Mathematics and its History 1

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famous mathematicians and their discoveries

580 BCE Samos was Thales' student.
discovered pythagorean theorem. discovered quantities that are not numerically computable. Founded a school. most notable success was explanation of musical harmony in term of whole-number ratios.

Pythagoras

300 BCE taught in Alexandria.
the "Elements" was the base of mathematical education for 2000 years.
constructed math by deduction from axioms.
proved that there were infinitely many prime numbers by contradiction.
contains constructions by ruler and compass only.

Euclid

250 CE in Alexandria.
general formula for generating Pythagorean triples: ax + by = c.
found methods to solve quadratic and cubic equations. Equations for which rational solutions are sought are called Diophantine - one relation in all equations.

Diophantus

1800 BCE may have known the pythagorean theorem before Pythagoras, found it on a clay tablet known as "Plimpton 322."

Babylonians

1620 introduced the coordinate system.
discovered conic sections.

Descartes

600 BCE Greek mathematician who discovered geometry and greek mathematics.

Thales

1619 wanted to explain the planets and their distance. discovery was ruined when they discovered Uranus was a planet.

Kepler

1882 proved that not only is pi irrational but it is also transcendental. there is no polynomial in which pi is the solution.

Lindemann

1796 discovered that the 17-gon is constructible by ruler and compass only.

Gauss

1600 2^2^h + 1, only five known primes.

Fermat primes

1700 found a factorization for h=5,
p_5_ = 2^32 + 1 is not a prime number.1700 found a factorization for h=5, p5=232+1 is not a prime number

Euler

m_k_ = 2^k - 1, not known if there are infinitely many of these primes.

mersenne primes

F(n+2) = F(n+1) + F(n)

Fibonacci sequence

x^2 - ny^2 = 1

Pell's equation

200 BCE Syracus. enraged a soldier by saying "stay away from my diagram!" and was murdered. his reputation rested on his mechanical inventions. found the area of a parabolic segment, which relied on an infinite process, the summation of an infinite geometric series.

Archimedes

450 BCE the limit wasn't known.

paradoxes of zeno

400 BCE theory of proportions, "method of exhaustion" - Book XII; theory of irrationals - book V

Eudoxus

415 BCE Book XIII - theory of regular polyhedra

Theaetus

integer triples (a,b,c) satisfying 1, such that
a^2 + b^2 = c^2, for example (3,4,5) (5,12,13) (8,15,17)

pythagorean triples

a quantity that is not a ratio of whole numbers

irrational number

a fractional number, ratio of integers Q

rational number

self-evident statement

axiom

visually evident statements

postulates

evident principles of logic

common notions

a=a

reflexive

a=b --> b=a

symmetric

a=b and b=c --> a=c

transitive

a solid that is convex and has congruent shapes on each side, there are only five.

regular polyhedron

a quantity that is not the root of any polynomial equation with rational coefficients

transcendental number

x^2/a^2 - y^2/b^2 = 1

hyperbola

x^2/a^2 + y^2/b^2 = 1

ellipse

y=ax^2

parabola

numbers with no rectangular representation
having no divisors other then 1 and itself

prime numbers

each natural number factors into primes in exactly one way

unique prime factorization

a method for finding the greatest common divisor of two natural numbers, book VII

Euclidean algorithm

a number transferred from geometric ideas to number theory

polygonal numbers

an integer of the form 2m, where m is an integer. can be divided easily into groups of two

even number

an integer of the form 2m + 1, where m is an integer

odd number

equals the sum of its divisors (including 1 but excluding itself)

perfect number

if p is a prime number that divides ab, then p divides a or b

prime divisor property

the Euclidean algorithm applied to line segments

anthyphairesis

a closed plane figure having three or more sides

polygon

two prime numbers that only have a common divisor of 1

coprime numbers

non-negative integer N

natural number

whole number that is negative or positive Z

integer

same sides and same angles, SSS SSA SAA

congruent triangles

only have 3 angles in common

similar triangles

counting, formulating, numbers

arithmetic

continuity, graphs, lines, curves

geometry

found in architecture, painting, and music. can be expressed as a continued fraction.

golden ratio

duplication of the cube
squaring of the circle
trisection of an angle

three main problems of the ancient greeks

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