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 King/Queen of Spam, who's the ruler of spam?
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Dreaming
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I'm annoyed because you can only have 255 smilies in one post. >=O Why, Why, Why do we press harder on a remote control when we know the batteries are getting dead? Why do banks charge a fee on 'insufficient funds' when they know there is not enough money? Why does someone believe you when you say there are four billion stars, but check when you say the paint is wet? Why doesn't glue stick to the bottle? Why do they use sterilized needles for death by lethal injection? Why doesn't Tarzan have a beard? Why does Superman stop bullets with his chest, but ducks when you throw a revolver at him? Why do Kamikaze pilots wear helmets? Whose idea was it to put an 'S' in the word 'lisp'? If people evolved from apes, why are there still apes? Why is it that no matter what color bubble bath you use the bubbles are always white? Is there ever a day that mattresses are not on sale? Why do people constantly return to the refrigerator with hopes that something new to eat will have materialized? Why do people keep running over a string a dozen times with their vacuum cleaner, then reach down, pick it up, examine it, then put it down to give the vacuum one more chance? Why is it that no plastic bag will open from the end on your first try? How do those dead bugs get into those enclosed light fixtures? When we are in the supermarket and someone rams our ankle with a shopping cart then apologizes for doing so, why do we say, 'It's all right?' Well, it isn't all right, so why don't we say, 'That hurt, you stupid idiot?' Why is it that whenever you attempt to catch something that's falling off the table you always manage to knock something else over? In winter why do we try to keep the house as warm as it was in summer when we complained about the heat? How come you never hear father-in-law jokes? Trigonometry was probably invented for use in sailing as a navigation method used with astronomy.[2] The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago.[citation needed] The common practice of measuring angles in degrees, minutes and seconds comes from the Babylonian's base sixty system of numeration. The Sulba Sutras written in India, between 800 BC and 500 BC, correctly computes the sine of π/4 (45°) as 1/√2 in a procedure for circling the square (the opposite of squaring the circle). The first recorded use of trigonometry came from the Hellenistic mathematician Hipparchus[1] circa 150 BC, who compiled a trigonometric table using the sine for solving triangles. Ptolemy further developed trigonometric calculations circa 100 AD. The ancient Sinhalese in Sri Lanka, when constructing reservoirs in the Anuradhapura kingdom, used trigonometry to calculate the gradient of the water flow. Archeological research also provides evidence of trigonometry used in other unique hydrological structures dating back to 4 BC.[3] The Indian mathematician Aryabhata in 499, gave tables of half chords which are now known as sine tables, along with cosine tables. He used zya for sine, kotizya for cosine, and otkram zya for inverse sine, and also introduced the versine. Another Indian mathematician, Brahmagupta in 628, used an interpolation formula to compute values of sines, up to the second order of the Newton-Stirling interpolation formula. In the 10th century, the Persian mathematician and astronomer Abul Wáfa introduced the tangent function and improved methods of calculating trigonometry tables. He established the angle addition identities, e.g. sin (a +  , and discovered the sine formula for spherical geometry: \frac{\sin(A)}{\sin(a)} = \frac{\sin( }{\sin( } = \frac{\sin©}{\sin©}. Also in the late 10th and early 11th centuries, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the formula \cos(a) \cos( = \frac{\cos(a+ + \cos(a- }{2}.. Indian mathematicians were the pioneers of variable computations algebra for use in astronomical calculations along with trigonometry. Lagadha (circa 1350-1200 BC) is the first person thought to have used geometry and trigonometry for astronomy, in his Vedanga Jyotisha. Persian mathematician Omar Khayyám (1048-1131) combined trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means. Khayyam solved the cubic equation x3 + 200x = 20x2 + 2000 and found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables. Detailed methods for constructing a table of sines for any angle were given by the Indian mathematician Bhaskara in 1150, along with some sine and cosine formulae. Bhaskara also developed spherical trigonometry. The 13th century Persian mathematician Nasir al-Din Tusi, along with Bhaskara, was probably the first to treat trigonometry as a distinct mathematical discipline. Nasir al-Din Tusi in his Treatise on the Quadrilateral was the first to list the six distinct cases of a right angled triangle in spherical trigonometry. In the 14th century, Persian mathematician al-Kashi and Timurid mathematician Ulugh Beg (grandson of Timur) produced tables of trigonometric functions as part of their studies of astronomy. The mathematician Bartholemaeus Pitiscus published an influential work on trigonometry in 1595 which may have coined the word "trigonometry". [edit] Overview In this right triangle: sin(A) = a/c; cos(A) = b/c; tan(A) = a/b. In this right triangle: sin(A) = a/c; cos(A) = b/c; tan(A) = a/b. By definition, one angle of a right triangle is 90 degrees. If one of the other angles is known, the third can be calculated since all three angles of any triangle must add up to 180 degrees. The shape of a right triangle is completely determined, up to similarity, by the angles. This means that once one of the other angles is known, the ratios of the various sides are always the same regardless of the overall size of the triangle. These ratios are given by the following trigonometric functions of the known angle: * The sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse. \sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}} * The cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse. \cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}} * The tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg. \tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{\sin A}{\cos A} The adjacent leg is the side that is adjacent to the angle but not the hypotenuse. The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec) and cotangent (cot), respectively. The inverse functions are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities. With these functions one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and an angle or two angles and a side or three sides are known. These laws are useful in all branches of geometry, since every polygon may be described as a finite combination