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.
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.
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.
1800 BCE may have known the pythagorean theorem before Pythagoras, found it on a clay tablet known as "Plimpton 322."
1620 introduced the coordinate system.
discovered conic sections.
600 BCE Greek mathematician who discovered geometry and greek mathematics.
1619 wanted to explain the planets and their distance. discovery was ruined when they discovered Uranus was a planet.
1882 proved that not only is pi irrational but it is also transcendental. there is no polynomial in which pi is the solution.
1796 discovered that the 17-gon is constructible by ruler and compass only.
1600 2^2^h + 1, only five known 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
m_k_ = 2^k - 1, not known if there are infinitely many of these primes.
F(n+2) = F(n+1) + F(n)
x^2 - ny^2 = 1
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.
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
415 BCE Book XIII - theory of regular polyhedra
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)
a quantity that is not a ratio of whole numbers
a fractional number, ratio of integers Q
visually evident statements
evident principles of logic
a=b --> b=a
a=b and b=c --> a=c
a solid that is convex and has congruent shapes on each side, there are only five.
a quantity that is not the root of any polynomial equation with rational coefficients
x^2/a^2 - y^2/b^2 = 1
x^2/a^2 + y^2/b^2 = 1
numbers with no rectangular representation
having no divisors other then 1 and itself
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
a number transferred from geometric ideas to number theory
an integer of the form 2m, where m is an integer. can be divided easily into groups of two
an integer of the form 2m + 1, where m is an integer
equals the sum of its divisors (including 1 but excluding itself)
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
a closed plane figure having three or more sides
two prime numbers that only have a common divisor of 1
non-negative integer N
whole number that is negative or positive Z
same sides and same angles, SSS SSA SAA
only have 3 angles in common
counting, formulating, numbers
continuity, graphs, lines, curves
found in architecture, painting, and music. can be expressed as a continued fraction.
duplication of the cube
squaring of the circle
trisection of an angle
three main problems of the ancient greeks