MA-634 APPLIED MATHEMATICS EXAM QUESTIONS AND
ANSWERS #15
Today, Diophantine analysis is the area of study where _____ solutions are sought for
equations, and Diophantine equations are polynomial equations with integer coefficients
to which only integer solutions are sought. It is usually rather difficult to tell whether a
given Diophantine equation is solvable. - correct answer integer (whole number)
Diophantus considered ____ or _____ solutions "useless", "meaningless", and even
"absurd". To give one specific example, he calls the equation 4 = 4x + 20 'absurd'
because it would lead to a negative value for x. One solution was all he looked for in a
quadratic equation. There is no evidence that suggests Diophantus even realized that
there could be two solutions to a quadratic equation. Diophantus would consider which
of the following equations absurd? - correct answer negative or irrational roots
From Pythagoras to Boethius, pure mathematics consisted of ______ while applied
mathematics consisted of _____. Mathematics could be characterized as the deductive
study of 'such abstractions as quantities and their consequences, namely figures and so
forth' (Acquinas ca. 1260). - correct answer artithmetic and geometry; music and
astronomy
Byzantine scientists preserved and continued the legacy of the great Ancient Greek
mathematicians and put mathematics in practice. In early Byzantine (5th to 7th century)
the architects and mathematician ____ and Anthemius of Tralles used complex
mathematical formulas to construct the great Hagia Sophia church, a technological
breakthrough for its time and for centuries afterwards due to its striking geometry, bold
design and height. - correct answer Isidore of Miletus
The death of ___ may be taken to mark the end of ancient mathematics in the Western
Roman Empire, as the death of ____ had marked the close of Alexandria as a
mathematical center. Work continued for a little longer in Athens due to commentator
Simplicius, but there was not a great original mathematician left. - correct answer
Boethius, Hypatia
Isadore of Miletus was one of the last directors of the Platonic Academy at Athens. The
school had lasted 900 years. When Justinian became emperor in the East in 527, he
was threatened by the learning at the academy and closed the schools and dispersed
the scholars. - correct answer True
Simplicius and others looked to Persia for haven and established "Athenian Academy in
Exile" The date 529 may be taken as the close of European mathematical development
in antiquity. After this Greek science developed in Near and Far Eastern countries for
the next 600 years. The spirit of mathematics ____. - correct answer languished
,Heron of Alexandria is best known in the history of mathematics for the formula, bearing
his name. The formula follows the geometric convention of his day, rather than the
trigonometric one of today, to find the area of any triangle, with only the side lengths.
To calculate the area, first you calculate the semiperimeter, that is half the sum of the
sides. Then subtract each side from the semiperimeter. Find the product of the resulting
differences and the semiperimeter and take the square root. - correct answer True
The civilizations of China and India are _____ than those of Greece and Rome, ____
than those in the Nile and Mesopotamian valleys. - correct answer older, although not
older
The Chou Pei, a Chinese astronomy text, indicates thatin China, as in Egypt, geometry
arose from _______; and, as in Babylonia, Chinese geometry was essentially only an
exercise in ______. There seem to be some indications in the Chou Pei of the
Pythagorean theorem, a theorem treated _______ by the Chinese. - correct answer
measuring magnitudes, lengths, areas, and volumes; arithmetic or algebra; algebraically
The idea of ___ numbers seems not to have causedmuch difficulty for the Chinese
since they were accustomed to calculating with two sets of rods — a red set for ____
and a black set for _____. Nevertheless, they did not accept the notion that a(n) ____
number might be a solution of an equation. - correct answer negative, positive
coefficients, negative coeficients, negative
The Precious Mirror opens with a diagram of the arithmetic triangle, previously
discoveredin the West by Pascal. In Chu's arrangement we have the coefficients of
binomial expansions through the eighth powergiven in rod numerals and a round zero
symbol. - correct answer False
According to Boyer, zero may have started in ____and transferred to India after the
______ had been established there. - correct answer the Greek world in Alexandria,
decimal positional system
With the introduction in Hindu notation, of the tenth numeral,a round goose egg for zero,
the modern system of numeration for integers was completed. Although the Medieval
Hindu forms of the ten numerals differ considerably from those in use today, the
principles of the system were established. The new numeration, which we generally call
the Hindu system, is merely a new combination of three basic principles, all of ancient
origin: _____. Not one of these three was due originally to the Hindus, but it presumably
is due to them that the three were first linked to form the modern system of numeration.
