MATH 3326 Quiz for Final Exam Review, History of
Mathematics: Covers chapters 1-6
Nicole Oresme (1320-1382)
Elementary proof of divergence of harmonic series.
Nicolo Tartaglia (1499-1557)
Formula for solving cubic equations.
Gerolamo Cardano (1501-1576)
Publishing formulas for solving cubic and quartic equations which were done by other
mathematicians.
Francois Viète (1540-1603)
Formula connecting the roots and coefficients of polynomials
John Napier (1550-1617)
Invention of logarithms
Bonaventura Cavalieri (1598-1647)
A principle to find the volume of solids in geometry
Girard Desargues (1591-1661)
The father of projective Geometry
Rene Descartes (1596-1650)
The father of Analytic Geometry
Pierre de Fermat (1606-1665)
Claim about a marvelous proof of the most famous problem which was unsolved until
1994. (Fermat's Last Theorem)
Blaise Pascal (1623-1662)
Very famous infinite triangle of numbers. Binomial Theorem.
Jacob Bernoulli (1655-1705)
Advances to Calculus
Sir Isaac Newton (1642-1726)
Love for infinite polynomials (power series)
Gottfried Wilhelm Leibniz (1646-1716)
Infinitesimal Calculus
Leonard Euler (1707-1773)
Konigsberg Bridges Problem
Sharaf al-Din al-Tusi (12th century)
Isolating unknown on one side of the equation and determining the maximum value of
the obtained expression.
Nicole Oresme (1323-1382)
General ideas about independent and dependent variables. In 1673 Leibniz was talking
about points on a curve, and used coordinates to do so. Everything was depending on
one of the coordinates, and it was here he used the word "curve slope".
Johenn Bernoulli, in 1698, he'd been following the above concept of Leibniz, and
Introduced the expressions made of one single variable. He called them "functions".
Alexis Claude Clairaut and Leonhard Euler in 1734
They introduced notation of f(x) for value of a function.
Bernard Bolzano in 1817
,He introduced definition of "limit" of a function.
(Real) Analysis - Theory of Calculus
A.Z. Cauchy
Karl Weierstress - definition of continuity of a point.
Example: Continuous function whose derivative is discontinuous.
Geometry (Ancient?)
Pythagorean theorem, ruler and compass constructions, constructing numbers
constructing numbers
: cube - which has double volume -, trisection of angle, squaring circle. These three
were impossible, although many people spent lots of time attempting them.
The first phase in the history of algebra was the search for solutions of
polynomial equations. The "degree of difficulty" of an equation corresponds
rather neatly to the
degree of the corresponding polynomial.
Linear equations are easily solved, and 2000 years ago the Chinese were even
able to solve n equations in n unknowns by the method we now call
"Gaussian elimination."
Quadratic equations are harder to solve, because they generally require
the square root operation. But the solution—essentially the same as that taught in high
schools today—was discovered independently in many cultures more than 1000 years
ago.
The first really hard case is the _________________ equation, whose solution
requires both square roots and cube roots.
Cubic.
The cubic equations discovery was made by
Italian mathematicians in the early 16th century and was a decisive breakthrough, and
equations quickly became the language of virtually all mathematics.
The obstacle in completely solving the problem of polynomial equations was
The "quintic" equation - the general equation of degree 5.
In the 1820s it finally became clear that the quintic equation
Is not solvable in the sense that equations of lower degrees are solvable.
The word "algebra" comes from
The Arabic word al-jabr meaning "restoring".
The word "algebra" passed into mathematics through
The book Al-jabr w'al muqabala (Science of restoring and opposition) of al-Khwarizmi in
830CE, a work on the solution of equations.
In this context
This chapter shows ___________________ being applied to number theory.
equations.
Diophantus had methods for finding ________________ __________________ of
quadratic and cubic equations.
Rational solutions.
When ____________ solutions are sought, even linear equations are not trivial.
Integer.
The first general solutions of linear equations in integers were found in
China and India, along with independent discoveries of the Euclidean algorithm.
, The Indians also rediscovered Pell's equation x^2 − Ny^2 = 1, and found
Methods of solving it for general natural number values of N.