of triangles. [edit] Extending the definitions Graphs of the functions sin(x) and cos(x), where the angle x is measured in radians. Graphs of the functions sin(x) and cos(x), where the angle x is measured in radians. The above definitions apply to angles between 0 and 90 degrees (0 and π/2 radians) only. Using the unit circle, one may extend them to all positive and negative arguments (see trigonometric function). The trigonometric functions are periodic, with a period of 360 degrees or 2π radians. That means their values repeat at those intervals. The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus and infinite series. With these definitions the trigonometric functions can be defined for complex numbers. The complex function cis is particularly useful \operatorname{cis} (x) = \cos x + i\sin x \! = e^{ix} See Euler's and De Moivre's formulas. [edit] Mnemonics Students often make use of mnemonics to remember the relationships and facts in trigonometry. For example, the sine, cosine and tangent ratios in right triangles can be remembered by representing all three ratios at once as a string of letters; SOH CAH TOA (sine-opposite-hypotenuse ::: cosine-adjacent-hypotenuse ::: tangent-opposite-adjacent), which can be pronounced as a single word. In addition, many remember similar letter sequences by creating sentences that consist of words that begin with the letters to be remembered, so that they are remembered in the correct order. For example, to remember Tan = Opposite/Adjacent, the letters TOA must be remembered in order. Any memorable phrase constructed of words beginning with the letters 'T, O, A' will serve, and often sentences are constructed to remember all three ratios at once. Other types of mnemonic describe facts in a simple, memorable way, such as "Plus to the right, minus to the left, positive height, negative depth" when referring to the trigonometric functions of a revolving line. [edit] Rule of quarters The rule of quarters makes it easy to remember the sine function of special angles: \begin{align} \sin (0^{\circ}) &= \sqrt{\frac{0}{4}} &= 0\\ \sin (30^{\circ}) &= \sqrt{\frac{1}{4}} &= \frac {1}{2}\\ \sin (45^{\circ}) &= \sqrt{\frac{2}{4}} &= \frac {\sqrt{2}}{2}\\ \sin (60^{\circ}) &= \sqrt{\frac{3}{4}} &= \frac {\sqrt{3}}{2}\\ \sin (90^{\circ}) &= \sqrt{\frac{4}{4}} &= 1 \end{align} [edit] Calculating trigonometric functions Main article: Generating trigonometric tables Trigonometric functions were among the earliest uses for mathematical tables. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. Slide rules had special scales for trigonometric functions. Today scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan and sometimes cis) and their inverses. Most allow a choice of angle measurement methods, degrees, radians and, sometimes, Grad. Most computer programming languages provide function libraries that include the trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers have built in instructions for calculating trigonometric functions. [edit] Applications of trigonometry Marine sextants like this are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer, the position of the ship can then be determined from several such measurements. Marine sextants like this are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer, the position of the ship can then be determined from several such measurements. Main article: Uses of trigonometry There are an enormous number of applications of trigonometry and trigonometric functions. For instance, the technique of triangulation is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves. Fields which make use of trigonometry or trigonometric functions include astronomy (especially, for locating the apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development. [edit] Common formulae Main article: Trigonometric identity Main article: Trigonometric function Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Many express important geometric relationships. For example, the Pythagorean identities are an expression of the Pythagorean Theorem. Here are some of the more commonly used identities, as well as the most important formulae connecting angles and sides of an arbitrary triangle. For more identities see trigonometric identity. [edit] Trigonometric identities Trigonometry History Usage Functions Inverse functions Further reading Reference List of identities Exact constants Generating trigonometric tables CORDIC Euclidean theory Law of sines Law of cosines Law of tangents Pythagorean theorem Calculus The Trigonometric integral Trigonometric substitution Integrals of functions Integrals of inverses [edit] Pythagorean identities \begin{align} \sin^2 A + \cos^2 A &= 1 \\ \tan^2 A + 1 &= \sec^2 A \\ 1+\cot^2 A &= \csc^2 A \end{align} [edit] Sum and product identities [edit] Sum to product: \begin{align} \sin A \pm \sin B &= 2\sin \left( \frac{A \pm B}{2}\right)\cos \left(\frac{A \mp B}{2} \right)\\ \cos A + \cos B &= 2\cos \left(\frac{A + B}{2} \right)\cos \left(\frac{A - B}{2}\right)\\ \cos A - \cos B &= -2\sin \left(\frac{A + B}{2} \right) \sin \left(\frac{A - B}{2}\right) \end{align} [edit] Product to sum: \begin{align} \cos A \,\cos B &= \frac{1}{2}[\cos(A + + \cos (A - ]\\ \sin A \,\sin B &= -\frac{1}{2}[\cos(A + - \cos (A - ]\\ \cos A \,\sin B &= \frac{1}{2}[\sin(A + - \sin (A - ]\\ \sin A \,\cos B &= \frac{1}{2}[\sin(A + + \sin (A - ] \end{align} [edit] Sine, cosine and tangent of a sum \begin{align} \sin(A \pm &= \sin A \cos B \pm \cos A \sin B \\ \cos(A \pm &= \cos A \cos B \mp \sin A \sin B \\ \tan(A \pm &= \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \end{align} [edit] Double-angle identities \begin{align} \sin 2A &= 2 \sin A \cos A \\ &= \frac{2 \tan A}{1 + \tan^2 A}\\ \cos 2A &= \cos^2 A - \sin^2 A \\ &= 2 \cos^2 A -1 \\ &= 1-2 \sin^2 A \\ &= {1 - \tan^2 A \over 1 + \tan^2 A}\\ \tan 2A &= \frac{2 \tan A}{1 - \tan^2 A}\\ &= \frac{2 \cot A}{\cot^2 A - 1}\\ &= \frac{2}{\cot A - \tan A} \end{align} [edit] Half-angle identities Note that \pm is correct, it means it may be either one, depending on the value of A/2. \begin{align} \sin \frac{A}{2} &= \pm \sqrt{\frac{1-\cos A}{2}} \\ \cos \frac{A}{2} &= \pm \sqrt{\frac{1+\cos A}{2}} \\\tan \frac{A}{2} &= \pm \sqrt{\frac{1-\cos A}{1+\cos A}} = \frac {\sin A}{1+\cos A} = \frac {1-\cos A}{\sin A} \end{align} [edit] Triangle identities Laws of Sines and Cosines Laws of Sines and Cosines\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \cos C=\frac{a^2+b^2-c^2}{2ab} In the following identities, A, B and C are the angles of a triangle and a, b and c are the lengths of sides of the triangle opposite the respective angles. [edit] Law of sines The law of sines (also know as the "sine rule") for an arbitrary triangle states: \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R where R is the radius of the circumcircle of the triangle. [edit] Law of cosines The law of cosines (also known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles: c^2=a^2+b^2-2ab\cos C ,\, or equivalently: \cos C=\frac{a^2+b^2-c^2}{2ab}\, [edit] Law of tangents The law of tangents: \frac{a+b}{a-b}=\frac{\tan\left[\tfrac{1}{2}(A+ \right]}{\tan\left[\tfrac{1}{2}(A- \right]} [edit] See also * Uses of trigonometry * trigonometric functions * List of basic trigonometry topics * Trigonometric identity * Trigonometry in Galois fields * List of triangle topics [edit] References 1. ^ trigonometry. Online Etymology Dictionary. 