- correct answer (1) a decimal base; (2) a positional notation; and (3) a ciphered form
for each of the ten numerals
When Latin translations of Al-Khowarizmi's work appeared in Europe, careless readers
began to attribute not only the book, but also the numeration, to the author. A loose
pronunciation of his name came to be known as ___. The title of his most important
book, Al-jabr, led to the word ___. - correct answer algorithm, algebra
, Around A.D. 265, Chinese mathematician Liu Huicreated a simple polygon-based
iterative algorithm forfinding the value of pi, which gave four accurate digits.Later,
around A.D. 480, Zu Chongzhi adopted Liu Hui'smethod and achieved seven digits of
accuracy.This record may have held for 800 years.
The method resembled the method of _____ made rigorous by ___ circa 350 BC -
correct answer exhaustion ,Eudoxus
Leonardo of Pisa (1180-1250), better known as Fibonacci, wrote Liber abaci - correct
answer when he was 20 years old,
Problems which advocated the use of Hindu-Arabic numerals, not
A book about the abacus, a thorough treatis on algebraic methods
Robert Bacon is cited in our text for the statement,"Neglect of ____ works injury to all
knowledge, since he who is ignorant of it cannot know the other sciences or the things
of this world." - correct answer mathematics
For purposes of the history of mathematics, our textbook author considers the beginning
of the Middle ages as year 529,roughly the ______, and the end at the year 1436,
roughlythe ______. - correct answer death of Boethius, death of al-Kashi
Even though he was a Christian scientist, Philoponus argued against many of the widely
held Aristotelian theories. For example, like Galileo later, he denied that the speed
acquired by a freely falling body is proportional to its weightas evidenced by his
statement: - correct answer If you let fall from the same height two weights of which one
is many times as heavy as the other, you will see that the ratio of the times required for
the motion does not depend on the ratio of the weights, but that the difference in time is
a very small one
It is one of the ironies of history that the chief advantage of positional notation —its
applicability to fractions — almost entirely escaped the users of the Hindu-Arabic
numerals for the first thousand years of their existence. In this respect Fibonacci was as
much to blame as anyone, for he used three types of fractions — common,
sexagesimal, and unit — but not decimal fractions. In the Liber abaci, in fact, the two
worst of these systems — ___ — are extensively used. - correct answer unit fractions
and common fractions
Without doubt the problem in the Liber abaci that has most inspired future
mathematicians was the following : How many pairs of rabbits will be produced in a
year, beginning with a single pair, if in every month each pair bears a new pair which
becomes productive from the second month on? This celebrated problem gives rise to
the "Fibonacci sequence" 1, 1, 2, 3, 5, 8, 13, 21, . . . , u„, . . . Which can be represented
generally as, ___ - correct answer s n = s n-1 + s n-2
ANSWERS #15
Today, Diophantine analysis is the area of study where _____ solutions are sought for
equations, and Diophantine equations are polynomial equations with integer coefficients
to which only integer solutions are sought. It is usually rather difficult to tell whether a
given Diophantine equation is solvable. - correct answer integer (whole number)
Diophantus considered ____ or _____ solutions "useless", "meaningless", and even
"absurd". To give one specific example, he calls the equation 4 = 4x + 20 'absurd'
because it would lead to a negative value for x. One solution was all he looked for in a
quadratic equation. There is no evidence that suggests Diophantus even realized that
there could be two solutions to a quadratic equation. Diophantus would consider which
of the following equations absurd? - correct answer negative or irrational roots
From Pythagoras to Boethius, pure mathematics consisted of ______ while applied
mathematics consisted of _____. Mathematics could be characterized as the deductive
study of 'such abstractions as quantities and their consequences, namely figures and so
forth' (Acquinas ca. 1260). - correct answer artithmetic and geometry; music and
astronomy
Byzantine scientists preserved and continued the legacy of the great Ancient Greek
mathematicians and put mathematics in practice. In early Byzantine (5th to 7th century)
the architects and mathematician ____ and Anthemius of Tralles used complex
mathematical formulas to construct the great Hagia Sophia church, a technological
breakthrough for its time and for centuries afterwards due to its striking geometry, bold
design and height. - correct answer Isidore of Miletus
The death of ___ may be taken to mark the end of ancient mathematics in the Western
Roman Empire, as the death of ____ had marked the close of Alexandria as a
mathematical center. Work continued for a little longer in Athens due to commentator
Simplicius, but there was not a great original mathematician left. - correct answer
Boethius, Hypatia
Isadore of Miletus was one of the last directors of the Platonic Academy at Athens. The
school had lasted 900 years. When Justinian became emperor in the East in 527, he
was threatened by the learning at the academy and closed the schools and dispersed
the scholars. - correct answer True
Simplicius and others looked to Persia for haven and established "Athenian Academy in
Exile" The date 529 may be taken as the close of European mathematical development
in antiquity. After this Greek science developed in Near and Far Eastern countries for
the next 600 years. The spirit of mathematics ____. - correct answer languished
,Heron of Alexandria is best known in the history of mathematics for the formula, bearing
his name. The formula follows the geometric convention of his day, rather than the
trigonometric one of today, to find the area of any triangle, with only the side lengths.