The first advance on Pell's equation was made by
Brahmagupta, who in 628CE found a way of "composing" solutions of x^2 − Ny^2 = k1
and x^2 − Ny^2 = k2 to produce a solution of x^2 − Ny^2 = k1k2
There is a curious formula of Brahmagupta that gives all triangles with
Rational sides and rational area.
In 1150CE, Bhaskara II found an extension of Brahmagupta's method that finds a
solution of x^2 − Ny^2 = 1 for any
Nonsquare natural number N. He illustrated it with the case N=61, for which the least
nontrivial solution is extraordinarily large.
Ancient Greece had an enormous influence on world mathematics and that most
of the fundamental concepts of mathematics
Can be found there.
The Greeks did not however
Discover everything first, nor did they do everything best.
The Pythagorean theorem and pythagorean triples were also known in
Ancient China and India. As far as we know these were independent discoveries, so it
seems that the Pythagorean theorem is mathematically universal.
Other cultural universals are the concept of pi - the ratio of diameter to
circumference in the circle - and
The Euclidean algorithm.
The Euclidean algorithm seems to arise whenever there is an interest in
Multiples, divisors, or integer solutions of linear and quadratic equations.
For Euclid, there were two quite separate applications of the Euclidean algorithm.
In the first, the algorithm was applied to
Integers and used to draw conclusions about divisibility and primes. In the second, the
algorithm was applied to line segments and was used as a criterion for irrationality: if the
algorithm does not terminate, then the ratio of the segments is irrational.
It is possible that the Greeks pushed the nonterminating Euclidean algorithm far
enough to see that it
Becomes periodic in certain cases; for example when the two line segments have
lengths 1 and square root of 2.
The first form of the Euclidean algorithm arose in China in the
Han dynasty, between 200BCE and 200CE. It was used by the Chinese to simplify
fractions - dividing numerator and denominator by their gcd - and also to find integer
solutions of linear equations.
The Chinese became highly skilled in such problems involving congruencies,
extending their methods to multiple congruences; this led them to an important
theorem, known today as the
Chinese Remainder Theorem.
Around the fifth and sixth centuries CE, similar linear Diophantine equations were
solved in
India, and perhaps with similar calendar problems in mind.
The Indians too the idea in
A different direction.
Mathematics: Covers chapters 1-6
Nicole Oresme (1320-1382)
Elementary proof of divergence of harmonic series.
Nicolo Tartaglia (1499-1557)
Formula for solving cubic equations.
Gerolamo Cardano (1501-1576)
Publishing formulas for solving cubic and quartic equations which were done by other
mathematicians.
Francois Viète (1540-1603)
Formula connecting the roots and coefficients of polynomials
John Napier (1550-1617)
Invention of logarithms
Bonaventura Cavalieri (1598-1647)
A principle to find the volume of solids in geometry
Girard Desargues (1591-1661)
The father of projective Geometry
Rene Descartes (1596-1650)
The father of Analytic Geometry
Pierre de Fermat (1606-1665)
Claim about a marvelous proof of the most famous problem which was unsolved until
1994. (Fermat's Last Theorem)
Blaise Pascal (1623-1662)
Very famous infinite triangle of numbers. Binomial Theorem.
Jacob Bernoulli (1655-1705)
Advances to Calculus
Sir Isaac Newton (1642-1726)
Love for infinite polynomials (power series)
Gottfried Wilhelm Leibniz (1646-1716)
Infinitesimal Calculus
Leonard Euler (1707-1773)
Konigsberg Bridges Problem
Sharaf al-Din al-Tusi (12th century)
Isolating unknown on one side of the equation and determining the maximum value of
the obtained expression.
Nicole Oresme (1323-1382)
General ideas about independent and dependent variables. In 1673 Leibniz was talking
about points on a curve, and used coordinates to do so. Everything was depending on
one of the coordinates, and it was here he used the word "curve slope".
Johenn Bernoulli, in 1698, he'd been following the above concept of Leibniz, and
Introduced the expressions made of one single variable. He called them "functions".
Alexis Claude Clairaut and Leonhard Euler in 1734
They introduced notation of f(x) for value of a function.
Bernard Bolzano in 1817
,He introduced definition of "limit" of a function.
(Real) Analysis - Theory of Calculus
A.Z. Cauchy
Karl Weierstress - definition of continuity of a point.
