Fermat's Last Theorem

Sunday, May 14, 2006

Fundamental Theorem of Algebra: Preliminaries

Cyclotomic integers allow Fermat's Last Theorem to be factored in the following way:

xⁿ + yⁿ = (x + y)(x + αy)(x + α²y)*...*(x + α^n-1y)

In the above factorization, n is an odd prime and α is a root of unity where αⁿ = 1 and αⁱ ≠ 1 for all 1 ≤ i ≤ n-1.

To establish this refactorization, I will be using the Fundamental Theorem of Algebra.

The main idea behind the Fundamental Theorem of Algebra is that for any given polynomial of a single variable of order n, there are at least n zeros to this equation in the complex domain.

This is an easy proof to state but difficult to prove. Rene Descartes and Jean le Rond d'Alembert knew about this result but it was not until Carl Friedrich Gauss that the fundamental theorem was rigorously proved. d'Alembert thought that the existence of a minimum point for a complex equation was obvious and needed no proof. I do not find this point so obvious so I begin with its proof.

The details in today's proof are taken from B. N. Delone's article on Algebra from Mathematics: Its Content, Methods, and Meaning by A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev and translated by S. H. Gould.

Theorem 1: Bolzano-Weierstrass Theorem

If a rectangle contains an infinite sequence of points (z₁, z₂, ..., z_n, ... in its interior, then there exists a point z₀ such that in any arbitrarily small neighborhood of z₀, there are infinitely many points of the sequence z₁, z₂, ..., z_n, ...

Proof:

(1) Let R₁ be a rectanlge that contains an infinite sequence of points.

(2) We divide it up into 4 equal parts using two lines parallel to each of its sides.

(3) At least one of these four parts contains infinitely many points, let us label it R₂

(4) We now divide up R₂ into 4 equal parts using two lines parallel to each of its sides.

(5) One of these four parts contains infinitely many points and we label it R₃

(6) In this way, we are able to generate a sequence of nested rectangles which we can label R₁, R₂, ..., R_n

(7) We can think of each side of this rectangle representing two nested intervals that exist on the x and y axises.

(8) Using Lemma 1 here, we there exists a point on the x-axis and a point on the y-axis that each of these nested intervals have in common.

(9) But this means that there is a point within the nested rectangles such that any arbitrary small neighborhood contains an infinity of points.

QED

Theorem 2: There exists a minimum point for c₀xⁿ + c₁x^n-1 + ... + c_n.

Let f(x) = c₀xⁿ + c₁x^n-1 + ... + c_n.

There exists a value x₀ such that w₀ = c₀(x₀)ⁿ + c₁(x₀)^n-1 + ... + c_n where absolute(w₀) is the minimum value.

Proof:

(1) Let g = absolute(f(0))

(2) Let G be a number greater than g.

(3) Let R be a number such that if absolute(x) > R, then absolute(f(x)) is greater than G

(4) Now, if f(0)=0, then x₀=0 and w₀=0 so we can assume going forward that f(0) ≠ 0

(5) If f(0) is greater than 0 and all f(x) ≥ g, then x₀=0 and w₀=g so we can assume that there exists at least 1 point x' such that absolute(f(x')) is less than g.

(6) Based on a nonzero g, we can set up the follow sequence:

0, g/n, 2g/n, ..., ng/n = g

(7) We can find a value i, c_n such that c_n = (i/n)g and all values absolute(f(x)) ≥ c_n.

(8) From i, we can also find a value c_n' such that c_n' = [(i+1)/n]g and there exists at least one value x' such that absolute(f(x')) is less than c_n'

(9) We can find c_n,i,c_n' regardless of the value of n so we can let n increase to infinity.

(10) Now for all values of n, we can assume that absolute(x) ≤ R since if absolute(x) is greater than R, then absolute(f(x)) is greater than G and therefore greater than g. For purposes here, let's call these values x_n

(11) So, we only need to consider the points x_n that lie inside a rectangle of sides 2R and with its center at the origin.

(12) By Theorem 1 above, there exists a point z₀ such that every neighborhood of z₀ contains infinitely many points of the sequence z₁, z₂, ..., z_n. Let us call this point x₀

(13) For any point x, we have:

absolute(f(x)) is greater than c_n = c_n' - g/n which is greater than absolute(f(x_n)) - g/n = absolute(f(x₀)) + absolute(f(x_n)) - absolute(f(x₀)) - g/n.

(14) This inequality is true for all values of n so as n increases toward infinity, we see that difference absolute(f(x_n)) - absolute(f(x₀)) becomes arbitrarily small in absolute value with g/n.