To calculate the area, first you calculate the semiperimeter, that is half the sum of the
sides. Then subtract each side from the semiperimeter. Find the product of the resulting
differences and the semiperimeter and take the square root. - correct answer True
The civilizations of China and India are _____ than those of Greece and Rome, ____
than those in the Nile and Mesopotamian valleys. - correct answer older, although not
older
The Chou Pei, a Chinese astronomy text, indicates thatin China, as in Egypt, geometry
arose from _______; and, as in Babylonia, Chinese geometry was essentially only an
exercise in ______. There seem to be some indications in the Chou Pei of the
Pythagorean theorem, a theorem treated _______ by the Chinese. - correct answer
measuring magnitudes, lengths, areas, and volumes; arithmetic or algebra; algebraically
The idea of ___ numbers seems not to have causedmuch difficulty for the Chinese
since they were accustomed to calculating with two sets of rods — a red set for ____
and a black set for _____. Nevertheless, they did not accept the notion that a(n) ____
number might be a solution of an equation. - correct answer negative, positive
coefficients, negative coeficients, negative
The Precious Mirror opens with a diagram of the arithmetic triangle, previously
discoveredin the West by Pascal. In Chu's arrangement we have the coefficients of
binomial expansions through the eighth powergiven in rod numerals and a round zero
symbol. - correct answer False
According to Boyer, zero may have started in ____and transferred to India after the
______ had been established there. - correct answer the Greek world in Alexandria,
decimal positional system
With the introduction in Hindu notation, of the tenth numeral,a round goose egg for zero,
the modern system of numeration for integers was completed. Although the Medieval
Hindu forms of the ten numerals differ considerably from those in use today, the
principles of the system were established. The new numeration, which we generally call
the Hindu system, is merely a new combination of three basic principles, all of ancient
origin: _____. Not one of these three was due originally to the Hindus, but it presumably
is due to them that the three were first linked to form the modern system of numeration.
- correct answer (1) a decimal base; (2) a positional notation; and (3) a ciphered form
for each of the ten numerals
When Latin translations of Al-Khowarizmi's work appeared in Europe, careless readers
began to attribute not only the book, but also the numeration, to the author. A loose
pronunciation of his name came to be known as ___. The title of his most important
book, Al-jabr, led to the word ___. - correct answer algorithm, algebra
, Around A.D. 265, Chinese mathematician Liu Huicreated a simple polygon-based
iterative algorithm forfinding the value of pi, which gave four accurate digits.Later,
around A.D. 480, Zu Chongzhi adopted Liu Hui'smethod and achieved seven digits of
accuracy.This record may have held for 800 years.
The method resembled the method of _____ made rigorous by ___ circa 350 BC -
correct answer exhaustion ,Eudoxus
Leonardo of Pisa (1180-1250), better known as Fibonacci, wrote Liber abaci - correct
answer when he was 20 years old,
Problems which advocated the use of Hindu-Arabic numerals, not
A book about the abacus, a thorough treatis on algebraic methods
Robert Bacon is cited in our text for the statement,"Neglect of ____ works injury to all
knowledge, since he who is ignorant of it cannot know the other sciences or the things
of this world." - correct answer mathematics
For purposes of the history of mathematics, our textbook author considers the beginning
of the Middle ages as year 529,roughly the ______, and the end at the year 1436,
roughlythe ______. - correct answer death of Boethius, death of al-Kashi
Even though he was a Christian scientist, Philoponus argued against many of the widely
held Aristotelian theories. For example, like Galileo later, he denied that the speed
acquired by a freely falling body is proportional to its weightas evidenced by his
statement: - correct answer If you let fall from the same height two weights of which one
is many times as heavy as the other, you will see that the ratio of the times required for
the motion does not depend on the ratio of the weights, but that the difference in time is
a very small one
It is one of the ironies of history that the chief advantage of positional notation —its
applicability to fractions — almost entirely escaped the users of the Hindu-Arabic
numerals for the first thousand years of their existence. In this respect Fibonacci was as
much to blame as anyone, for he used three types of fractions — common,
sexagesimal, and unit — but not decimal fractions. In the Liber abaci, in fact, the two
worst of these systems — ___ — are extensively used. - correct answer unit fractions
and common fractions
Without doubt the problem in the Liber abaci that has most inspired future
mathematicians was the following : How many pairs of rabbits will be produced in a
year, beginning with a single pair, if in every month each pair bears a new pair which
becomes productive from the second month on? This celebrated problem gives rise to
the "Fibonacci sequence" 1, 1, 2, 3, 5, 8, 13, 21, . . . , u„, . . . Which can be represented
generally as, ___ - correct answer s n = s n-1 + s n-2