2. ^ Christopher M. Linton (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy. Cambridge University Press. 3. ^ http://www.lankalibrary.com/phpBB/viewt ... 821eded404 [edit] External links Find more information on Trigonometry by searching Wikipedia's sister projects Dictionary definitions from Wiktionary Textbooks from Wikibooks Quotations from Wikiquote Source texts from Wikisource Images and media from Commons News stories from Wikinews Learning resources from Wikiversity * Trigonometric Delights, by Eli Maor, Princeton University Press, 1998. Ebook version, in PDF format, full text presented. * Trigonometry by Alfred Monroe Kenyon and Louis Ingold, The Macmillan Company, 1914. In images, full text presented. * Trigonometry on PlainMath.net Trigonometry Articles from PlainMath.Net * Trigonometry on Mathwords.com index of trigonometry entries on Mathwords.com * Benjamin Banneker's Trigonometry Puzzle at Convergence * Trigonometry
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I am the dreamer
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| The Dreamer |
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Dreaming
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There are four unifying principles of biology[citation needed]:
* Cell theory. All living organisms are made of at least one cell, the basic unit of function in all organisms. In addition, the core mechanisms and chemistry of all cells in all organisms are similar, and cells emerge only from preexisting cells that multiply through cell division.
* Evolution. Through natural selection and genetic drift, a population's inherited traits change from generation to generation.
* Gene theory. A living organism's traits are encoded in DNA, the fundamental component of genes. In addition, traits are passed on from one generation to the next by way of these genes. All information flows from the genotype to the phenotype, the observable physical or biochemical characteristics of the organism. Although the phenotype expressed by the gene may adapt to the environment of the organism, that information is not transferred back to the genes. Only through the process of evolution do genes change in response to the environment.
* Homeostasis. The physiological processes that allow an organism to maintain its internal environment notwithstanding its external environment.
[edit] Cell theory
Main article: Cell theory
The cell is the fundamental unit of life. Cell theory states that all living things are composed of one or more cells, or the secreted products of those cells, for example, shell and bone. Cells arise from other cells through cell division, and in multicellular organisms, every cell in the organism's body is produced from a single cell in a fertilized egg. Furthermore, the cell is considered to be the basic part of the pathological processes of an organism.[2]
[edit] Evolution
Main article: Evolution
A central organizing concept in biology is that life changes and develops through evolution and that all lifeforms known have a common origin (see Common descent). This has led to the striking similarity of units and processes discussed in the previous section. Introduced into the scientific lexicon by Jean-Baptiste de Lamarck in 1809,Charles Darwin established evolution fifty years later as a viable theory by articulating its driving force, natural selection (Alfred Russel Wallace is recognized as the co-discoverer of this concept as he helped research and experiment with the concept of evolution). Darwin theorized that species and breeds developed through the processes of natural selection as well as by artificial selection or selective breeding[3]. Genetic drift was embraced as an additional mechanism of evolutionary development in the modern synthesis of the theory.
The evolutionary history of the species— which describes the characteristics of the various species from which it descended— together with its genealogical relationship to every other species is called its phylogeny. Widely varied approaches to biology generate information about phylogeny. These include the comparisons of DNA sequences conducted within molecular biology or genomics, and comparisons of fossils or other records of ancient organisms in paleontology. Biologists organize and analyze evolutionary relationships through various methods, including phylogenetics, phenetics, and cladistics. For a summary of major events in the evolution of life as currently understood by biologists, see evolutionary timeline.
Up into the 19th century, it was commonly believed that life forms could appear spontaneously under certain conditions (see spontaneous generation). This misconception was challenged by William Harvey's diction that "all life [is] from [an] egg" (from the Latin "Omne vivum ex ovo"), a foundational concept of modern biology. It simply means that there is an unbroken continuity of life from its initial origin to the present time.
A group of organisms shares a common descent if they share a common ancestor. All organisms on the Earth both living and extinct have been or are descended from a common ancestor or an ancestral gene pool. This last universal common ancestor of all organisms is believed to have appeared about 3.5 billion years ago. Biologists generally regard the universality of the genetic code as definitive evidence in favor of the theory of universal common descent (UCD) for all bacteria, archaea, and eukaryotes (see: origin of life).
Evolution does not always give rise to progressively more complex organisms. For example, the process of dysgenics has been observed among the human population.[4]
[edit] Gene theory
Schematic representation of DNA, the primary genetic material.
Schematic representation of DNA, the primary genetic material.
Main article: Gene
Biological form and function are created from and passed on to the next generation by genes, which are the primary units of inheritance. Physiological adaption to an organism's environment cannot be coded into its genes and cannot be inherited by its offspring (see Lamarckism). Remarkably, widely different organisms, including bacteria, plants, animals, and fungi, all share the same basic machinery that copies and transcribes DNA into proteins. For example, bacteria with inserted human DNA will correctly yield the corresponding human protein.