(15) Consequently, all absolute(f(x_n)) ≥ absolute(f(x₀)) so x₀ is the minimum point.

QED

References

A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent'ev, Mathematics: Its Content, Methods, and Meaning.

Saturday, May 13, 2006

Cyclotomic Integers: Factoring Fermat's Last Theorem

Today's blog continues the discussion of Kummer's proof of Fermat's Last Theorem for regular primes. If you would like to review the historical context for this proof, start here.

The major reason why cyclotomic integers are interesting in relation to Fermat's Last Theorem is because they enable us to factor Fermat's Last Theorem in the following way:

zⁿ = xⁿ + yⁿ = (x + y)(x + αy)(x + α²y) .... (x + α^n-1y)

Below I will show how I can derive this factoring using the Fundamental Theorem of Algebra.

Lemma 1: Let α be a primitive root of unity such that n is an odd prime and αⁿ = 1, and let x,y,z be integers such that xⁿ + yⁿ = zⁿ, then:

zⁿ = xⁿ + yⁿ = (x + y)(x + αy)(x + α²y) .... (x + α^n-1y)

Proof:

(1) We know that xⁿ - 1 has n root from the Fundamental Theorem of Algebra.

(2) We also note that for all αⁱ where 0 ≤ i ≤ n-1, we have (αⁱ)ⁿ = 1.

NOTE: αⁿ = 1 so it is really the same as α⁰.

(3) Based on #2, the Fundamental Theorem of Algebra gives us:

xⁿ - 1 = (x - 1)*(x - α)*(x - α²)*...*(x - α^n-1)

QED

Theorem 1: if n is odd, then zⁿ = xⁿ + yⁿ = (x + y)(x + αy)(x + α²y) .... (x + α^n-1y)

Proof:

(1) aⁿ - 1 = (a - 1)*(a - α)*(a - α²)*...*(a - α^n-1) [From Lemma 1 above]

(2) Since a can be any value, let a = -x/y so that:

(-x/y)ⁿ - 1 = [(-x/y) - 1]*[(-x/y) - α]*...*[(-x/y) - α^n-1] = -(x)ⁿ/yⁿ - 1

(3) If we multiply (-y)ⁿ=-(yⁿ) to both sides, we get:

xⁿ + yⁿ = (x + y)*(x + yα)*...*(x + α^n-1y)

QED

Cyclotomic Integers: Division Algorithm

Friday, May 12, 2006

Cyclotomic Integers: Units and Primes

Tuesday, May 09, 2006

Basic Properties of Cyclotomic Integers

Today's blog continues the discussion of Kummer's proof of Fermat's Last Theorem for regular primes. If you would like to review the historical context for this proof, start here.

Today, I will review the basic properties of cyclotomic integers. Today's content comes directly from Chapter 4 of Harold M. Edwards' Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory.

1. Notation

For Kummer's notation, he used λ to represent the odd prime number and α to represent the root of unity so that we have:

Definition 1:
α^λ = 1

2. Standard Form of Cyclotomic Integers

Lemma 1:
If a₀, a₁, ... a_λ-1 are integers, then all cyclotomic integers for a given value of λ can be represented in the following form:

a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1

Proof:

(1) Let's assume that we have cyclotomic integer = a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1 + a_λα^λ

(2) By definition 1 above, α^λ = 1

(3) So that we have:

(a₀ + a_λ) + a₁α + a₂α² + ... + a_λ-1α^λ-1

(4) We can do the same thing for any power of αⁱ where i ≥ λ

(5) So we can conclude that all values can be reduced to the form required.

QED

Lemma 2: For any given value of λ, 1 + α+α² + ... + α^λ-1 = 0

Proof:

(1) Since α^λ = 1, we have:

1 + α+α² + ... + α^λ-1 =α^λ + α+α² + ... + α^λ-1 =

= α(α^λ-1 + 1 + α+α² + ... + α^λ-2)

(2) Now, we know that α ≠ 0 since 0^λ = 0 which contradicts with definition 1.

(3) We also know that α ≠ 1 since α is a λth root of unity [using Euler's Identity, see here], we know that α = e^2iπ/λ

(4) So, therefore, 1 + α+α² + ... + α^λ-1= 0

QED

Corollary 2.1: for any given integer c, a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1= (a₀ + c) + (a₁ + c)α + (a₂ + c)α² + ... + (a_λ-1 + c)α^λ-1.

Proof:

(1) 1 + α + α² + ... + α^λ-1= 0 [From Lemma 2 above]

(2) c + cα + cα² + ... + cα^λ-1= c*0 = 0

(3) So that:

a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1= a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1 + 0 =

= a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1 + c + cα + cα² + ... + cα^λ-1=

= (a₀ + c) + (a₁ + c)α + (a₂ + c)α² + ... + (a_λ-1 + c)α^λ-1.

