Pythagoras

The Life of Pythagoras

Pythagoras of Samos lived in the 5th century BC and traveled throughout much of the Mediterranean Sea. He was a well-educated child, under the guidance of a number of teachers and philosophers, including Pherekydes, Thales, and Anaximander. Pythagoras was talented in many areas besides mathematics. Among his studies included astronomy, geometry, cosmology, music, and literature such as poetry and the works of Homer. He is often called the first pure mathematician because of his early mathematical interest and wo rks.

Later in life, Pythagoras would start a school at the southeastern part of Italy. The philosophical and religious school included an inner and outer circle of followers, the mathematikoi and the akousmatics respectively. Historians and mathematicians would call them, in general, the Pythagoreans. The Pythagoreans were extremely strict and secretive. As a result, today we do not have concrete evidence of Pythagoras's work. And we are not able differentiate his work from his followers.

It is because of this school that mathematics continued to flourish. The Pythagoreans believed that "All things are number." This stems from the correlation they found between mathematics, music and astronomy. For example, the number seven has special meaning because of the seven wandering stars or planets from which the week is derived. They believed that each number had certain qualities, such as masculine or feminine and incomplete or perfect. Masculine numbers were those that were odd and the feminine numbers were even. It is interesting to note that odd numbers were not only equated with males but also with divinity and harmony. However, the feminine even numbers are the only perfect numbers and the first even number, two, was considered the number of opinion. After joining the first feminine number, two, and the first masculine number, three, we get the number of marriage, five. These are just a few of the many attributes the Pythagoreans would assign to numbers.

Pythagoras' Interest in Perfect Numbers

Pythagoras and his supporters were interested in numbers for their unique properties. One of the important number characteristics they looked at was the idea of an ideal number, a perfect one in nature. Pythagoreans studied these numbers not for the subject of mathematics, but for their mystical properties.

A perfect number is one that equals the sum of its divisors, not including the number itself. In the past, a perfect number was defined in terms of its aliquot parts, instead of divisors. An aliquot part is really the same as a divisor or part. An aliquont part of a number is a proper quotent of the number. They are positive integers.

In order to find a perfect number, one must start with a prime number. An integer, n, is considered prime if and only if its divisiors are one and itself. Such examples are 2, 3, 5, 7, and 11. A prime in the form of [2^(k)-1] is called a Mersenne Prime.

A perfect number can be found by using this therom: If [2^(k)-1] is prime, then [2^(k-1)][2^(k)-1] is even and perfect. In fact, all perfect numbers have this form, and therefore, this is how one can find a perfect number. One can draw a conclusion that there are just as many perfect numbers as there are Mersenne Primes. Since all perfect numbers are in this form, so far in history, there have been no odd perfect numbers found.

Euclid's Elements houses the first recorded results with perfect numbers. In Book IX, Proposition 36 states:

If as many numbers as we please beginning from a unit be set out continuously in

double proportion, until the sum of all becomes a prime, and if the sum multiplied

into the last make some sumber, the product will be perfect.

This is done in a sequence of numbers where each is twice the preceeding number:

1 + 2 + 4 + 8 + 16 + 32 + ... + 2^(k-1) = 2^(k)-1.

For example:

If we take k = 2, 1 + 2 = 3, which is prime!

(the sum)*(the last) = 6. This is a perfect number.

If we take k = 3, 1 + 2 + 4 = 7 which is prime!

(the sum)*(the last) = 28. This is a perfect number.

If we take k = 4, 1 + 2 + 4 + 8 = 15 which is not prime!

If we take k = 5, 1 + 2 + 4 + 8 + 16 = 31 which is prime!

(the sum)*(the last) = 496. This is a perfect number!

We use mathematical induction to find the divisors of a perfect number and the general formula.

Assume that 2^(k)-1 is prime, then show that [2^(k-1)][2^(k)-1] is pefect.

First we show that the sum of all divisors of [2^(k-1)][2^(k)-1] is 2{[2^(k-1)][2^(k)-1]}. Which can be simplified to...

= 2^(+1)[2^(k-1)][2^(k)-1]

=[2^(k)-1]2^(k)

This says that if all the divisors are summed, it will be two times the number itself. When summing all the factors of a perfect number EXCEPT the number itself, you get the number.