The total complement of genes in an organism or cell is known as its genome which is stored on one or more chromosomes. A chromosome is a single, long DNA strand on which thousands of genes, depending on the organism, are encoded. When a gene is active, the DNA code is transcribed into an RNA copy of the gene's information. A ribosome then translates the RNA into a structural protein or catalytic protein.
[edit] Homeostasis
Main article: Homeostasis
Homeostasis is the ability of an open system to regulate its internal environment to maintain a stable condition by means of multiple dynamic equilibrium adjustments controlled by interrelated regulation mechanisms. All living organisms, whether unicellular or multicellular, exhibit homeostasis. Homeostasis exists at the cellular level, for example cells maintain a stable internal acidity (pH); and at the level of the organism, for example warm-blooded animals maintain a constant internal body temperature. Homeostasis is a term that is also used in association with ecosystems, for example, the atmospheric concentration of carbon dioxide on Earth has been regulated by the concentration of plant life on Earth because plants remove more carbon dioxide from the atmosphere during the daylight hours than they emit to the atmosphere at night. Tissues and organs can also maintain homeostasis.
See also: Health.
[edit] Research
Main article: List of biology disciplines
[edit] Structural
Schematic of typical animal cell depicting the various organelles and structures.
Schematic of typical animal cell depicting the various organelles and structures.
Main articles: Molecular biology, Cell biology, Genetics, and Developmental biology
Molecular biology is the study of biology at a molecular level. This field overlaps with other areas of biology, particularly with genetics and biochemistry. Molecular biology chiefly concerns itself with understanding the interactions between the various systems of a cell, including the interrelationship of DNA, RNA, and protein synthesis and learning how these interactions are regulated.
Cell biology studies the physiological properties of cells, as well as their behaviors, interactions, and environment. This is done both on a microscopic and molecular level. Cell biology researches both single-celled organisms like bacteria and specialized cells in multicellular organisms like humans.
Understanding cell composition and how they function is fundamental to all of the biological sciences. Appreciating the similarities and differences between cell types is particularly important in the fields of cell and molecular biology. These fundamental similarities and differences provide a unifying theme, allowing the principles learned from studying one cell type to be extrapolated and generalized to other cell types.
Genetics is the science of genes, heredity, and the variation of organisms. Genes encode the information necessary for synthesizing proteins, which in turn play a large role in influencing (though, in many instances, not completely determining) the final phenotype of the organism. In modern research, genetics provides important tools in the investigation of the function of a particular gene, or the analysis of genetic interactions. Within organisms, genetic information generally is carried in chromosomes, where it is represented in the chemical structure of particular DNA molecules.
Developmental biology studies the process by which organisms grow and develop. Originating in embryology, modern developmental biology studies the genetic control of cell growth, differentiation, and "morphogenesis," which is the process that gives rise to tissues, organs, and anatomy. Model organisms for developmental biology include the round worm Caenorhabditis elegans, the fruit fly Drosophila melanogaster, the zebrafish Brachydanio rerio, the mouse Mus musculus, and the weed Arabidopsis thaliana.
[edit] Physiological
Main articles: Physiology and Anatomy
Physiology studies the mechanical, physical, and biochemical processes of living organisms by attempting to understand how all of the structures function as a whole. The theme of "structure to function" is central to biology. Physiological studies have traditionally been divided into plant physiology and animal physiology, but the principles of physiology are universal, no matter what particular organism is being studied. For example, what is learned about the physiology of yeast cells can also apply to human cells. The field of animal physiology extends the tools and methods of human physiology to non-human species. Plant physiology also borrows techniques from both fields.
Anatomy is an important branch of physiology and considers how organ systems in animals, such as the nervous, immune, endocrine, respiratory, and circulatory systems, function and interact. The study of these systems is shared with medically oriented disciplines such as neurology and immunology.
[edit] Evolution
In population genetics the evolution of a population of organisms is sometimes depicted as if travelling on a fitness landscape. The arrows indicate the preferred flow of a population on the landscape, and the points A, B, and C are local optima. The red ball indicates a population that moves from a very low fitness value to the top of a peak.
In population genetics the evolution of a population of organisms is sometimes depicted as if travelling on a fitness landscape. The arrows indicate the preferred flow of a population on the landscape, and the points A, B, and C are local optima. The red ball indicates a population that moves from a very low fitness value to the top of a peak.
Main articles: Evolutionary biology, Evolution, Evolutionary synthesis, and Natural selection
Evolution is concerned with the origin and descent of species, as well as their change over time, and includes scientists from many taxonomically-oriented disciplines. For example, it generally involves scientists who have special training in particular organisms such as mammalogy, ornithology, botany, or herpetology, but use those organisms as systems to answer general questions about evolution. Evolutionary biology is mainly based on paleontology, which uses the fossil record to answer questions about the mode and tempo of evolution, as well as the developments in areas such as population genetics and evolutionary theory. In the 1980s, developmental biology re-entered evolutionary biology from its initial exclusion from the modern synthesis through the study of evolutionary developmental biology. Related fields which are often considered part of evolutionary biology are phylogenetics, systematics, and taxonomy.
Up into the 19th century, it was believed that life forms were being continuously created under certain conditions (see spontaneous generation). This misconception was challenged by William Harvey's diction that "all life [is] from [an] egg" (from the Latin "Omne vivum ex ovo"), a foundational concept of modern biology. It simply means that there is an unbroken continuity of life from its initial origin to the present time.
A group of organisms shares a common descent if they share a common ancestor. All organisms on the Earth have been and are descended from a common ancestor or an ancestral gene pool. This last universal common ancestor of all organisms is believed to have appeared about 3.5 billion years ago. Biologists generally regard the universality of the genetic code as definitive evidence in favor of the theory of universal common descent (UCD) for all bacteria, archaea, and eukaryotes (see: origin of life).