QED

3. Conjugates

Since each cyclotomic value can be represented as:

a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1

Kummer used the following shorthand to represent a cyclotomic integer:

f(α), g(α), φ(α), F(α), etc.

One important point that we find is that if f(α) = g(α), then f(α²) = g(α²) and so on up until λ - 1.

Lemma 2.5: Conjugates preserve relations between equations

That is, if f(α) = g(α), then f(αⁱ) = g(αⁱ) where i is a positive number less than λ, αⁱ ≠ 1 and α^λ = 1.

Proof:

(1) Let f(α) = a₀ + a₁α + ... + a_λ-1α^λ-1

(2) For any value f(αⁱ) we see that:

f(αⁱ) = a₀ + a₁αⁱ + ... + a_λ-1α^i*(λ-1)

(3) In step #1, let j be the possible values ranging from 1 to λ -1. Combining this with step #2, we get:

f(αⁱ) = ∑ a_jα^j*i

(4) To prove this lemma, we need to show each element j*i is congruent to a unique value of i modulo λ

In other words, we are trying to prove that each element of the f(αⁱ) is distinct.

(5) This turns out to be the case from Lemma 1 here.

QED

For this reason, we say that f(α), f(α²), ...., and f(α^λ-1) are conjugates of each other.

4. Norm

Definition 2: Norm of a cyclotomic integer f(α)

Nf(α) = f(α)*f(α²)*...*f(α^λ-1)

I will now use this definition in the following proofs.

Lemma 3: Nf(α) = Nf(αⁱ) for all values of i between 1 and λ-1.

Proof:

(1) Nf(αⁱ) = f(αⁱ)*f(α^2*i)*...*f(α^i(λ-1))

(2) Now, each value i, 2*i, 3*i, ... (λ-1)*i maps to a distinct value of 1,2,3,...,(λ-1) modulo λ (see Lemma 1 here)

(3) So in each case, i,2*i, etc. maps to a₁*λ+1, a₂*λ+2, etc.

(4) So we get Nf(αⁱ) = f(α^a₀*λ+1)*f(α^a₁*λ+2)*...*f(α^{a_λ-1*λ+λ-1}) where a_i is a nonnegative integer.

(5) Since α^n*λ=1, we get:

Nf(αⁱ) = f(α)*f(α²)*...*f(α^λ-1)

QED

Lemma 4: α^j = α^λ-j

Proof:

(1) From roots of unity and Euler's Formula, we know that:

α = e^(i2π/λ) = cos(2π/λ) + isin(2π/λ)

(2) We also know that the complex conjugate of a + bi is a - bi, so the complex conjugate for α is:

α = cos(2π/λ) - isin(2π/λ)

(3) Likewise, we know that the complex conjugate for α^j is:

α^j = cos(2jπ/λ) - isin(2jπ/λ)

(4) Using Euler's Formula, we see that:

e^-2jπ/λ = cos(-2jπ/λ) + isin(-2jπ/λ)

(5) Since cos(-x) = cos(x) and sin(-x) = -sin(x) [see here], we can use (#4) to get:

e^-2jπ/λ = cos(2jπ/λ) - isin(2jπ/λ)

which is from #3, the complex conjugate for α^j

(6) Now, e^-2jπ/λ = (e^2π/λ)^-j =

= α^-j = α^-j*α^λ =

= α^{λ - j}

QED

Corollary 4.1: f(α^j) = f(α^λ-j)

Proof:

(1) From Lemma 1, we have:

f(α) = a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1

(2) From this,

f(α^j) = a₀ + a₁α^j + a₂α^2*j + ... a_λ-1α^j*(λ-1)

(3) Now, from Lemma 4, we know that:

f(α^j) = a₀ + a₁α^λ-j + a₂α^{λ - 2*j} + ... a_λ-1α^{λ - j*(λ-1)}

(4) And, we know that:

f(α^λ-j) = a₀ + a₁α^λ-j + a₂α^(λ-j)*2 + ... a_λ-1α^{(λ - j)*(λ - 1)}

(5) Now,

n*λ - j*n ≡ λ - j*n (mod λ) [See here if you need a review of modular arithmetic]

(6) So that we see that step #3 and step #4 are equal so that:

f(α^j) = f(α^λ-j)

QED

Corollary 4.2: f(α^j)*f(α^λ-j) is a nonnegative real number

Proof:

(1) f(α^j) * f(α^λ-j) = f(α^j)* f(α^j) [From Corollary 4.1 above]

(2) So that:
f(α^j) * f(α^λ-j) = (a₀ + a₁α^j + ... + a_λ-1α^j*(λ-1))(a₀ + a₁α^j + ... + a_λ-1α^j*(λ-1)) =

= (a₀)² + (a₁)²(α^j*α^j) + ... + (a_λ-1)²*α^j*(λ-1)*α^j*(λ-1))

(3) Since each α*α is a nonnegative number, the conclusion follows.

