Infinity refers to something that is boundless or endless. Ancient Greek philosophers discussed the philosophical nature of infinity. In the 17th century, mathematicians began using infinity in calculus but it remained unclear if infinity could be considered a number. In the late 19th century, Georg Cantor studied different sizes of infinite sets and numbers. Mathematicians now define and study infinity precisely, while philosophers and physicists explore the concept in other domains.
In modern mathematics the concept of the limit arises from the twofold requirement to specify the nature of the set of real numbers and to remove the many critiques to the Newtonian definition of the derivative.
In Cauchy’s definition the limit is associated with a function’s behaviour when we approach a fixed point or when this point increases indefinitely.
A satisfactory mathematical approach to the limit concept and the computational rules appears only at the end of the XIX century.
More recently this fundamental concept was introduced in all mathematical fields, not only in the study of functions of several real variables but also in the study of general abstract spaces such as metric and topological spaces.
From Rejection to Revolution: The Challenges and Impact of the Discovery of Z...Suresh Mandal
The number zero is a fundamental concept in mathematics, and its discovery had a profound impact on the development of mathematics, science, and technology. Although zero is such a basic and essential concept to modern society, it was not always so widely accepted. This article will explore the discovery of zero, including its history, significance, and the challenges faced in its acceptance.
History of Zero
The concept of zero dates back to ancient civilizations, such as the Babylonians and the Mayans. However, the concept of zero as a number did not develop until later in history. The earliest known example of zero as a number is found in the Bakhshali manuscript, a mathematical text written on birch bark that dates back to the third or fourth century AD. The manuscript, which was discovered in 1881 in what is now Pakistan, contains examples of arithmetic and algebraic formulas that involve zero.
The concept of zero also emerged independently in different parts of the world. In India, the concept of zero was developed by mathematicians known as Brahmagupta and Aryabhata. Brahmagupta wrote the first known text on zero, known as the Brahmasphutasiddhanta, in the seventh century AD. The text includes rules for using zero in mathematical calculations, such as addition and subtraction. Aryabhata, a mathematician and astronomer, also wrote extensively on zero in his works.
The concept of zero also appeared in the Islamic world, where it was adopted and developed by mathematicians such as Al-Khwarizmi, who is considered the father of algebra. Al-Khwarizmi wrote extensively on zero in his book, Al-Jabr wa-al-Muqabala (The Compendious Book on Calculation by Completion and Balancing), which was translated into Latin and became one of the foundational texts of algebra in Europe.
Significance of Zero
The discovery of zero as a number was a major breakthrough in the history of mathematics. Before the concept of zero was developed, numbers were represented by positional systems, where the value of a digit was determined by its position in a number. For example, the number 123 represented one hundred, two tens, and three ones. This system was limited, however, because it did not allow for the representation of empty places, or spaces where there were no digits.
Zero solved this problem by providing a symbol for an empty place. With the inclusion of zero as a number, it became possible to represent numbers of any size, including negative numbers and fractions. The concept of zero also revolutionized mathematical calculations, making it possible to perform complex calculations that were not possible before.
The significance of zero extended beyond mathematics. It had important implications in the development of science, including physics and astronomy. For example, the concept of zero was essential in the development of the decimal system, which is used in many scientific calculations. The decimal system is based on the use of zero as a placeholder,
In modern mathematics the concept of the limit arises from the twofold requirement to specify the nature of the set of real numbers and to remove the many critiques to the Newtonian definition of the derivative.
In Cauchy’s definition the limit is associated with a function’s behaviour when we approach a fixed point or when this point increases indefinitely.
A satisfactory mathematical approach to the limit concept and the computational rules appears only at the end of the XIX century.
More recently this fundamental concept was introduced in all mathematical fields, not only in the study of functions of several real variables but also in the study of general abstract spaces such as metric and topological spaces.
From Rejection to Revolution: The Challenges and Impact of the Discovery of Z...Suresh Mandal
The number zero is a fundamental concept in mathematics, and its discovery had a profound impact on the development of mathematics, science, and technology. Although zero is such a basic and essential concept to modern society, it was not always so widely accepted. This article will explore the discovery of zero, including its history, significance, and the challenges faced in its acceptance.
