: it contained in itself the first four integers - one, two, three, and four [1 + 2 + 3 + 4 = 10] - and these written in dot notation formed a perfect triangle. • All ideas could be represented by numbers. 1 - Reason because reason could produce only one consistent body of truth 2 - Man 3 - Woman 4 - Justice because it could be produced by two equal numbers 2+2=4 and 2*2=4 5 - Marriage since it was the union of 2 and 3 6 - Creation

5. Even numbers beyond the first were considered feminine and earthly, odd numbers were masculine and divine. [2 p 89] Today we particularly remember Pythagoras for his famous geometry theorem. Although the theorem, now known as Pythagoras's theorem, was known to the Babylonians 1000 years earlier he may have been the first to prove it. [3] Here is a list of theorems attributed to Pythagoras, or rather more generally to the Pythagoreans. (i) The sum of the angles of a triangle is equal to two right angles. Also the Pythagoreans knew the generalization which states that a polygon with n sides has sum of interior angles 2n-4 right angles and sum of exterior angles equal to four right angles. (ii) The theorem of Pythagoras - for a right angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. We should note here that to Pythagoras the square on the hypotenuse would certainly not be thought of as a number multiplied by itself, but rather as a geometrical square constructed on the side. To say that the sum of two squares is equal to a third square meant that the two squares could be cut up and reassembled to form a square identical to the third square. (iii) Constructing figures of a given area and geometrical algebra. For example they solved equations such as a (a - x) = x2 by geometrical means.

6. (iv) The discovery of irrationals. This is certainly attributed to the Pythagoreans but it does seem unlikely to have been due to Pythagoras himself. This went against Pythagoras's philosophy the all things are numbers, since by a number he meant the ratio of two whole numbers. However, because of his belief that all things are numbers it would be a natural task to try to prove that the hypotenuse of an isosceles right angled triangle had a length corresponding to a number. (v) The five regular solids. It is thought that Pythagoras himself knew how to construct the first three but it is unlikely that he would have known how to construct the other two. (vi) In astronomy Pythagoras taught that the Earth was a sphere at the center of the Universe. He also recognized that the orbit of the Moon was inclined to the equator of the Earth and he was one of the first to realize that Venus as an evening star was the same planet as Venus as a morning star. [3] [1] Simon, Singh. Fermat's Enigma. Walker & Company New York 1997. [2] Burton, David M. The History of Mathematics. McGraw Hill 2003. [3] Web site: MacTutor History. http://turnbull.mcs.st-and.ac.uk/~history(9/2000).

7. One of the subjects the Pythagoreans studied were called figurative numbers. The two types we will examine here are triangular and square numbers. The number of symbols that can be represented by the numbers can arranged in a square as shown below. The first 4 triangular numbers t1=1, t2=3, t3=6 and t4=10                     3 1 6 10 The first 4 square numbers s1=1, s2=4, s3=9 and s4=16 1 4 9 16

8. One of the key questions that was asked was to classify and describe all of the triangular numbers. The Pythagoreans realized the following. and In the case of triangular numbers the Pythagoreans were able to classify them by using an array model type of argument.                                         2=1×2 12=3×4 20=4×5 6=2×3 In general two triangular numbers form a rectangle. To determine the sum of the square numbers 12+22+32+...+n2 was difficult for the methods of algebra known at the time. It was posed by them but is best known as being resolved in the 1st century. It was realized that the sums of higher powers could be constructed from the sums of lower powers. In the case of square numbers the following rectangles were constructed.

9.                                    In general we can account for all the stars in the rectangle by adding up the squares and what is left in each row and equating that with the dimensions of the rectangle. We then apply the formula for

10. The Pythagoreans also noticed some other interesting relationships built on the geometric arrangement of numbers. An example of this would be the sum of odd numbers are square numbers.                          In the further development of mathematics the problem of finding a formula for the sums of powers of integers would become more important. This is especially true when it came to develop the ideas associated with integral calculus. In general mathematicians wanted to be able to find the sum of the kth powers of the first n integers. In other words a formula for 1k+2k+3k+…+nk. A more modern algebraic method was developed using the binomial theorem to expand sums. The idea is that if you know all of the formulas for sum powers less than k you can use those to develop a formula for the sum of the kth powers. … Knowing Find

11. Here is how the sum of cubes (3rd powers) is developed knowing the following: To get the 3rd powers form a telescoping sum with the 4th powers. Do some algebra to simplify the expression you get. Notice that the sum of the cubes are the squares of the triangular numbers.