QED

Lemma 5: For any cyclotomic integer f(α), its norm is a nonnegative rational integer.

Proof:

(1) Using Lemma 1 above, we know that:

Nf(α) = a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1

(2) By Lemma 3 above, we can substitute any conjugate α^j and get the same norm so that:

Nf(α^j) = Nf(α)

(3) But by changing to a conjugate, we keep the same coefficients but get the following:

Nf(α^j) = a₀ + a₁α^j + a₂α^j*2 + ... + a_λ-1α^(λ-1)*j

(4) Combining the two equations gets us:

a₀ + a₁α^j + a₂α^j*2 + ... + a_λ-1α^(λ-1)*j= a₀ + a₁α + a₂α² + ... + a_λ-1αλ-1

(5) Subtracting one from the other gives us:

a₀ - a₀ + (a₁ - a_j)α^j + ... = 0

(6) Since we know that each of these j,2*j,...,(λ-1)*j matches up with a value 1,2,...,λ-1, we know that:

a₁ = a_j

(7) Further, since j can be any value from 2 thru λ-1, we can conclude the following:

a₁ = a₂ = a₃ = ... = a_λ-1

(8) So that:
Nf(α) = a₀ + a₁(α + α² + ... + α^λ-1)

(9) From Lemma 2, we know that:

1 + α+α² + ... + α^λ-1 = 0

so that:

α+α² + ... + α^λ-1= -1

(10) So, we apply (#9) to (#8) to give us:

Nf(α) = a₀ - a₁
(11) We know that it is nonnegative since:

Nf(α) = [f(α¹)*f(α^λ-1)]*[f(α²)*f(α^λ-2)]*...

(12) From Corollary 4.2 above, we know that multiplication of (λ-1)/2 pairs of nonnegative values will result in a nonnegative value.

QED

Lemma 6: f(α)g(α) = h(α) → Nf(α)*Ng(α) = Nh(α)

Proof:

(1) Let f(α)g(α) = h(α)

(2) By Definition 2 above:

Nf(α) = f(α)*f(α²)*...*f(α^λ-1)

Ng(α) = g(α)*g(α²)*...*g(α^λ-1)

Nh(α) = h(α)*h(α²)*...*h(α^λ-1)

(3) Using step #1 gives us:

Nh(α) = f(α)*g(α)*f(α²)*g(α²)*...*f(α^λ-1)*g(α^λ-1) =

= f(α)*f(α²)*...*f(α^λ-1) *g(α)*g(α²)*...*g(α^λ-1) =

= Nf(α)*Ng(α)

QED

Sunday, May 07, 2006

Fermat's Last Theorem: Proof for regular primes

One of the highpoints of the 19th century mathematics is Kummer's proof of Fermat's Last Theorem for regular primes.

Kummer's theory of ideal numbers is one of the foundations of algebraic number theory. In future blogs, I will talk about some of the other very important proofs that came out at this time (impossibility of a general method for quintic equations, transcendence of π, and the fundamental theorem of algebra) and show how Dedekind reinterpreted many of these developments into the modern concepts of ideals, rings, groups, and fields.

Kummer's proof comes down to three major points.

(A) For certain primes (which Kummer called "regular primes"), cyclotomic integers can be said to have a form of unique factorization. [See here for discussion on ideal numbers and how they "save" unique factorization for cyclotomic integers]

(B) For a regular prime λ, there is no solution to x^λ + y^λ = z^λ where x,y,z are pairwise relatively prime all prime to λ

(C) For a regular prime λ, there is no solution to x^λ + y^λ = z^λ where x,y, z are pairwise relatively prime and where λ divides z.

For the full proof, go here.

References

Harold M. Edwards, Fermat's Last Theorem

Sunday, April 30, 2006

Eudoxus of Cnidas

All original works of Eudoxus of Cnidas have been lost. All that we know is found in Euclid's Elements and referred to by commentators on Eudoxus. From this information, Eudoxus is now viewed as the second greatest mathematician of the ancient world after Archimedes.

I have spoken about the formal definition of the limit in a previous blog. Today's blog is part of the effort to review the background and details that make up, Euler's Identity, one of the most astounding mathematical formulas of all time:

e^iπ + 1 = 0

For those who are not familiar with Euler's Identity, you may wish to start here.