The two major traditional taxonomically-oriented disciplines are botany and zoology. Botany is the scientific study of plants. Botany covers a wide range of scientific disciplines that study the growth, reproduction, metabolism, development, diseases, and evolution of plant life. Zoology involves the study of animals, including the study of their physiology within the fields of anatomy and embryology. The common genetic and developmental mechanisms of animals and plants is studied in molecular biology, molecular genetics, and developmental biology. The ecology of animals is covered under behavioral ecology and other fields.[5]
[edit] Taxonomy
A phylogenetic tree of all living things, based on rRNA gene data, showing the separation of the three domains bacteria, archaea, and eukaryotes as described initially by Carl Woese. Trees constructed with other genes are generally similar, although they may place some early-branching groups very differently, presumably owing to rapid rRNA evolution. The exact relationships of the three domains are still being debated.
A phylogenetic tree of all living things, based on rRNA gene data, showing the separation of the three domains bacteria, archaea, and eukaryotes as described initially by Carl Woese. Trees constructed with other genes are generally similar, although they may place some early-branching groups very differently, presumably owing to rapid rRNA evolution. The exact relationships of the three domains are still being debated.
Main article: Taxonomy
Classification is the province of the disciplines of systematics and taxonomy. Taxonomy places organisms in groups called taxa, while systematics seeks to define their relationships with each other. This classification technique has evolved to reflect advances in cladistics and genetics, shifting the focus from physical similarities and shared characteristics to phylogenetics.
Traditionally, living things have been divided into five kingdoms:[6]
Monera -- Protista -- Fungi -- Plantae -- Animalia
However, many scientists now consider this five-kingdom system to be outdated. Modern alternative classification systems generally begin with the three-domain system:[7]
Archaea (originally Archaebacteria) -- Bacteria (originally Eubacteria) -- Eukarya
These domains reflect whether the cells have nuclei or not, as well as differences in the cell exteriors.
Further, each kingdom is broken down continuously until each species is separately classified. The order is:
1. Domain
2. Kingdom
3. Phylum
4. Class
5. Order
6. Family
7. Genus
8. Species
The scientific name of an organism is obtained from its genus and species. For example, humans would be listed as Homo sapiens. Homo would be the genus and sapiens is the species. Whenever writing the scientific name of an organism, it is proper to capitalize the first letter in the genus and put all of the species in lowercase; in addition the entire term would be put in italics or underlined. The term used for classification is called taxonomy.
There is also a series of intracellular parasites that are progressively "less alive" in terms of metabolic activity:
Viruses -- Viroids -- Prions
The dominant classification system is called Linnaean taxonomy, which includes ranks and binomial nomenclature. How organisms are named is governed by international agreements such as the International Code of Botanical Nomenclature (ICBN), the International Code of Zoological Nomenclature (ICZN), and the International Code of Nomenclature of Bacteria (ICNB). A fourth Draft BioCode was published in 1997 in an attempt to standardize naming in these three areas, but it has yet to be formally adopted. The Virus International Code of Virus Classification and Nomenclature (ICVCN) remains outside the BioCode.
[edit] Environmental
Main articles: Ecology, Ethology, Behavior, and Biogeography
Ecology studies the distribution and abundance of living organisms, and the interactions between organisms and their environment. The environment of an organism includes both its habitat, which can be described as the sum of local abiotic factors such as climate and ecology, as well as the other the organisms that share its habitat. Ecological systems are studied at several different levels, from individuals and populations to ecosystems and the biosphere. As can be surmised, ecology is a science that draws on several disciplines.
Ethology studies animal behavior (particularly of social animals such as primates and canids), and is sometimes considered a branch of zoology. Ethologists have been particularly concerned with the evolution of behavior and the understanding of behavior in terms of the theory of natural selection. In one sense, the first modern ethologist was Charles Darwin, whose book "The Expression of the Emotions in Man and Animals" influenced many ethologists.
Biogeography studies the spatial distribution of organisms on the Earth, focusing on topics like plate tectonics, climate change, dispersal and migration, and cladistics.
Every living thing interacts with other organisms and its environment. One reason that biological systems can be difficult to study is that so many different interactions with other organisms and the environment are possible, even on the smallest of scales. A microscopic bacterium responding to a local sugar gradient is responding to its environment as much as a lion is responding to its environment when it searches for food in the African savannah. For any given species, behaviors can be co-operative, aggressive, parasitic or symbiotic. Matters become more complex when two or more different species interact in an ecosystem. Studies of this type are the province of ecology.