The divisors for 2^(k-1) are 1, 2, 2^2, 2^3,2^4,...,2^(k-1). (see below)

1 + 2 + 4 + 8 + 16 + 32 + ... + 2^(k-1) = 2^(k)-1. This says the sum of all divisors of 2^(k-1) is equal to 2^(k)-1.

Next we look at the divisors of 2^(k)-1. Since we assume that it is prime, the only divisors are 1 and 2^(k)-1 itself. We multiply it by 2^(k-1) to get all of its factors. The sum of these is [2^(k)-1][2^(k)-1].

Now we sum all the divisors of the two factors (the sum of all divisors of [2^(k-1)][2^(k)-1] ) and get

2^(k)-1 + [2^(k)-1][2^(k)-1].

This can be simplified. = [ 2^(k)-1][1 + 2^(k)-1]

= [2^(k)-1]2^(k)

We see here that the two are indeed equal!

Below, we use Maple V Release 5, to find prime and perfect numbers.

> with(numtheory):

> perfect :=(2^(x-1))*(2^x-1):

> perf := proc(x)
local p;
p := 2^x -1;
if isprime(p) then (2^(x-1))*p
else `not perfect` fi end:

> perf(6);

> perf(7);

> subs(x=7,perfect);

> perf(8);

> perf(9);

> perf(10);

> perf(11);

> perf(12);

> perf (13);

> seq([i,perf(i)],i=14..23);

Pythagoras' Theorem

Pythagoras was well known for the many mathematical ideas that he developed. One area that he studied was triangle theory. Pythagoras studied and learned the different concepts of triangles and how the angles, sides and different triangles were related. By using sticks to make figures and sand to draw in he was able to develop a popular mathematical equation that is used today in many applications. This equation is known as Pythagoras' Theorem and is mathematically expressed in the equation below.

> A^2+B^2=C^2:

> Pyth :=sqrt(A^2+B^2):

The basic idea of this equation is to solve for unknown lengths of sides of any 90 degree triangle. If any two lengths are known from the triangle, the third side can be determined. This is also very useful in making predictions about triangles when designing a structure or solving a free body diagram in engineering. The variables A, B and C of this equation represent the length of the triangle's sides. See the drawing below.

> tri := plots[polygonplot]([[0,0],[3,0],[0,4]],color=white,
axes=none,scaling=constrained):

> txt := plots[textplot]({[1.5,-.1,`B`],[-.1,2,`A`],[1.5,2.2,`C`]}):

> plots[display]([tri,txt]);

>

Here is a quick example on how to apply Pythagoras' Theorem. Let's say we have a right triangle and each side A and B have the respective lengths 9m and 4m. We can use the equation the solve for the unknown hypotenuse's length.

> evalf(subs(A=9,B=4,Pyth));

or

> evalf(sqrt(9^2+4^2));

This will give us a length of 9.85m.

Now you might say to yourself, how do I know this is the right answer and that this equation works all of the time? Well having to physically make the triangle and measure each side every time you applied the equation would be an impossible burden. So, lucky for us, there are a few proofs developed to ensure the equations consistency.

The first step in proving Pythagoras' Theorem is to draw any triangle and than connect a line at point C perpendicular to side c at D . See the picture below.

> trii := plots[polygonplot]([[0,0],[3,0],[0,3],[-3,0],[0,0],[0,3]],
axes=none,scaling=constrained):

> lable := plots[textplot]({[1.5,-.1,`cx`],[-1,-.1,`x`],[1.5,1.9,`a`],[3.1,0,`B`],[0,3.2,`C`],[0,-.3,`D`],[-3.2,0,`A`],[-2,1.4,`b`],[.2,-.5,`c`]}):

> plots[display]([trii,lable]);

This develops two triangles ACD and DCB that are similar and ACB is a right triangle.

Therefore the below equations hold true.

> (AD/AC)=(AC/AB);

> (x/b)=(b/c);

> b^2=(cx);

Also triangles BCD and ABC are similar equiangularly and this implies the below equations.

> (BD/BC)=(BC/AB);

> (cx)/a=a/c;

> a^2=c^2-(cx);

The combination of the two final equations from the equations above solve the proof.

> a^2+b^2=c^2-cx+cx;

Another popular proof of Pythagoras' Theorem to consider is Euclid's proof. This method is explained at the end of the first book of the Elements. See the drawing below.