Eudoxus was born around 408 B.C. in Cnidas which is resides in modern Turkey. Not much is known about Eudoxus's early life. We do believe that he studied in Italy with Archytas who was one of the followers of Pythagorus. Archytas was fascinated by the problem of duplicating the cube (the problem of finding a compass/ruler construction for a cube root). It is also probable that Eudoxus studied Pythagorean number theory and music theory.

Later, Eudoxus studied medicine in Sicily under Philiston and later in Athens under Theomedan. While in Athens, he attended lectures by Plato and his followers at the newly created Academy.

Professor Heath writes that Eudoxus at this time was very poor (Heath quoted in MacTutor):
... so poor was he that he took up his abode at the Piraeus and trudged to Athens and back on foot each day.

Later, he went to Egypt to study astronomy with the priests at Heliopolis. He stayed in Egypt for over a year and after this time, he decided to open a school of his own in Cyzikus which is in Asia Minor near the shore the Sea of Marmara. The school became a big success.

There is evidence that after his school became established, Eudoxus went with some of his followers to visit Athens. At this time, there seems to have been a rivalry between Plato and Eudoxus. Eudoxus was critical of Plato's mathematical abilities and Plato seemed jealous of the popularity of Eudoxus's school.

Eudoxus later returned to Cnidas where he came one of the town leaders. He continued to write texts on theology, astronomy, and geography and built an observatory. Hipparchus refers to astronomic data captured by Eudoxus at this observatory that helped explain the rising and setting of constellations.

Eudoxus's contributions to mathematics are legendary. He proposed a theory of proportion to deal with irrational numbers which is found in Euclid's Elements, Book V. All rational numbers can be compared by finding a common unit. The problem is that with irrational numbers there is no such common unit. Eudoxus instead proposed the use of ranges so that a comparison occurs by applying a common multiple and then comparing the resultant value.

In modern terms, for real numbers a,b,c,d: a/b = c/d if for every pair of integers m, n:

(a) if ma is less than nb, then mc is less than nd.

(b) if ma = nb, then mc = nd

(c) if ma is greater than nb, then mc is greater than nd.

This work on proportions was very important in the modern definition of real numbers (see here).

In addition to his theory of proportion, he proposed what is today termed the Method of Exhaustion. This generalized the work done by Antiphon on estimating the area of a circle by using inscribed polygons. Eudoxus's theory is found in Euclid's Elements, Book X, Proposition 1. In Euclid, Eudoxus's Method of Exhaustion is used to show that the area of two circles is in proportion to the square of their diameters.

Eudoxus also offered a solution to the duplication of the cube using curved lines. Eratosthenes who wrote a history of the problem talks about Eudoxus's solution. Unfortunately, the exact details of his solution have been lost. Paul Tannery was able to come up with a proof that is consistent with information known about Eudoxus. The proof uses an algebraic curve known as a kampyle curve. Today, this solution is known as the kampyile of Eudoxus.

In his own lifetime, Eudoxus was most famous for his planetary theory where he was able to emulate the planetary motions using 27 spheres that turn upon each other. These spheres of Eudoxus are mentioned by Aristotle in his famous work Metaphysics.

Although his book Tour of the Earth has been lost, there still exist over 100 quotes in secondary sources. The book surveys all the different peoples known to Eudoxus and reviews their history, their political systems, and their culture. He wrote deeply about Egyptian society and about the Pythagoreans in Italy.

References

Eudoxus, MacTutor Biography
David Joyce, Euclid's Elements

Tuesday, April 18, 2006

Claudius Ptolemy

Claudius Ptolemy was born around 85AD in Egypt. He lived during a time of Hellenized Egypt but very little is known of his personal life.

His reputation rests on his unprecedented contributions to astronomy and geography and to the controversy that these works later caused as astronomers reacted to the famous work by Copernicus. In Ptolemy's universe, the earth sits at the center.

Ptolemy lived in Alexandria, Egypt. We know very little of his education. He mentions the astronomic data of Theon the mathematician who was probably Theon of Smyrna, one of his teachers. Many of his early works are dedicated to Syrus who may have also been one of his teachers.

Many of Ptolemy's major works still exist. His magnus opus is the Almagest which is Arabic for "the greatest". Its original name was the Mathematical Compilations. The Greeks soon started calling it The Greatest Compilations. Later on, the Arabic version became the basis of the Latin translation which is why Ptolemy's work is largely known as the Almagest.