[edit] History
Main articles: History of biology and History of medicine
Although the concept of biology as a single coherent field arose in the 19th century, the biological sciences emerged from traditions of medicine and natural history reaching back to Galen and Aristotle in ancient Greece. During the Renaissance and early modern period, biological thought was revolutionized by a renewed interest in empiricism and the discovery of many novel organisms. Prominent in this movement were Vesalius and Harvey, who used experimentation and careful observation in physiology, and naturalists such as Linnaeus and Buffon who began to classify the diversity of life and the fossil record, as well as the development and behavior of organisms. Microscopy revealed the previously unknown world of microorganisms, laying the groundwork for cell theory. The growing importance of natural theology, partly a response to the rise of mechanical philosophy, encouraged the growth of natural history.[8][9]
Over the 18th and 19th centuries, biological sciences such as botany and zoology became increasingly professional scientific disciplines. Lavoisier and other physical scientists began to connect the animate and inanimate worlds through physics and chemistry. Explorer-naturalists such as Alexander von Humboldt investigated the interaction between organisms and their environment, and the ways this relationship depends on geography—laying the foundations for biogeography, ecology and ethology. Naturalists began to reject essentialism and consider the importance of extinction and the mutability of species. Cell theory provided a new perspective on the fundamental basis of life. These developments, as well as the results from embryology and paleontology, were synthesized in Charles Darwin's theory of evolution by natural selection. The end of the 19th century saw the fall of spontaneous generation and the rise of the germ theory of disease, though the mechanism of inheritance remained a mystery.[5][10][8]
In the early 20th century, the rediscovery of Mendel's work led to the rapid development of genetics by Thomas Hunt Morgan and his students, and by the 1930s the combination of population genetics and natural selection in the "neo-Darwinian synthesis". New disciplines developed rapidly, especially after Watson and Crick proposed the structure of DNA. Following the establishment of the Central Dogma and the cracking of the genetic code, biology was largely split between organismal biology—the fields that deal with whole organisms and groups of organisms—and the fields related to cellular and molecular biology. By the late 20th century, new fields like genomics and proteomics were reversing this trend, with organismal biologists using molecular techniques, and molecular and cell biologists investigating the interplay between genes and the environment, as well as the genetics of natural populations of organisms.[11][12][13][14]
[edit] See also
* List of biology topics
* List of basic biology topics
* List of biologists
[show]
v • d • e
Topics related to biology
People and history Biologist - Notable biologists - History of biology - Nobel Prize in Physiology or Medicine - Timeline of biology and organic chemistry - List of geneticists and biochemists
Institutions, publications Bachelor of Science - Publications
Terms and phrases Omne vivum ex ovo - In vivo - In vitro - In utero - In silico
Related disciplines Medicine (Physician) - Physical anthropology - Environmental science - Life Sciences - Biotechnology
Other List of conservation topics - Altricial and Precocial development strategies
[edit] References
1. ^ King, TJ & Roberts, MBV (1986). Biology: A Functional Approach. Thomas Nelson and Sons. ISBN 978-0174480358.
2. ^ Mazzarello, P (1999). "A unifying concept: the history of cell theory". Nature Cell Biology 1: E13-E15. doi:10.1038/8964.
3. ^ Darwin, Charles (1859). On the Origin of Species, 1st, John Murray
4. ^ Lynn, Richard; Van Court, Marilyn (2004). "New evidence of dysgenic fertility for intelligence in the United States". Intelligence 32 (2): p. 193. Ablex Pub.. ISSN 0160-2896.
5. ^ a b Futuyma, DJ (2005). Evolution. Sinauer Associates. ISBN 978-0878931873.
6. ^ Margulis, L; Schwartz, KV (1997). Five Kingdoms: An Illustrated Guide to the Phyla of Life on Earth, 3rd edition, WH Freeman & Co. ISBN 978-0716731832.
7. ^ Woese C, Kandler O, Wheelis M (1990). "Towards a natural system of organisms: proposal for the domains Archaea, Bacteria, and Eukarya.". Proc Natl Acad Sci U S A 87 (12): 4576-9. ISSN 0027-8424. PMID 2112744.
8. ^ a b Mayr, E (1985). The Growth of Biological Thought. Belknap Press. ISBN 978-0674364462.
9. ^ Magner, LN (2002). A History of the Life Sciences. TF-CRC. ISBN 978-0824708245.
10. ^ Coleman, W (1978). Biology in the Nineteenth Century: Problems of Form, Function and Transformation. Cambridge University Press. ISBN 978-0521292931.
11. ^ Allen, GE (1978). Life Science in the Twentieth Century. Cambridge University Press. ISBN 978-0521292962.
12. ^ Fruton, JS (1999). Proteins, Enzymes, Genes: The Interplay of Chemistry and Biology. Yale University Press. ISBN 978-0300076080.
13. ^ Morange, M & Cobb, M (2000). A History of Molecular Biology. Harvard University Press. ISBN 978-0674001695.
14. ^ Smocovitis, VB (1996). Unifying Biology. Princeton University Press. ISBN 978-0691033433.
[edit] Further reading
* Alberts, Bruce; Johnson, A, Lewis, J, Raff, M, Roberts, K & Walter, P (2002). Molecular Biology of the Cell, 4th edition, Garland. ISBN 978-0815332183.
* Begon, Michael; Townsend, CR & Harper, JL (2005). Ecology: From Individuals to Ecosystems, 4th edition, Blackwell Publishing Limited. ISBN 978-1405111171.
* Campbell, Neil (2004). Biology, 7th edition, Benjamin-Cummings Publishing Company. ISBN 0-8053-7146-X.
* Colinvaux, Paul (1979). Why Big Fierce Animals are Rare: An Ecologist's Perspective, reissue edition, Princeton University Press. ISBN 0691023646.
* Hoagland, Mahlon (2001). The Way Life Works, reprint edition, Jones and Bartlett Publishers inc. ISBN 076371688X.
* Janovy, John Jr. (2004). On Becoming a Biologist, 2nd edition, Bison Books. ISBN 0803276206.
* Johnson, George B. (2005). Biology, Visualizing Life. Holt, Rinehart, and Winston. ISBN 0-03-016723-X.
[edit] External links
Wikibooks
Wikibooks has more on the topic of
Biology
Look up Biology in
Wiktionary, the free dictionary.
Wikiversity
At Wikiversity you can learn more about Biology at:
The School of Biology
Biology Portal
* The Dolan DNA Learning Center: The source for timely information about your life
* OSU's Phylocode
* The Tree of Life: A multi-authored, distributed Internet project containing information about phylogeny and biodiversity.
* MIT video lecture series on biology
* A wiki site for protocol sharing run from MIT.
* Biology and Bioethics.
* Biology online wiki dictionary.
* Biology Video Sharing Community.