> with(geometry):

> point(A,0,0),point(B,1,0),point(C,1,1),point(D,0,1):

> square(EC,[A,B,C,D]):

> point(E,0,1),point(F,.5,1),point(G,.5,1.5),point(H,0,1.5):

> square(SM,[E,F,G,H]):

> rotation(SMM,SM,(1/3)*Pi,'counterclockwise',E):

> point(K,.134,1),point(L,1,1.866),point(M,.134,1.866),point(F,.25,1.433):

> square(LG,[K,C,L,M]):

> rotation(LGG,LG,(1/6)*Pi,'clockwise',C):

> point(Z,1.433,1.75),point(Y,-.433,1.25), point(E,.25,0):

> point(Q,1.434,1.76),point(S,.25,1):

> segment(l,[E,F]),segment(l4,[Y,C]),segment(l5,[F,A]):

> f :=t->triangle(tri3,[C,B,point(W,.25,1.433-.433*t)],color=yellow):

> g :=t->triangle(tri2,[Q,C,point(K,.25*t,1+.433*t)],color=yellow):

> frame := t->draw([EC(color='green',filled=true),f(t)(printtext=false),g(t)(printtext=false),SMM(color='blue',filled=true,printtext=false),LGG(color='plum',filled=true,printtext=false),l(color='black'),l4(color='black'),l5(color='black'),Q,D,S],axes=none,scaling=constrained,printtext=true):

> plots[display]([seq( frame(i/20),i=0..20)],insequence=true,axes=none,scaling=constrained);

Given: Any right triangle formed, DCF, by aranging the squares as shown. Whant to prove: A^2+B^2=C^2.

Proof: The area of triangle FCB is equal to half the area formed by line EF and the left side of the green triangle, EBCS.

This is because they have the same base CB between same parallels, CB and EF. Also triangle DCQ is equal to half the area of the plum square. This, like previously, is because of their same base, CQ and between parallels. Triangle FCB and triangle DCQ are congruent by their similar sides and angles. This implies that the area of SCEB is equal to the area of the plum square, which equals CF^2. Similarly, if we join C and Y, and join F and A we find that the area of DSAE equals the area of the blue square, DF^2. By adding the area of SCEB and the area of DSAE we get (DF)^2+(FC)^2. Since, we proved earlier that the area of the plum square equals the area of SCEB and the area of the blue square equals the area of DSAE, this implies (DC)^2=(DF)^2+(FC)^2, and therefore A^2+B^2=C^2 holds true.

Pythagorean Triples

A Pythagorean triple is a set of three integers a, b, and c that satisfy the equation a^2 + b^2 = c^2. There is a method that produces the infinite number of integers that satisfies this equation. Although triples were known to the Babylonians as early as 1900 B. C., it was not until Pythagoras' time that a method for finding triples was recorded. Pythagoras was the first to receive credit for a formal proof of how these triples are generated.

Pythagoras' method included a geometric representation of the square numbers created by a square array of dots. By adding another row and column of dots, another perfect square is formed. This configuration is called a "gnomon."

Starting with a square figure that represents n^2, the dots in the column and row added around it will be 2n + 1, and the square will then be (n + 1)^2. Now, let 2n +1 = m^2 which gives n = (m^2 - 1)/2 and n + 1 = (m^2 +1)/2. This yields m^2 +((m^2 - 1)/2)^2 = ((m^2 +1)/2)^2 , with m an odd integer.

The question arises whether or not all triples are produced by this equation. Since Pythagoras' formula allows only the use of odd numbers for m, this formula generates all primitive triples (triples with no common factors) whose lowest terms are the odd numbers. In other words, all triples with an odd number as the lowest a or b in the Pythagorean theorem can be generated from this formula which are all the primitive triples.

Plato is credited with the second formula which let m be any natural number. The equation was derivable from Pythagoras' by multiplying by four to get (2m)^2 +(m^2 -1)^2 = (m^2 + 1)^2. This equation, however, did not come about until 160 years after Pythagoras'.

Unlike Pythagoras' formula, the m in Plato's formula can be any natural number. So all triples with lowest term that is an even number greater than 2 are generated. However, this will produce multiples of primitive triples along with primitive triples. When m is even, Plato's formula generates primitive triples, but when m is odd , multiples of 2 of triples are generated.

Below is a spreadsheet of both Pythagoras' and Plato's version of the formula to produce Pythagorean triples. It will demonstrate how effective each formula is.

Using Pythagoras's Formula

Using Platos's Formula