History of Zero
The concept of zero dates back to ancient civilizations, such as the Babylonians and the Mayans. However, the concept of zero as a number did not develop until later in history. The earliest known example of zero as a number is found in the Bakhshali manuscript, a mathematical text written on birch bark that dates back to the third or fourth century AD. The manuscript, which was discovered in 1881 in what is now Pakistan, contains examples of arithmetic and algebraic formulas that involve zero.
The concept of zero also emerged independently in different parts of the world. In India, the concept of zero was developed by mathematicians known as Brahmagupta and Aryabhata. Brahmagupta wrote the first known text on zero, known as the Brahmasphutasiddhanta, in the seventh century AD. The text includes rules for using zero in mathematical calculations, such as addition and subtraction. Aryabhata, a mathematician and astronomer, also wrote extensively on zero in his works.
The concept of zero also appeared in the Islamic world, where it was adopted and developed by mathematicians such as Al-Khwarizmi, who is considered the father of algebra. Al-Khwarizmi wrote extensively on zero in his book, Al-Jabr wa-al-Muqabala (The Compendious Book on Calculation by Completion and Balancing), which was translated into Latin and became one of the foundational texts of algebra in Europe.
Significance of Zero
The discovery of zero as a number was a major breakthrough in the history of mathematics. Before the concept of zero was developed, numbers were represented by positional systems, where the value of a digit was determined by its position in a number. For example, the number 123 represented one hundred, two tens, and three ones. This system was limited, however, because it did not allow for the representation of empty places, or spaces where there were no digits.
Zero solved this problem by providing a symbol for an empty place. With the inclusion of zero as a number, it became possible to represent numbers of any size, including negative numbers and fractions. The concept of zero also revolutionized mathematical calculations, making it possible to perform complex calculations that were not possible before.
The significance of zero extended beyond mathematics. It had important implications in the development of science, including physics and astronomy. For example, the concept of zero was essential in the development of the decimal system, which is used in many scientific calculations. The decimal system is based on the use of zero as a placeholder,
From Rejection to Revolution: The Challenges and Impact of the Discovery of Z...Suresh Mandal
From Rejection to Revolution: The Challenges and Impact of the Discovery of Zero in Mathematics, Science, and Technology.
Posted on February 15, 2023
The number zero is a fundamental concept in mathematics, and its discovery had a profound impact on the development of mathematics, science, and technology. Although zero is such a basic and essential concept to modern society, it was not always so widely accepted. This article will explore the discovery of zero, including its history, significance, and the challenges faced in its acceptance.
History of Zero
The concept of zero dates back to ancient civilizations, such as the Babylonians and the Mayans. However, the concept of zero as a number did not develop until later in history. The earliest known example of zero as a number is found in the Bakhshali manuscript, a mathematical text written on birch bark that dates back to the third or fourth century AD. The manuscript, which was discovered in 1881 in what is now Pakistan, contains examples of arithmetic and algebraic formulas that involve zero.
The concept of zero also emerged independently in different parts of the world. In India, the concept of zero was developed by mathematicians known as Brahmagupta and Aryabhata. Brahmagupta wrote the first known text on zero, known as the Brahmasphutasiddhanta, in the seventh century AD. The text includes rules for using zero in mathematical calculations, such as addition and subtraction. Aryabhata, a mathematician and astronomer, also wrote extensively on zero in his works.
The concept of zero also appeared in the Islamic world, where it was adopted and developed by mathematicians such as Al-Khwarizmi, who is considered the father of algebra. Al-Khwarizmi wrote extensively on zero in his book, Al-Jabr wa-al-Muqabala (The Compendious Book on Calculation by Completion and Balancing), which was translated into Latin and became one of the foundational texts of algebra in Europe.
Significance of Zero
The discovery of zero as a number was a major breakthrough in the history of mathematics. Before the concept of zero was developed, numbers were represented by positional systems, where the value of a digit was determined by its position in a number. For example, the number 123 represented one hundred, two tens, and three ones. This system was limited, however, because it did not allow for the representation of empty places, or spaces where there were no digits.
Zero solved this problem by providing a symbol for an empty place. With the inclusion of zero as a number, it became possible to represent numbers of any size, including negative numbers and fractions. The concept of zero also revolutionized mathematical calculations, making it possible to perform complex calculations that were not possible before.
