Arithmetic
Arithmetic or arithmetics in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though professional mathematicians often treat arithmetic as a synonym for number theory. It is the oldest and simplest branch of mathematics, used widely by almost everyone from simple daily counting to more advanced science and business.
Arithmetic operationsThe traditional arithmetic operations are addition, subtraction, multiplication and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject. Arithmetic is performed according to an order of operations.
Number theoryThe term arithmetic is also used to refer to number theory. This includes the properties of integers related to primality, divisibility, and the solution of equations by integers, as well as modern research which is an outgrowth of this study. It is in this context that one runs across the fundamental theorem of arithmetic and arithmetic functions. A Course in Arithmetic by Serre reflects this usage, as do such phrases as first order arithmetic or arithmetical algebraic geometry.
Arithmetic in educationPrimary education in mathematics often places a strong focus on arithmetic, as further studies in mathematics as well as science benefit from an understanding of arithmetic. There are exceptions, such as New Math, which taught arithemtic indirectly using set theory.
The arithmetic of natural numbers, integers, rational numbers (in the form of vulgar fractions), and real numbers (using the decimal place-value system known as algorism) is typically studied by schoolchildren, who learn manual algorithms for arithmetic. However, in adult life, many people prefer to use tools such as calculators, computers, or the abacus to perform the more complex arithmetical computations.
Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions.
Mathematics, in nearly every society, is used in fields such as the natural sciences, engineering, measuring land, predicting astronomical events, medicine, accounting, and economics. Some of these fields often give rise to and make use of new mathematical discoveries. Mathematical study often involves discovering and cataloging patterns, without regard for application.
The word "mathematics" comes from the Greek μάθημα (máthēma) meaning science, knowledge, or learning, and μαθηματικός (mathēmatikós), meaning fond of learning. It is often abbreviated maths in Commonwealth English and math in North American English.
HistoryThe evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g., addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry.
Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse.
From the beginnings of recorded history, the major disciplines within mathematics arose out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change.
Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both.
Mathematical discoveries have been made throughout history and continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proof."
Inspiration, pure and applied mathematics, and aestheticsMathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. The remarkable fact that even the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called " the unreasonable effectiveness of mathematics."
As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science.
Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics.
Notation, language, and rigorMost of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict grammar (under the influence of computer science, more often now called syntax) and encodes information that would be difficult to write in any other way.
Mathematical language also is hard for beginners. Even common words, such as or and only, have more precise meanings than in everyday speech. Mathematicians, like lawyers, strive to be as unambiguous as possible. Also confusing to beginners, words such as open and field have been given specialized mathematical meanings, and mathematical jargon includes technical terms such as "homeomorphism" and integrable. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".
Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions, of which many instances have occurred in the history of the subject (for example, in mathematical analysis). The level of rigor expected in mathematics has varied over time; the Greeks expected detailed arguments, but by the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since errors can be made in a computation, such proofs may not be sufficiently rigorous.
Axioms in traditional thought were 'self-evident truths', but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.
Is mathematics a science?Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".
If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [1]
In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.
Overview of fields of mathematicsAs noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e., arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics).
The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory.
The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space.
The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics.
Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behaviour.
Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science.
An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a role. It is used in all the sciences. (Many statisticians, however, do not consider themselves to be mathematicians, but rather part of an allied group.) Numerical analysis investigates computational methods for efficiently solving a broad range of mathematical problems that are typically much too large for a human's capacity; it includes the study of rounding errors or other sources of error in computation
IntroductionThe term quantum ( Latin, "how much") refers to the discrete units that the theory assigns to certain physical quantities, such as the energy of an atom at rest (see Figure 1, at right). The discovery that waves could be measured in particle-like small packets of energy called quanta led to the branch of physics that deals with atomic and subatomic systems which we today call Quantum Mechanics. The foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Albert Einstein, Niels Bohr, Werner Heisenberg, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Wolfgang Pauli and others. Some fundamental aspects of the theory are still actively studied.
Quantum mechanics is a more fundamental theory than Newtonian mechanics and classical electromagnetism, in the sense that it provides accurate and precise descriptions for many phenomena that these "classical" theories simply cannot explain on the atomic and subatomic level. It is necessary to use quantum mechanics to understand the behaviour of systems at atomic length scales and smaller. For example, if Newtonian mechanics governed the workings of an atom, electrons would rapidly travel towards and collide with the nucleus. However, in the natural world the electron normally remains in a stable orbit around a nucleus seeming to defy classical electromagnetism.
Quantum mechanics was initially developed to explain the atom, especially the spectra of light emitted by different atomic species. The quantum theory of the atom developed as an explanation for the electron's staying in its orbital, which couldn't be explained by Newton's laws of motion and by classical electromagnetism.