The Almagest was one of Ptolemy's earliest works. It presents the Greek theory about the motions of the Sun, the moon, and the planets. This stood as the standard text on astronomy until Copernicus released his heliocentric theory in 1543.

It is a very ambitious work. Ptolemy writes (taken from MacTutor Biography):
We shall try to note down everything which we think we have discovered up to the present time; we shall do this as concisely as possible and in a manner which can be followed by those who have already made some progress in the field. For the sake of completeness in our treatment we shall set out everything useful for the theory of the heavens in the proper order, but to avoid undue length we shall merely recount what has been adequately established by the ancients. However, those topics which have not been dealt with by our predecessors at all, or not as usefully as they might have been, will be discussed at length to the best of our ability.
His geocentric astronomic model was based on Aristotle even though Aristarchus of Samos had argued for a heliocentric model. After presenting this model, he then introduces trigonometric models to predict the motions of the orbits. He presents tables of chords (which are similar to the sine function) and uses the concept of "epicycles" (circles-within-circles) to model the planetary movements. It is his theory of planets which is his greatest contribution. In his day, there were only 5 planets recognized and he presented detailed mathematical models for explaining their motions.

Although Hipparchus gets credit for much of the theory that Ptolemy presents, it is Ptolemy who was able to apply that theory to the planetary data.

The historian Toomer wrote (from MacTutor web site):
As a didactic work the "Almagest" is a masterpiece of clarity and method, superior to any ancient scientific textbook and with few peers from any period. But it is much more than that. Far from being a mere 'systemisation' of earlier Greek astronomy, as it is sometimes described, it is in many respects an original work.
There have been many who questioned whether Ptolemy deserves his reputation for being the greatest astronomer of his day. Tycho Brahe noticed that all of Ptolemy's star catalogue data was consistently off by 1 degree longitude. Although Ptolemy claimed that he had gathered the data. Brahe believed that Ptolemy had copied the data from another source. Some scholars have argued that Hipparchus is the true genius and Ptolemy merely copied Hipparchus's work. It is very hard to assess the truth of any of these claims since almost of all of Hipparchus's work is lost.

Regardless of these criticisms, it is undeniable that Ptolemy played the lead role in the geocentric theory of the solar system that ruled the day until the publication of Copernicus's famous work. Hipparchus may be the father of trigonometry but it is Ptolemy who became the lead voice in the application of trigonometry to the motions of the planets, sun, and moon.

References

Biography of Ptolemy, MacTutor

Sunday, April 16, 2006

Hipparchus of Rhodes

Hipparchus is considered by many to be the father of trigonometry because he was the first to organize measurements in relation to angles in a trigonometric table. Unfortunately, all but one of his math writings have been lost and the one text that remains is a minor work. Most of what we know today comes from the Greek astronomer Ptolemy.

Hipparchus was born around 190 B.C. in the town of Nicaea and died around 120 B.C. in Rhodes. He may have studied geometry in Alexandria. Unfortunately, not much more is known about his life.

His influence on astronomy and mathematics has been significant. He produced the first trigonometric table based on chords (the table lists the length of the chord that corresponds to a given angle). He discovered the precession of the equinoxes and accurately determined the length of a year to within 6.5 minutes. It is believed that Hipparchus created the first star catalogue which may have consisted of 650 stars. It is believed that he created this around 134 B.C. when a new star was said to have burst across the sky.

He was able to calculate the distance of the moon from the earth and developed a theory of the moon's motions based on epicycles. He also used epicycles to model the Sun's motions. He was perhaps the first to be able to predict solar and lunar eclipses. Hipparchus may have invented the astrolabe and used an equatorial ring to observe the sun's equinoxes.

Most of what is known about Hipparchus is found in Ptolemy's magnus opus Almagest. It was not Ptolemy's intentions to preserve the memory of Hipparchus. In fact, Ptolemy seems to assume that his reader has access to Hipparchus's original works.

References

Hipparchus, Wikipedia
Hipparchus, MacTutor

Monday, April 03, 2006

Archimedes

Archimedes was born around 287 B.C. in Syracuse. His father was an astronomer and it is believed that Archimedes was a relative of the King of Syracuse.

As a young man, Archimedes traveled to Egypt for it is said that while there, he invented what is today known as the Archimedes' screw.

It is believed that he studied with the students of Euclid in Alexandria because he mastered the ideas of Euclid and he references one Alexandrian mathematician, Conon of Samos, as a close friend. In his work On Spirals, he mentions a practice in Alexandria where mathematicians would give results but would not include proofs for those results. Archimedes disapproved of this practice and on one occasion sent false results to see how the
Alexandrian mathematicians responded.