The significance of zero extended beyond mathematics. It had important implications in the development of science, including physics and astronomy. For example, the concept of zero was essential in t
This report discusses about Logical Empiricism, or Logical Positivism – from its origins, who founded this "movement", its influences, weaknesses, and its contribution to education in general.
These are the slides of a talk by Rens Bod presented on January 18, 2012 at WERELD BEELD, Amsterdam University College. The title is: How the Humanities Changed the World, Or why we should stop worrying and love the history of the humanities.
The humanities are under severe pressure worldwide. While the humanities have been viewed for centuries as the pinnacle of education, during the last forty years or so the study of art, history, literature, language and music is typically seen as a luxury, both by policy makers and the public. The humanities are an ornamentation of life but useless for technology, economy and industry. Humanities scholars have been unable to come up with a convincing answer to their marginalization. Arguments in favour of the humanities are defensive and get lost in mantra-like repetitions like: the humanistic disciplines are important for self-cultivation (Bildung), they are relevant for cultural and historical consciousness, and they form the basis for critical thinking and democracy. While these arguments may all be true, most scholars overlook the possibility that the assumption behind the image problem itself may be wrong.
Humanities scholars seem to have taken for granted that the humanities are economically irrelevant. Yet a quick glance over the history of the humanities shows the opposite: humanistic insights not only radically changed the world but they also resulted in concrete applications. As if humanities scholars have no idea of their own history – or decided to neglect a part of it -- these applications are attributed to the sciences. Here something has to be rectified, where the attack is the best defense.
Science and contribution of mathematics in its developmentFernando Alcoforado
Mathematics is the science of logical reasoning that has its development linked to research, interest in discovering the new and investigate highly complex situations. The escalation of Mathematics began in ancient times when it was aroused the interest by the calculations and numbers according to the need of man to relate the natural events to their daily lives. Today, Mathematics is the most important science of the modern world because it is present in all scientific areas.
Chatty Kathy - UNC Bootcamp Final Project Presentation - Final Version - 5.23...John Andrews
SlideShare Description for "Chatty Kathy - UNC Bootcamp Final Project Presentation"
Title: Chatty Kathy: Enhancing Physical Activity Among Older Adults
Description:
Discover how Chatty Kathy, an innovative project developed at the UNC Bootcamp, aims to tackle the challenge of low physical activity among older adults. Our AI-driven solution uses peer interaction to boost and sustain exercise levels, significantly improving health outcomes. This presentation covers our problem statement, the rationale behind Chatty Kathy, synthetic data and persona creation, model performance metrics, a visual demonstration of the project, and potential future developments. Join us for an insightful Q&A session to explore the potential of this groundbreaking project.
Project Team: Jay Requarth, Jana Avery, John Andrews, Dr. Dick Davis II, Nee Buntoum, Nam Yeongjin & Mat Nicholas
From Rejection to Revolution: The Challenges and Impact of the Discovery of Z...Suresh Mandal
From Rejection to Revolution: The Challenges and Impact of the Discovery of Zero in Mathematics, Science, and Technology.
Posted on February 15, 2023
The number zero is a fundamental concept in mathematics, and its discovery had a profound impact on the development of mathematics, science, and technology. Although zero is such a basic and essential concept to modern society, it was not always so widely accepted. This article will explore the discovery of zero, including its history, significance, and the challenges faced in its acceptance.
History of Zero
The concept of zero dates back to ancient civilizations, such as the Babylonians and the Mayans. However, the concept of zero as a number did not develop until later in history. The earliest known example of zero as a number is found in the Bakhshali manuscript, a mathematical text written on birch bark that dates back to the third or fourth century AD. The manuscript, which was discovered in 1881 in what is now Pakistan, contains examples of arithmetic and algebraic formulas that involve zero.
The concept of zero also emerged independently in different parts of the world. In India, the concept of zero was developed by mathematicians known as Brahmagupta and Aryabhata. Brahmagupta wrote the first known text on zero, known as the Brahmasphutasiddhanta, in the seventh century AD. The text includes rules for using zero in mathematical calculations, such as addition and subtraction. Aryabhata, a mathematician and astronomer, also wrote extensively on zero in his works.