Quantum mechanics uses orbitals as complex number wave functions, and more generally, complex state vectors to explain such effects. These are related to classical physics largely through probability. Probability in the context of quantum mechanics has to do with the likelihood of finding a system in a particular state at a certain time, for example, finding an electron, in a particular region around the nucleus at a particular time. Therefore, electrons cannot be pictured as localized particles in space but rather should be thought of as "clouds" of negative charge spread out over the entire orbit. These clouds represent the regions around the nucleus where the probability of "finding" an electron is the largest. This probability cloud obeys a quantum mechanical principle called Heisenberg's Uncertainty Principle, which states that there is an uncertainty in the classical position of any subatomic particle, including the electron; so instead of describing where an electron or other particle is, the entire range of possible values is used, describing a probability distribution. So in normal atoms with electrons in stable orbits, the probability of the electron being within the nucleus is nearly zero according to the Uncertainty Principle (it is nearly zero as the nucleus has a volume and is not a point). Therefore, quantum mechanics, translated to Newton's equally deterministic description, leads to a probabilistic description of nature.
The other exemplar that led to quantum mechanics was the study of electromagnetic waves such as light. When it was found in 1900 by Max Planck that the energy of waves could be described as consisting of small packets or quanta, Albert Einstein exploited this idea to show that an electromagnetic wave such as light could be described by a particle called the photon with a discrete energy dependent on its frequency. This led to a theory of unity between subatomic particles and electromagnetic waves called wave-particle duality in which particles and waves were neither one nor the other, but had certain properties of both.. It was from this view that the uncertainty principle, the foundation of quantum mechanics, arose.
While quantum mechanics describes the world of the very small, it also is needed to explain certain " macroscopic quantum systems" such as superconductors and superfluids.
Broadly speaking, quantum mechanics incorporates four classes of phenomena that classical physics cannot account for: (i) the quantization (discretization) of certain physical quantities, (ii) wave-particle duality, (iii) the uncertainty principle, and (iv) quantum entanglement. Each of these phenomena will be described in greater detail in subsequent sections.
Most physicists believe that quantum mechanics provides a correct description for the physical world under almost all circumstances. However, the effects of quantum mechanics are generally not significant when considering the observable Universe as a whole. This is because although atoms and subatomic particles are the building blocks of matter, when analyzing the universe on large scales one finds that the dominant force becomes gravity governed by Einstein's theory of general relativity. In some cases, both general relativity and quantum mechanics converge. As an example, general relativity is unable to explain what will happen if a subatomic particle hits the singularity of a black hole which is a phenomenon predicted by general relativity and involves gravity in the macro world. Only quantum mechanics can provide the answer stating that the particle will have an uncertainty in position according to the Heisenberg Uncertainty Principle already discussed, such that it might not really reach the singularity and thus escape the possible collapse to infinite density.
It is believed that the theories of general relativity and quantum mechanics, the two great achievements of physics in the 20th century, contradict one another for two main reasons. One is that the former is an essentially deterministic theory and the latter is essentially indeterministic. And two, general relativity relies mainly on the force of gravity while quantum mechanics relies mainly on the other three fundamental forces being the strong, the weak, and the electromagnetic. The question of how to resolve this contradiction remains an area of active research (see, for example, the article on quantum gravity).
In certain situations, the laws of classical physics approximate the laws of quantum mechanics to a high degree of precision. This is often expressed by saying that quantum mechanics "reduces" to classical mechanics and classical electromagnetism. The situation in which this reduction occurs is called the correspondence, or classical limit.
Quantum mechanics can be formulated in either a relativistic or non-relativistic manner. Relativistic quantum mechanics ( quantum field theory) provides the framework for some of the most accurate physical theories known, though non-relativistic quantum mechanics is also frequently used for reasons of convenience. We will use the terms quantum mechanics, quantum physics, and quantum theory synonymously, to refer to both relativistic and non-relativistic quantum mechanics. It should be noted, however, that certain authors refer to "quantum mechanics" in the more restricted sense of non-relativistic quantum mechanics. Also, in quantum mechanics, the use of the term particle refers to an elementary or subatomic particle.
The ceremony marked the end of a journey spanning thousands of years and many miles. The remains and artifacts, including beads, tools and weapons, were originally excavated from various sites in the Ottawa Valley, including Morrison and the Allumette Islands. They had been part of the Canadian Museum of Civilization’s research collection for decades, some since the late 1800s. Elders from various Algonquin communities conferred on an appropriate reburial, eventually deciding on traditional redcedar and birchbark boxes lined with redcedar chips, muskrat and beaver pelts.
