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51 terms

Mathematics and its History 1

famous mathematicians and their discoveries
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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