The concept of zero also appeared in the Islamic world, where it was adopted and developed by mathematicians such as Al-Khwarizmi, who is considered the father of algebra. Al-Khwarizmi wrote extensively on zero in his book, Al-Jabr wa-al-Muqabala (The Compendious Book on Calculation by Completion and Balancing), which was translated into Latin and became one of the foundational texts of algebra in Europe.
Significance of Zero
The discovery of zero as a number was a major breakthrough in the history of mathematics. Before the concept of zero was developed, numbers were represented by positional systems, where the value of a digit was determined by its position in a number. For example, the number 123 represented one hundred, two tens, and three ones. This system was limited, however, because it did not allow for the representation of empty places, or spaces where there were no digits.
Zero solved this problem by providing a symbol for an empty place. With the inclusion of zero as a number, it became possible to represent numbers of any size, including negative numbers and fractions. The concept of zero also revolutionized mathematical calculations, making it possible to perform complex calculations that were not possible before.
The significance of zero extended beyond mathematics. It had important implications in the development of science, including physics and astronomy. For example, the concept of zero was essential in t
This report discusses about Logical Empiricism, or Logical Positivism – from its origins, who founded this "movement", its influences, weaknesses, and its contribution to education in general.
These are the slides of a talk by Rens Bod presented on January 18, 2012 at WERELD BEELD, Amsterdam University College. The title is: How the Humanities Changed the World, Or why we should stop worrying and love the history of the humanities.
The humanities are under severe pressure worldwide. While the humanities have been viewed for centuries as the pinnacle of education, during the last forty years or so the study of art, history, literature, language and music is typically seen as a luxury, both by policy makers and the public. The humanities are an ornamentation of life but useless for technology, economy and industry. Humanities scholars have been unable to come up with a convincing answer to their marginalization. Arguments in favour of the humanities are defensive and get lost in mantra-like repetitions like: the humanistic disciplines are important for self-cultivation (Bildung), they are relevant for cultural and historical consciousness, and they form the basis for critical thinking and democracy. While these arguments may all be true, most scholars overlook the possibility that the assumption behind the image problem itself may be wrong.
Humanities scholars seem to have taken for granted that the humanities are economically irrelevant. Yet a quick glance over the history of the humanities shows the opposite: humanistic insights not only radically changed the world but they also resulted in concrete applications. As if humanities scholars have no idea of their own history – or decided to neglect a part of it -- these applications are attributed to the sciences. Here something has to be rectified, where the attack is the best defense.
Science and contribution of mathematics in its developmentFernando Alcoforado
Mathematics is the science of logical reasoning that has its development linked to research, interest in discovering the new and investigate highly complex situations. The escalation of Mathematics began in ancient times when it was aroused the interest by the calculations and numbers according to the need of man to relate the natural events to their daily lives. Today, Mathematics is the most important science of the modern world because it is present in all scientific areas.
Chatty Kathy - UNC Bootcamp Final Project Presentation - Final Version - 5.23...John Andrews
SlideShare Description for "Chatty Kathy - UNC Bootcamp Final Project Presentation"
Title: Chatty Kathy: Enhancing Physical Activity Among Older Adults
Description:
Discover how Chatty Kathy, an innovative project developed at the UNC Bootcamp, aims to tackle the challenge of low physical activity among older adults. Our AI-driven solution uses peer interaction to boost and sustain exercise levels, significantly improving health outcomes. This presentation covers our problem statement, the rationale behind Chatty Kathy, synthetic data and persona creation, model performance metrics, a visual demonstration of the project, and potential future developments. Join us for an insightful Q&A session to explore the potential of this groundbreaking project.
Project Team: Jay Requarth, Jana Avery, John Andrews, Dr. Dick Davis II, Nee Buntoum, Nam Yeongjin & Mat Nicholas
Adjusting primitives for graph : SHORT REPORT / NOTESSubhajit Sahu
Graph algorithms, like PageRank Compressed Sparse Row (CSR) is an adjacency-list based graph representation that is
Multiply with different modes (map)
1. Performance of sequential execution based vs OpenMP based vector multiply.
2. Comparing various launch configs for CUDA based vector multiply.
Sum with different storage types (reduce)