Example: Continuous function whose derivative is discontinuous.
Geometry (Ancient?)
Pythagorean theorem, ruler and compass constructions, constructing numbers
constructing numbers
: cube - which has double volume -, trisection of angle, squaring circle. These three
were impossible, although many people spent lots of time attempting them.
The first phase in the history of algebra was the search for solutions of
polynomial equations. The "degree of difficulty" of an equation corresponds
rather neatly to the
degree of the corresponding polynomial.
Linear equations are easily solved, and 2000 years ago the Chinese were even
able to solve n equations in n unknowns by the method we now call
"Gaussian elimination."
Quadratic equations are harder to solve, because they generally require
the square root operation. But the solution—essentially the same as that taught in high
schools today—was discovered independently in many cultures more than 1000 years
ago.
The first really hard case is the _________________ equation, whose solution
requires both square roots and cube roots.
Cubic.
The cubic equations discovery was made by
Italian mathematicians in the early 16th century and was a decisive breakthrough, and
equations quickly became the language of virtually all mathematics.
The obstacle in completely solving the problem of polynomial equations was
The "quintic" equation - the general equation of degree 5.
In the 1820s it finally became clear that the quintic equation
Is not solvable in the sense that equations of lower degrees are solvable.
The word "algebra" comes from
The Arabic word al-jabr meaning "restoring".
The word "algebra" passed into mathematics through
The book Al-jabr w'al muqabala (Science of restoring and opposition) of al-Khwarizmi in
830CE, a work on the solution of equations.
In this context
This chapter shows ___________________ being applied to number theory.
equations.
Diophantus had methods for finding ________________ __________________ of
quadratic and cubic equations.
Rational solutions.
When ____________ solutions are sought, even linear equations are not trivial.
Integer.
The first general solutions of linear equations in integers were found in
China and India, along with independent discoveries of the Euclidean algorithm.
, The Indians also rediscovered Pell's equation x^2 − Ny^2 = 1, and found
Methods of solving it for general natural number values of N.
The first advance on Pell's equation was made by
Brahmagupta, who in 628CE found a way of "composing" solutions of x^2 − Ny^2 = k1
and x^2 − Ny^2 = k2 to produce a solution of x^2 − Ny^2 = k1k2
There is a curious formula of Brahmagupta that gives all triangles with
Rational sides and rational area.
In 1150CE, Bhaskara II found an extension of Brahmagupta's method that finds a
solution of x^2 − Ny^2 = 1 for any
Nonsquare natural number N. He illustrated it with the case N=61, for which the least
nontrivial solution is extraordinarily large.
Ancient Greece had an enormous influence on world mathematics and that most
of the fundamental concepts of mathematics
Can be found there.
The Greeks did not however
Discover everything first, nor did they do everything best.
The Pythagorean theorem and pythagorean triples were also known in
Ancient China and India. As far as we know these were independent discoveries, so it
seems that the Pythagorean theorem is mathematically universal.
Other cultural universals are the concept of pi - the ratio of diameter to
circumference in the circle - and
The Euclidean algorithm.
The Euclidean algorithm seems to arise whenever there is an interest in
Multiples, divisors, or integer solutions of linear and quadratic equations.
For Euclid, there were two quite separate applications of the Euclidean algorithm.
In the first, the algorithm was applied to
Integers and used to draw conclusions about divisibility and primes. In the second, the
algorithm was applied to line segments and was used as a criterion for irrationality: if the
algorithm does not terminate, then the ratio of the segments is irrational.
It is possible that the Greeks pushed the nonterminating Euclidean algorithm far
enough to see that it
Becomes periodic in certain cases; for example when the two line segments have
lengths 1 and square root of 2.
The first form of the Euclidean algorithm arose in China in the
Han dynasty, between 200BCE and 200CE. It was used by the Chinese to simplify
fractions - dividing numerator and denominator by their gcd - and also to find integer
solutions of linear equations.
The Chinese became highly skilled in such problems involving congruencies,
extending their methods to multiple congruences; this led them to an important
theorem, known today as the
Chinese Remainder Theorem.
Around the fifth and sixth centuries CE, similar linear Diophantine equations were
solved in
India, and perhaps with similar calendar problems in mind.
The Indians too the idea in
A different direction.
