applied mathematics The study and use of the mathematical techniques to solve practical problems is called
applied mathematics. The field has various branches
including STATISTICS, PROBABILITY, mechanics, mathematical physics, and the topics that derive from them,
but the distinction from PURE MATHEMATICS might not
be sharp. For instance, the general study of VECTORs
and VECTOR SPACEs can be viewed as either an abstract
study or a practical one if one later has in mind to use
this theory to analyze force diagrams in mechanics.
Many research universities of today possess two
departments of mathematics, one considered pure and
the other applied. Students can obtain advanced degrees
in either field.
approximation A numerical answer to a problem
that is not exact but is sufficient for all practical purposes is called an approximation. For example, noting
that 2
10 is approximately 1,000 allows us to quickly
estimate the value of 2
100 = (2
10 )
10 as 10
30 . Students are
often encouraged to use the fraction 22/7 as an approximate value for π.
Mathematicians use the notation “≈” to denote
approximately equal to. Thus, for example, π ≈ 22/7.
Physicists and engineers often approximate functions by their TAYLOR SERIES with the higher-order
terms dropped. For example,
, at
least for small values of x. The theory of INTEGRAL
CALCULUS begins by approximating areas under curves
as sums of areas of rectangles.
See also ERROR; FACTORIAL; NUMERICAL DIFFERENTIATION; NUMERICAL INTEGRATION.
Arabic mathematics Mathematical historians of
today are grateful to the Arabic scholars of the past for
preserving, translating, and honoring the great Indian,
Greek, and Islamic mathematical works of the scholars
before them, and for their own significant contributions
to the development of mathematics. At the end of the
eighth century, with the great Library of Alexandria
destroyed, Caliph al-Ma’mun set up a House of Wisdom in Baghdad, Iraq, which became the next prominent center of learning and research, as well as the
repository of important academic texts. Many scholars
were employed by the caliph to translate the mathematical works of the past and develop further the ideas they
contained. As the Islamic empire grew over the following seven centuries, the culture of intellectual pursuit
also spread. Many scholars of 12th-century Europe, and
later, visited the Islamic libraries of Spain to read the
texts of the Arabic academics and to learn of the
advances that had occurred in the East during the dark
ages of the West. A significant amount of mathematical
material was transmitted to Europe via these means.
One of the first Greek texts to be translated at the
House of Wisdom was EUCLID’s famous treatise, THE
ELEMENTS. This work made a tremendous impact on
the Arab scholars of the period, and many of them,
when conducting their own research, formulated theorems and proved results precisely in the style of Euclid.
Members of the House of Wisdom also translated the
works of ARCHIMEDES OF SYRACUSE, DIOPHANTUS OF
ALEXANDRIA, MENELAUS OF ALEXANDRIA, and others,
and so they were certainly familiar with all the great
Greek advances in the topics of GEOMETRY, NUMBER
THEORY, mechanics, and analysis. They also translated
the works of Indian scholars,
–
ARYABHATA and
BH
–
ASKARA, for instance, and were familiar with the theory of TRIGONOMETRY, methods in astronomy, and further topics in geometry and number theory. Any Arab
scholar who visited the House of Wisdom had, essentially, the entire bulk of human mathematical knowledge
available to him in his own language.
Arab mathematician MUHAMMAD IBN M –
US
–
A
AL-KHW
–
ARIZM
– ı (ca. 800) wrote a number of original
texts that were enormously influential. His first piece
simply described the decimal place-value system he had
learned from Indian sources. Three hundred years later,
when translated into Latin, this work became the primary source for Europeans who wanted to learn the
new system for writing and manipulating numbers. But
more important was al-Khw –
arizm – ı ’s piece Hisab al-jabr
w’al-muq¯ abala (Calculation by restoration and reduction), from which the topic of “algebra” (“al-jabr”)
arose. Al-Khw –
arizm – ı was fortunate to have all sources
of mathematical knowledge available to him. He began
to see that the then-disparate notions of “number” and
“geometric magnitude” could be unified as one whole
by developing the concept of algebraic objects. This represented a significant departure from Greek thinking, in
which mathematics is synonymous with geometry.
Al-Khw –
arizm – ı’s insight provided a means to study both
arithmetic and geometry under a single framework, and
sin
!