1. Performance of vector element sum using float vs bfloat16 as the storage type.
Sum with different modes (reduce)
1. Performance of sequential execution based vs OpenMP based vector element sum.
2. Performance of memcpy vs in-place based CUDA based vector element sum.
3. Comparing various launch configs for CUDA based vector element sum (memcpy).
4. Comparing various launch configs for CUDA based vector element sum (in-place).
Sum with in-place strategies of CUDA mode (reduce)
1. Comparing various launch configs for CUDA based vector element sum (in-place).
Techniques to optimize the pagerank algorithm usually fall in two categories. One is to try reducing the work per iteration, and the other is to try reducing the number of iterations. These goals are often at odds with one another. Skipping computation on vertices which have already converged has the potential to save iteration time. Skipping in-identical vertices, with the same in-links, helps reduce duplicate computations and thus could help reduce iteration time. Road networks often have chains which can be short-circuited before pagerank computation to improve performance. Final ranks of chain nodes can be easily calculated. This could reduce both the iteration time, and the number of iterations. If a graph has no dangling nodes, pagerank of each strongly connected component can be computed in topological order. This could help reduce the iteration time, no. of iterations, and also enable multi-iteration concurrency in pagerank computation. The combination of all of the above methods is the STICD algorithm. [sticd] For dynamic graphs, unchanged components whose ranks are unaffected can be skipped altogether.
2. WHAT IS
INFINITY?
ITS HISTORY
EXAMPLES
OTHER
TOPICS
ADDITIONAL
INFORMATION
KINDS OR
TYPES
CREDITS
Since the time of the ancient Greeks, the
philosophical nature of infinity was the
subject of many discussions among
philosophers. In the 17th century, with
the introduction of the infinity symbol[1]
and the infinitesimal calculus,
mathematicians began to work with
infinite series and what some
mathematicians (including l'Hôpital and
Bernoulli)[2] regarded as infinitely small
quantities, but infinity continued to be
associated with endless processes.
3. WHAT IS
INFINITY?
ITS HISTORY
EXAMPLES
OTHER
TOPICS
ADDITIONAL
INFORMATION
KINDS OR
TYPES
CREDITS
As mathematicians struggled with the foundation of
calculus, it remained unclear whether infinity could be
considered as a number or magnitude and, if so, how
this could be done.[1] At the end of the 19th century,
Georg Cantor enlarged the mathematical study of
infinity by studying infinite sets and infinite numbers,
showing that they can be of various sizes.[1][3] For
example, if a line is viewed as the set of all of its points,
their infinite number (i.e., the cardinality of the line) is
larger than the number of integers.[4] In this usage,
infinity is a mathematical concept, and infinite
mathematical objects can be studied, manipulated, and
used just like any other mathematical object.
Ancient cultures had
various ideas about the
nature of infinity.
The ancient
Indians and
the Greeks did not
define infinity in
precise formalism as
does modern
mathematics, and
instead approached
infinity as a
philosophical concept.
5. WHAT IS
INFINITY?
ITS HISTORY
EXAMPLES
OTHER
TOPICS
ADDITIONAL
INFORMATION
KINDS OR
TYPES
CREDITS
THE MATHEMATICAL INFINITY
For example, 1 + 1/2 + 1/3 + … is an
infinite series. Infinity is a means to
describe anything that is endless,
making it impossible to measure. It
refers to unending time, a series of
numbers that continues forever, or a
perpetual series of operations.
6. WHAT IS
INFINITY?
ITS HISTORY
EXAMPLES
OTHER
TOPICS
ADDITIONAL
INFORMATION
KINDS OR
TYPES
CREDITS
THE PHYSICAL INFINITY
'Physical infinity' moves from
mathematics to the real world
and tackles questions such as 'is
space infinite? ' In many areas of
physics, the presence of an
infinite quantity (often called a
singularity) is construed as a
warning that the theory is losing
touch with reality.
7. WHAT IS
INFINITY?
ITS HISTORY
EXAMPLES
OTHER
TOPICS
ADDITIONAL
INFORMATION
KINDS OR
TYPES
CREDITS
THE METAPHYSICAL INFINITY
Infinity in Physical Science. From
a metaphysical perspective, the
theories of mathematical physics
seem to be ontologically
committed to objects and their
properties. If any of those
objects or properties are infinite,
then physics is committed to
there being infinity within the
physical world.