[edit] Journal links
* PLos Biology A peer-reviewed, open-access journal published by the Public Library of Science
* International Journal of Biological Sciences A biological journal publishes peer-reviewed scientific papers of significance
* Perspectives in Biology and Medicine
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Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".[2] Other practitioners of mathematics maintain that mathematics is the science of pattern, and that mathematicians seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere.[3][4] Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.[5]
Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Knowledge and use of basic mathematics have always been an inherent and integral part of individual and group life. Refinements of the basic ideas are visible in mathematical texts originating in ancient Egypt, Mesopotamia, ancient India, ancient China, and ancient Greece. Rigorous arguments appear in Euclid's Elements. The development continued in fitful bursts until the Renaissance period of the 16th century, when mathematical innovations interacted with new scientific discoveries, leading to an acceleration in research that continues to the present day.[6]
Today, mathematics is used throughout the world in many fields, including natural science, engineering, medicine, and the social sciences such as economics. Applied mathematics, the application of mathematics to such fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although applications for what began as pure mathematics are often discovered later.[7]
Contents
[hide]
* 1 Etymology
* 2 History
* 3 Inspiration, pure and applied mathematics, and aesthetics
* 4 Notation, language, and rigor
* 5 Mathematics as science
* 6 Fields of mathematics
o 6.1 Quantity
o 6.2 Structure
o 6.3 Space
o 6.4 Change
o 6.5 Foundations and philosophy
o 6.6 Discrete mathematics
o 6.7 Applied mathematics
* 7 Common misconceptions
o 7.1 Mathematics and physical reality
* 8 See also
* 9 Notes
* 10 References
* 11 External links
Etymology
The word "mathematics" (Greek: μαθηματικά or mathēmatiká) comes from the Greek μάθημα (máthēma), which means learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times. Its adjective is μαθηματικός (mathēmatikós), related to learning, or studious, which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), in Latin ars mathematica, meant the mathematical art.
The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle, and meaning roughly "all things mathematical".[8] In English, however, the noun mathematics takes singular verb forms. It is often shortened to math in English-speaking North America and maths elsewhere.[dubious – discuss]
History
A quipu, a counting device used by the Inca.
A quipu, a counting device used by the Inca.
Main article: History of mathematics
The evolution of mathematics might be seen as 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 have something in common was a breakthrough in human thought. In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like time — days, seasons, years. Arithmetic (addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to knowledge of geometry.
Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called quipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus.
Mayan numerals
Mayan numerals
From the beginnings of recorded history, the major disciplines within mathematics arose out of the need to do calculations relating to 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 has since been greatly 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 proofs."[9]
Inspiration, pure and applied mathematics, and aesthetics
Sir Isaac Newton (1643-1727), an inventor of infinitesimal calculus.
Sir Isaac Newton (1643-1727), an inventor of infinitesimal calculus.
Main article: Mathematical beauty
Mathematics 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 was one of the infinitesimal calculus inventors, although nearly all of the notation used in infinitesimal calculus was contributed by Leibniz with the exception of a dot above a variable to signify differentiation with respect to time. Feynman invented the 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. Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science.
For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant 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. Mathematicians often strive to find proofs of theorems that are particularly elegant, a quest Paul Erdős often referred to as finding proofs from "The Book" in which God had written down his favorite proofs. The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.
Notation, language, and rigor
The infinity symbol ∞ in several typefaces.
The infinity symbol ∞ in several typefaces.
Main article: Mathematical notation
Most of the mathematical notation in use today was not invented until the 16th century.[10] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. In the 18th century, Euler was responsible for many of the notations in use today. 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 syntax and encodes information that would be difficult to write in any other way.
Mathematical language also is hard for beginners. Words such as or and only have more precise meanings than in everyday speech. Also confusing to beginners, words such as open and field have been given specialized mathematical meanings. Mathematical jargon includes technical terms such as homeomorphism and integrable. 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.[11] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at 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 large computations are hard to verify, such proofs may not be sufficiently rigorous.[12] 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 impossible. 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.[13]
Mathematics as science
Carl Friedrich Gauss, himself known as the "prince of mathematicians", referred to mathematics as "the Queen of the Sciences".
Carl Friedrich Gauss, himself known as the "prince of mathematicians", referred to mathematics as "the Queen of the Sciences".
Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".[14] In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means (field of) knowledge. Indeed, this is also the original meaning in English, and there is no doubt that mathematics is in this sense a science. The specialization restricting the meaning to natural science is of later date. If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. Albert Einstein has stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[15]
Many philosophers believe that mathematics is not experimentally falsifiable,[citation needed] and thus not a science according to the definition of Karl Popper. However, in the 1930s important work in mathematical logic showed that mathematics cannot be reduced to logic, and Karl Popper concluded that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."[16] Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself.
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.[17] 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. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the scientific method. In his 2002 book A New Kind of Science, Stephen Wolfram argues that computational mathematics deserves to be explored empirically as a scientific field in its own right.
The opinions of mathematicians on this matter are varied. Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics.
Mathematical awards are generally kept separate from their equivalents in science. The most prestigious award in mathematics is the Fields Medal,[18][19] established in 1936 and now awarded every 4 years. It is often considered, misleadingly, the equivalent of science's Nobel Prizes. The Wolf Prize in Mathematics, instituted in 1979, recognizes lifetime achievement, and another major international award, the Abel Prize, was introduced in 2003. These are awarded for a particular body of work, which may be innovation, or resolution of an outstanding problem in an established field. A famous list of 23 such open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Solution of each of these problems carries a $1 million reward, and only one (the Riemann hypothesis) is duplicated in Hilbert's problems.
Fields of mathematics
An abacus, a simple calculating tool used since ancient times
An abacus, a simple calculating tool used since ancient times
As 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), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty.
Quantity
The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, whence such popular results as Fermat's last theorem. Number theory also holds two widely-considered unsolved problems: the twin prime conjecture and Goldbach's conjecture.
As the number system is further developed, the integers are recognized as a subset of the rational numbers ("fractions"). These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalized to complex numbers. These are the first steps of a hierarchy of numbers that goes on to include quarternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of counting to infinity. Another area of study is size, which leads to the cardinal numbers and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets.