Now, an inconspicuous rock mound marks the reburial site where close to 90 boxes of various sizes are buried. Although negotiations were at times tense between the Kitigan Zibi community and museum, they were able to reach agreement (source: Canadian Geographic Online).
Arithmetic operationsThe traditional arithmetic operations are addition, subtraction, multiplication and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject. Arithmetic is performed according to an order of operations.
Number theoryThe term arithmetic is also used to refer to number theory. This includes the properties of integers related to primality, divisibility, and the solution of equations by integers, as well as modern research which is an outgrowth of this study. It is in this context that one runs across the fundamental theorem of arithmetic and arithmetic functions. A Course in Arithmetic by Serre reflects this usage, as do such phrases as first order arithmetic or arithmetical algebraic geometry.
Arithmetic in educationPrimary education in mathematics often places a strong focus on arithmetic, as further studies in mathematics as well as science benefit from an understanding of arithmetic. There are exceptions, such as New Math, which taught arithemtic indirectly using set theory.
The arithmetic of natural numbers, integers, rational numbers (in the form of vulgar fractions), and real numbers (using the decimal place-value system known as algorism) is typically studied by schoolchildren, who learn manual algorithms for arithmetic. However, in adult life, many people prefer to use tools such as calculators, computers, or the abacus to perform the more complex arithmetical computations.
Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions.
Mathematics, in nearly every society, is used in fields such as the natural sciences, engineering, measuring land, predicting astronomical events, medicine, accounting, and economics. Some of these fields often give rise to and make use of new mathematical discoveries. Mathematical study often involves discovering and cataloging patterns, without regard for application.
The word "mathematics" comes from the Greek μάθημα (máthēma) meaning science, knowledge, or learning, and μαθηματικός (mathēmatikós), meaning fond of learning. It is often abbreviated maths in Commonwealth English and math in North American English.
HistoryThe evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g., addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry.
Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse.
From the beginnings of recorded history, the major disciplines within mathematics arose out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change.
Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both.
Mathematical discoveries have been made throughout history and continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proof."
Inspiration, pure and applied mathematics, and aestheticsMathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. The remarkable fact that even the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called " the unreasonable effectiveness of mathematics."
As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science.
Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics.
Notation, language, and rigorMost of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict grammar (under the influence of computer science, more often now called syntax) and encodes information that would be difficult to write in any other way.
Mathematical language also is hard for beginners. Even common words, such as or and only, have more precise meanings than in everyday speech. Mathematicians, like lawyers, strive to be as unambiguous as possible. Also confusing to beginners, words such as open and field have been given specialized mathematical meanings, and mathematical jargon includes technical terms such as "homeomorphism" and integrable. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".
Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions, of which many instances have occurred in the history of the subject (for example, in mathematical analysis). The level of rigor expected in mathematics has varied over time; the Greeks expected detailed arguments, but by the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since errors can be made in a computation, such proofs may not be sufficiently rigorous.
Axioms in traditional thought were 'self-evident truths', but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.
Is mathematics a science?Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".
If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [1]
In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.
Overview of fields of mathematicsAs noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e., arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics).
The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory.
The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space.
The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics.
Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behaviour.
Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science.
An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a role. It is used in all the sciences. (Many statisticians, however, do not consider themselves to be mathematicians, but rather part of an allied group.) Numerical analysis investigates computational methods for efficiently solving a broad range of mathematical problems that are typically much too large for a human's capacity; it includes the study of rounding errors or other sources of error in computation
IntroductionThe term quantum ( Latin, "how much") refers to the discrete units that the theory assigns to certain physical quantities, such as the energy of an atom at rest (see Figure 1, at right). The discovery that waves could be measured in particle-like small packets of energy called quanta led to the branch of physics that deals with atomic and subatomic systems which we today call Quantum Mechanics. The foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Albert Einstein, Niels Bohr, Werner Heisenberg, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Wolfgang Pauli and others. Some fundamental aspects of the theory are still actively studied.
Quantum mechanics is a more fundamental theory than Newtonian mechanics and classical electromagnetism, in the sense that it provides accurate and precise descriptions for many phenomena that these "classical" theories simply cannot explain on the atomic and subatomic level. It is necessary to use quantum mechanics to understand the behaviour of systems at atomic length scales and smaller. For example, if Newtonian mechanics governed the workings of an atom, electrons would rapidly travel towards and collide with the nucleus. However, in the natural world the electron normally remains in a stable orbit around a nucleus seeming to defy classical electromagnetism.
Quantum mechanics was initially developed to explain the atom, especially the spectra of light emitted by different atomic species. The quantum theory of the atom developed as an explanation for the electron's staying in its orbital, which couldn't be explained by Newton's laws of motion and by classical electromagnetism.