Throughout his life, Archimedes made numerous engineering inventions. Despite his rising fame, Archimedes was most proud of his work in pure mathematics. Plutarch writes:
Archimedes possessed so high a spirit, so profound a soul, and such treasures of scientific knowledge, that though these inventions had now obtained him the renown of more than human sagacity, he yet would not deign to leave behind him any commentary or writing on such subjects; but, repudiating as sordid and ignoble the whole trade of engineering, and every sort of art that lends itself to mere use and profit, he placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life; studies, the superiority of which to all others is unquestioned, and in which the only doubt can be whether the beauty and grandeur of the subjects examined, of the precision and cogency of the methods and means of proof, most deserve our admiration.
Archimedes is responsible for contributing the term "eureka" into the English language. Eureka is Greek for "I found it." Archimedes was working on a problem for King Heiron when he figured out the solution while taking a bath. He was so excited, that he ran around Syracuse naked shouting "Eureka! Eureka!" which means "I have found it! I have found it!".

In his day, Archimedes was very well known. There are many references to him from contemporary writers. Interestingly, he was not famous for his mathematics but for his many mechanical devices that could be used in warfare. Plutarch writes about the use of these war machines when Marcellus led the Roman army against Syracuse in 212 BC:

... when Archimedes began to ply his engines, he at once shot against the land forces all sorts of missile weapons, and immense masses of stone that came down with incredible noise and violence; against which no man could stand; for they knocked down those upon whom they fell in heaps, breaking all their ranks and files. In the meantime huge poles thrust out from the walls over the ships and sunk some by great weights which they let down from on high upon them; others they lifted up into the air by an iron hand or beak like a crane's beak and, when they had drawn them up by the prow, and set them on end upon the poop, they plunged them to the bottom of the sea; or else the ships, drawn by engines within, and whirled about, were dashed against steep rocks that stood jutting out under the walls, with great destruction of the soldiers that were aboard them. A ship was frequently lifted up to a great height in the air (a dreadful thing to behold), and was rolled to and fro, and kept swinging, until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall.

Ultimately, the Romans succeeded in taking Syracuse and Archimedes was killed during the invasion. Plutarch writes about three versions of the death of Archimedes. The most famous version is that he didn't notice the Roman soldier because he was intense on a math problem. In another version, he is confronted by the same soldier who tells him that he has come to kill him. Archimedes pleads for his life by saying that he has many mathematical works that have not yet been finished. The soldier is unmoved by this plea and kills him. In the third version, Archimedes carries his equipment to show Marcellus what can be done. The soldiers stop him and slay him because they believe that Archimedes has gold hidden inside the equipment.

Cicero writes that when Marcellus returned to Rome, he brought with him two inventions of Archimedes: a star globe, that mapped the sky onto a sphere, and the other was an orrery, an object that predicts the motions of the sun, moon, and planets.

Today, historians agree that Archimedes, Newton, and Gauss are the greatest mathematicians who have ever lived.

He made great advances in the method of integration that enabled him to find areas, volumes, and surface areas of geometric forms. He gave one of the first accurate approximations of π and found a method for estimating square roots. He invented a method for representing very large numbers. He invented the compound pulley, the law of the lever, he found a method for estimating the weight of an object immersed in a liquid, he came up with the concept of center of gravity. His favorite theorem was that the volume of a sphere is two-thirds the volume of a circumscribed cylinder.

Heath writes about Archimedes:
The treatises are, without exception, monuments of mathematical exposition; the gradual revelation of the plan of attack, the masterly ordering of the propositions, the stern elimination of everything not immediately relevant to the purpose, the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.
References

Archimedes, Wikipedia
Archimedes, MacTutor
T. L. Heath, A history of Greek mathematics II (Oxford, 1931).
Plutarch, Marcellus, MIT's on-line version

Saturday, April 01, 2006

Euclid of Alexandria

Euclid of Alexandria is perhaps the greatest math teacher of all time. His textbook The Elements is the perhaps the most influential math book ever published.

Much of what we know about Euclid must be deduced from the various clues that are available in the writings of others. We know for example that he must have lived around the time of Ptolemy I in Alexandria from 325 - 265 BC from Proclus who writes:

He lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that there was no royal road to geometry.