!
x x
x
x
≈ −
+
3
5
3
5
Arabic mathematics 17
applied mathematics. The field has various branches
including STATISTICS, PROBABILITY, mechanics, mathematical physics, and the topics that derive from them,
but the distinction from PURE MATHEMATICS might not
be sharp. For instance, the general study of VECTORs
and VECTOR SPACEs can be viewed as either an abstract
study or a practical one if one later has in mind to use
this theory to analyze force diagrams in mechanics.
Many research universities of today possess two
departments of mathematics, one considered pure and
the other applied. Students can obtain advanced degrees
in either field.
approximation A numerical answer to a problem
that is not exact but is sufficient for all practical purposes is called an approximation. For example, noting
that 2
10 is approximately 1,000 allows us to quickly
estimate the value of 2
100 = (2
10 )
10 as 10
30 . Students are
often encouraged to use the fraction 22/7 as an approximate value for π.
Mathematicians use the notation “≈” to denote
approximately equal to. Thus, for example, π ≈ 22/7.
Physicists and engineers often approximate functions by their TAYLOR SERIES with the higher-order
terms dropped. For example,
, at
least for small values of x. The theory of INTEGRAL
CALCULUS begins by approximating areas under curves
as sums of areas of rectangles.
See also ERROR; FACTORIAL; NUMERICAL DIFFERENTIATION; NUMERICAL INTEGRATION.
Arabic mathematics Mathematical historians of
today are grateful to the Arabic scholars of the past for
preserving, translating, and honoring the great Indian,
Greek, and Islamic mathematical works of the scholars
before them, and for their own significant contributions
to the development of mathematics. At the end of the
eighth century, with the great Library of Alexandria
destroyed, Caliph al-Ma’mun set up a House of Wisdom in Baghdad, Iraq, which became the next prominent center of learning and research, as well as the
repository of important academic texts. Many scholars
were employed by the caliph to translate the mathematical works of the past and develop further the ideas they
contained. As the Islamic empire grew over the following seven centuries, the culture of intellectual pursuit
also spread. Many scholars of 12th-century Europe, and
later, visited the Islamic libraries of Spain to read the
texts of the Arabic academics and to learn of the
advances that had occurred in the East during the dark
ages of the West. A significant amount of mathematical
material was transmitted to Europe via these means.
One of the first Greek texts to be translated at the
House of Wisdom was EUCLID’s famous treatise, THE
ELEMENTS. This work made a tremendous impact on
the Arab scholars of the period, and many of them,
when conducting their own research, formulated theorems and proved results precisely in the style of Euclid.
Members of the House of Wisdom also translated the
works of ARCHIMEDES OF SYRACUSE, DIOPHANTUS OF
ALEXANDRIA, MENELAUS OF ALEXANDRIA, and others,
and so they were certainly familiar with all the great
Greek advances in the topics of GEOMETRY, NUMBER
THEORY, mechanics, and analysis. They also translated
the works of Indian scholars,
–
ARYABHATA and
BH
–
ASKARA, for instance, and were familiar with the theory of TRIGONOMETRY, methods in astronomy, and further topics in geometry and number theory. Any Arab
scholar who visited the House of Wisdom had, essentially, the entire bulk of human mathematical knowledge
available to him in his own language.
Arab mathematician MUHAMMAD IBN M –
US
–
A
AL-KHW
–
ARIZM
– ı (ca. 800) wrote a number of original
texts that were enormously influential. His first piece
simply described the decimal place-value system he had
learned from Indian sources. Three hundred years later,
when translated into Latin, this work became the primary source for Europeans who wanted to learn the
new system for writing and manipulating numbers. But
more important was al-Khw –
arizm – ı ’s piece Hisab al-jabr
w’al-muq¯ abala (Calculation by restoration and reduction), from which the topic of “algebra” (“al-jabr”)
arose. Al-Khw –
arizm – ı was fortunate to have all sources
of mathematical knowledge available to him. He began
to see that the then-disparate notions of “number” and
“geometric magnitude” could be unified as one whole
by developing the concept of algebraic objects. This represented a significant departure from Greek thinking, in
which mathematics is synonymous with geometry.
Al-Khw –
arizm – ı’s insight provided a means to study both
arithmetic and geometry under a single framework, and
sin
!
!
x x
x
x
≈ −
+
3
5
3
5
Arabic mathematics 17