12. The most famous result in Geometry is the Pythagorean Theorem. Much of modern mathematics developed from this idea. This states in a right triangles the sum of the squares of the legs is equal to the square of the hypotenuse. As mentioned before this was known before the time of Pythagoras, but he was tried to establish this result in terms of proof. The algebraic methods to establish such a result were still primitive. Since the Pythagoreans did not write down their knowledge the method could not be certain. People think it would proceed using a dissection method as follows. The areas of both of the squares can both be represented as the sum of the areas of smaller squares and rectangles. The square on the left is 4 squares and two rectangles so the area would be: a2+b2+2ab The square on the right is one square and a rectangle and the area would be: c2+ 4 (ab/2) If we equate these expressions we get: a2+b2+2ab = c2+ 4 (ab/2) The idea of "canceling" as we do in modern algebra is quite strange. What was done was to deduct the area of the four triangles from both figures giving the result: a2+b2 = c2 Such proofs of addition of areas are so simple they may have been done by other cultures. The Pythagoreans were interested in the study of arithmetic relations. The problem of finding all right triangles whose sides were of integral length is known as the Pythagorean problem.

13. The Pythagorean Theorem is an idea that relates the sides of a right triangle that is thousands of years old. There are some words to identify certain sides of a right triangle that are used. The hypotenuse of a right triangle is the side opposite the right angle. The legs of a right triangle are the sides that form the right angle. The Pythagorean Theorem The Pythagorean Theorem states that the squares of the legs of any right triangle added together is equal to the square of the hypotenuse. Labeling the lengths of the legs as a and b and the length of the hypotenuse as c we get the relation: a2 + b2 = c2. c b a The justification for this (some people refer to this as a proof) has a long history. Many different people have contributed to this and seeking logical justification of the Pythagorean Theorem was one of the driving forces in the development of mathematics. There are books filled with hundreds of arguments (proofs) as to why the Pythagorean Theorem is true. We will give one here. This is very old some people attribute it to Pythagoras himself. The method used here is to compute the area in two different ways and apply some algebra.

14. b a 1. Start with any right triangle with sides of length a, b (legs) and c (hypotenuse). 2. Make 4 identical (congruent) copies of this triangle and place them vertex to vertex to make a square. 3. Each angle of the inside figure is a right angle since: m1 + m2 + 90 = m1 + m2 + m3 90 = m3 4. Compute the area of the big square by squaring the length of the side. 5. Compute the area of the big square by adding the area of 4 triangles and the little square. 6. The two expression both compute the same area. Set them equal and cancel the term 2ab from each side. 90 2 1 3 c a b c b 1 c c a b c a a b 4. Area of Big Square = (a+b)2 = (a+b)(a+b) = a2 + 2ab + b2 5. Area of Big Square = Area of 4 triangles + Area of small square = 4·(½)·a·b + c2 = 2ab + c2 Set equal: a2+2ab+b2 = 2ab + c2 a2 + b2 = c2

15. Pythagorean Problem: Find all positive integer solutions of the Pythagorean equation x2+y2=z2. The ordered triple (x,y,z) satisfying the equation is called a Pythagorean triple. Pythagoras expressed a partial solution of the problem as follows: x=2n+1 y=2n2+2nz=2n2+2n+1 It is not clear if he was the originator of this or not. The method he used to establish this was far ahead of its time. He reasoned you could produce a square from the next smaller number by using an algebraic fact. (2k-1)+(k-1)2=k2 Consider the number 2k-1 as a perfect square. The number 2k-1 is odd and there are infinitely many perfect squares that are odd. So this is possible at least. Let m2=2k-1 and solve for k. k=(m2+1)/2 and k-1=(m2-1)/2 This was not trivial do to the algebra methods of the time. If we substitute we get: m2+((m2-1)/2)2=((m2+1)/2)2 This has solution: x=my=(m2-1)/2 z=(m2+1)/2 Remember that x,y and z will only be integers if m is odd. Let m=2n+1. x=2n+1 y=((2n+1)2-1)/2 =(4n2+4n+1-1)/2 =2n2+2nz=((2n+1)2+1)/2 =(4n2+4n+1+1)/2 =2n2+2n+1