(from David Joyce's Web Site)

It is believed that Euclid must have attended Plato's Academy in Athens because of his deep knowledge about the works Eudoxus and Theaetetus. This is also clear since the organization of the Elements is an attempt to understand the Platonic ideal shapes. The Greek philosopher Proclus writes:
In his aim he [Euclid] was a Platonist, being in sympathy with this philosophy, whence he made the end of the whole "Elements" the construction of the so-called Platonic figures.
(from Euclid, MacTutor Biography)

It is probable that Euclid ran a school of mathematics in Alexandria. Pappus mentions that the Greek mathematician Apollonius learned geometry from the students of Euclid in Alexandria (Donald Lancon, here). There is a story told by Stobaeus in the fifth century:
... someone who had begun to learn geometry with Euclid, when he had learnt the first theorem, asked Euclid "What shall I get by learning these things?" Euclid called his slave and said "Give him threepence since he must make gain out of what he learns".
(from Heath, a History of Greek Mathematics)

Euclid had a reputation for being fair-minded, polite, and a serious scholar. Pappus writes that Euclid was:
... most fair and well disposed towards all who were able in any measure to advance mathematics, careful in no way to give offence, and although an exact scholar not vaunting himself.
(from Euclid, Mac Tutor Biography)

For some reason, Euclid did not write a preface to his great work as did many of the other Greek mathematicians. As a result of this, we are forced to rely on writings that came hundreds of years after his death.

It is believed that when Euclid was writing the Elements, he borrowed content from other textbooks that came earlier. For example, he presents definitions for oblong, rhombus, and rhomboid which are never used (Euclid, MacTutor).

The Elements consists of 13 books. The first two books focus on triangles, parallel lines, parallelograms, rectangles and squares. Included in these book is the parallel postulate, which is the starting point for non-Euclidean geometry, the famous Pythagorean Theorem, a construction for squaring a rectangle.

Books three and four deal with the properties of circles. Book five presents Eudoxus's theory of proportions. Book six deals with similar triangles.

Books seven to nine deal with number theory including Euclid's algorithm for greatest common divisor. Books ten deals with irrational numbers which is taken from Theaetetus's theory.

Books eleven through thirteen cover three-dimensional geometry. Book twelve ends with the proof that circles are to each other as the square of their diameters and spheres to each other as the cubes of their diameters. Book thirteen ends with the proof that there can be only 5 types of regular polyhedra (three-dimensional shapes with equal sides) which are the: tetrahedon (4-sided), cube (6-sided), octahedron (8-sided), dodecahedron (12-sided), and icosahedron (20-sided). These are of course the Platonic solids (see here for details):

1. Tetrahedron

2. Cube

3. Octahedron

4. Dodecahedron

5. Icosahedron

Euclid wrote other works besides the Elements including a work on perspective called Optics. Many of his works are lost including a work he did on conics which predates Apollonius.

The Elements was first published in book form in 1482. Since then, there have been over 1000 editions of this classic math book. It set the standard for rigor and clarity of mathematics and one of the chief themes in the history of mathematics is the effort to place other areas of mathematics on the same firm foundations at Euclid's Elements. Today, Euclid is known as the father of geometry.

References

Euclid, MacTutor Biography
Euclid, Wikipedia
Euclid, David Joyce
David Joyce, Euclid's Elements
Sir Thomas Heath, History of Mathematics
An Introduction to the Works of Euclid, Donald Lancon

Friday, March 31, 2006

More on Euler's Identity and Roots of Unity

When Leonhard Euler came up with his Formula and his Identity, he stood on the shoulders of many giants. In the next few blogs, I plan to focus a bit on some of the giants that Euler stood upon including: Euclid, Archimedes, Hipparchus, Ptolemy, Napier and Bernoulli.

Thinking about Euler's identity in the context of Fermat's Last Theorem raises some questions which I think need to be answered:

Pi, Sine, and Cosine are based on Euclid's plane geometry. What validity can it have for number theory which is independent of Euclidean or non-Euclidean geometry?
How is it possible for a number to be put to the power of an imaginary number? How can this construction possibly have any meaning?

It turns out that trigonometric functions can be defined independently of Euclid (see here). It also turns out that it is possible to use the Maclaurin Series to define a exponents so that they can include any complex power including i (see here )

One of the goals of this blog is to provide a complete set of proofs for each of the propositions that I present or to state those propositions as postulates. Implicit in the use of sine, cosine, and pi is a set of assumptions that are often not thought about:

How can we be sure that all right triangles regardless of their size have the same ratio between their sides so that for a given angle, there is one and only one sine or cosine value? (See here for the answer)
How can we be sure that pi is really constant? It may be obvious but where's the proof? How can we be sure that for all circles, the ratio of circumference-to-diameter is the same? (See here for the answer)

While these questions are very elementary, I think that it is still important to ask them.

In understanding Roots of Unity and Cyclotomic Integers, I think it is worthwhile to review the achievements of DeMoivre, Taylor, and Maclaurin.

I think it is important and valuable to understand all of these details before exploring Kummer's proof.