Quantum mechanics uses orbitals as complex number wave functions, and more generally, complex state vectors to explain such effects. These are related to classical physics largely through probability. Probability in the context of quantum mechanics has to do with the likelihood of finding a system in a particular state at a certain time, for example, finding an electron, in a particular region around the nucleus at a particular time. Therefore, electrons cannot be pictured as localized particles in space but rather should be thought of as "clouds" of negative charge spread out over the entire orbit. These clouds represent the regions around the nucleus where the probability of "finding" an electron is the largest. This probability cloud obeys a quantum mechanical principle called Heisenberg's Uncertainty Principle, which states that there is an uncertainty in the classical position of any subatomic particle, including the electron; so instead of describing where an electron or other particle is, the entire range of possible values is used, describing a probability distribution. So in normal atoms with electrons in stable orbits, the probability of the electron being within the nucleus is nearly zero according to the Uncertainty Principle (it is nearly zero as the nucleus has a volume and is not a point). Therefore, quantum mechanics, translated to Newton's equally deterministic description, leads to a probabilistic description of nature.
The other exemplar that led to quantum mechanics was the study of electromagnetic waves such as light. When it was found in 1900 by Max Planck that the energy of waves could be described as consisting of small packets or quanta, Albert Einstein exploited this idea to show that an electromagnetic wave such as light could be described by a particle called the photon with a discrete energy dependent on its frequency. This led to a theory of unity between subatomic particles and electromagnetic waves called wave-particle duality in which particles and waves were neither one nor the other, but had certain properties of both.. It was from this view that the uncertainty principle, the foundation of quantum mechanics, arose.
While quantum mechanics describes the world of the very small, it also is needed to explain certain " macroscopic quantum systems" such as superconductors and superfluids.
Broadly speaking, quantum mechanics incorporates four classes of phenomena that classical physics cannot account for: (i) the quantization (discretization) of certain physical quantities, (ii) wave-particle duality, (iii) the uncertainty principle, and (iv) quantum entanglement. Each of these phenomena will be described in greater detail in subsequent sections.
Most physicists believe that quantum mechanics provides a correct description for the physical world under almost all circumstances. However, the effects of quantum mechanics are generally not significant when considering the observable Universe as a whole. This is because although atoms and subatomic particles are the building blocks of matter, when analyzing the universe on large scales one finds that the dominant force becomes gravity governed by Einstein's theory of general relativity. In some cases, both general relativity and quantum mechanics converge. As an example, general relativity is unable to explain what will happen if a subatomic particle hits the singularity of a black hole which is a phenomenon predicted by general relativity and involves gravity in the macro world. Only quantum mechanics can provide the answer stating that the particle will have an uncertainty in position according to the Heisenberg Uncertainty Principle already discussed, such that it might not really reach the singularity and thus escape the possible collapse to infinite density.
It is believed that the theories of general relativity and quantum mechanics, the two great achievements of physics in the 20th century, contradict one another for two main reasons. One is that the former is an essentially deterministic theory and the latter is essentially indeterministic. And two, general relativity relies mainly on the force of gravity while quantum mechanics relies mainly on the other three fundamental forces being the strong, the weak, and the electromagnetic. The question of how to resolve this contradiction remains an area of active research (see, for example, the article on quantum gravity).
In certain situations, the laws of classical physics approximate the laws of quantum mechanics to a high degree of precision. This is often expressed by saying that quantum mechanics "reduces" to classical mechanics and classical electromagnetism. The situation in which this reduction occurs is called the correspondence, or classical limit.
Quantum mechanics can be formulated in either a relativistic or non-relativistic manner. Relativistic quantum mechanics ( quantum field theory) provides the framework for some of the most accurate physical theories known, though non-relativistic quantum mechanics is also frequently used for reasons of convenience. We will use the terms quantum mechanics, quantum physics, and quantum theory synonymously, to refer to both relativistic and non-relativistic quantum mechanics. It should be noted, however, that certain authors refer to "quantum mechanics" in the more restricted sense of non-relativistic quantum mechanics. Also, in quantum mechanics, the use of the term particle refers to an elementary or subatomic particle.
The ceremony marked the end of a journey spanning thousands of years and many miles. The remains and artifacts, including beads, tools and weapons, were originally excavated from various sites in the Ottawa Valley, including Morrison and the Allumette Islands. They had been part of the Canadian Museum of Civilization’s research collection for decades, some since the late 1800s. Elders from various Algonquin communities conferred on an appropriate reburial, eventually deciding on traditional redcedar and birchbark boxes lined with redcedar chips, muskrat and beaver pelts.
Now, an inconspicuous rock mound marks the reburial site where close to 90 boxes of various sizes are buried. Although negotiations were at times tense between the Kitigan Zibi community and museum, they were able to reach agreement (source: Canadian Geographic Online).
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