1, 2, 3\,\! -2, -1, 0, 1, 2\,\! -2, \frac{2}{3}, 1.21\,\! -e, \sqrt{2}, 3, \pi\,\! 2, i, -2+3i, 2e^{i\frac{4\pi}{3}}\,\!
Natural numbers Integers Rational numbers Real numbers Complex numbers
Structure
Many mathematical objects, such as sets of numbers and functions, exhibit internal structure. The structural properties of these objects are investigated in the study of groups, rings, fields and other abstract systems, which are themselves such objects. This is the field of abstract algebra. An important concept here is that 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. Vector calculus expands the field into a fourth fundamental area, that of change.
Number theory Abstract algebra Group theory Order theory
Space
The study of space originates with geometry - in particular, Euclidean geometry. Trigonometry combines space and numbers, and encompasses the well-known Pythagorean theorem. 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 and 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 have been the greatest growth area in 20th century mathematics, and includes the long-standing Poincaré conjecture and the controversial four color theorem, whose only proof, by computer, has never been verified by a human.
Geometry Trigonometry Differential geometry Topology Fractal geometry
Change
Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and real-valued functions is known as real analysis, with complex analysis the equivalent field for the complex numbers. The Riemann hypothesis, one of the most fundamental open questions in mathematics, is drawn from complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. 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 behavior.
Calculus Vector calculus Differential equations Dynamical systems Chaos theory
Foundations and philosophy
In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed, as well as category theory which is still in development.
Mathematical logic is concerned with setting mathematics on a rigid axiomatic framework, and studying the results of such a framework. As such, it is home to Gödel's second incompleteness theorem, perhaps the most widely celebrated result in logic, which (informally) implies that any formal system that contains basic arithmetic, if sound (meaning that all theorems that can be proven are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system). Gödel showed how to construct, whatever the given collection of number-theoretical axioms, a formal statement in the logic that is a true number-theoretical fact, but which does not follow from those axioms. Therefore no formal system is a true axiomatization of full number theory. Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science.
p \Rightarrow q \,
Mathematical logic Set theory Category theory
Discrete mathematics
Discrete mathematics is the common name for the fields of mathematics most generally useful in theoretical computer science. This includes computability theory, computational complexity theory, and information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most powerful known model - the Turing machine. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with rapid advance of computer hardware. Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence concepts such as compression and entropy.
As a relatively new field, discrete mathematics has a number of fundamental open problems. The most famous of these is the "P=NP?" problem, one of the Millennium Prize Problems.[20]
\begin{matrix} (1,2,3) & (1,3,2) \\ (2,1,3) & (2,3,1) \\ (3,1,2) & (3,2,1) \end{matrix}
Combinatorics Theory of computation Cryptography Graph theory
Applied mathematics
Applied mathematics considers the use of abstract mathematical tools in solving concrete problems in the sciences, business, and other areas. 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. Most experiments, surveys and observational studies require the informed use of statistics. (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 too large for human numerical capacity; it includes the study of rounding errors or other sources of error in computation.
Mathematical physics
Mathematical fluid dynamics
Numerical analysis
Optimization
Probability
Statistics
Financial mathematics
Game theory
Common misconceptions
Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Mathematicians publish many thousands of papers embodying new discoveries in mathematics every month.
Mathematics is not numerology, nor is it accountancy; nor is it restricted to arithmetic.
Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
* misunderstanding of the implications of mathematical rigor;
* attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, often in the belief that the journal is biased against the author;
* lack of familiarity with, and therefore underestimation of, the existing literature.
The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne.
Mathematics and physical reality
Mathematical concepts and theorems need not correspond to anything in the physical world. Insofar as a correspondence does exist, while mathematicians and physicists may select axioms and postulates that seem reasonable and intuitive, it is not necessary for the basic assumptions within an axiomatic system to be true in an empirical or physical sense. Thus, while many systems of axioms are derived from our perceptions and experiments, they are not dependent on them.
For example, we could say that the physical concept of two apples may be accurately modeled by the natural number 2. On the other hand, we could also say that the natural numbers are not an accurate model because there is no standard "unit" apple and no two apples are exactly alike. The modeling idea is further complicated by the possibility of fractional or partial apples. So while it may be instructive to visualize the axiomatic definition of the natural numbers as collections of apples, the definition itself is not dependent upon nor derived from any actual physical entities.
Nevertheless, mathematics remains extremely useful for solving real-world problems. This fact led physicist Eugene Wigner to write an article titled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".
See also
Mathematics Portal
* List of basic mathematics topics
* Lists of mathematics topics
* Mathematics portal
* Philosophy of mathematics
* Mathematics education
* Mathematical game
* Mathematical model
* Mathematical problem
* Mathematics competitions
* Dyscalculia
Notes
1. ^ No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art depends on the artist's imagination (see Euclid).
2. ^ Peirce, p.97
3. ^ Steen, L.A. (April 29, 1988). The Science of Patterns. Science, 240: 611–616. and summarized at Association for Supervision and Curriculum Development.
4. ^ Devlin, Keith, Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Library) 1996, ISBN 9780716750475
5. ^ Jourdain
6. ^ Eves
7. ^ Peterson
8. ^ The Oxford Dictionary of English Etymology, Oxford English Dictionary
9. ^ Sevryuk
10. ^ Earliest Uses of Various Mathematical Symbols (Contains many further references)
11. ^ See false proof for simple examples of what can go wrong in a formal proof. The history of the Four Color Theorem contains examples of false proofs accepted by other mathematicians.
12. ^ Ivars Peterson, The Mathematical Tourist, Freeman, 1988, ISBN 0-7167-1953-3. p. 4 "A few complain that the computer program can't be verified properly," (in reference to the Haken-Apple proof of the Four Color Theorem).
13. ^ Patrick Suppes, Axiomatic Set Theory, Dover, 1972, ISBN 0-486-61630-4. p. 1, "Among the many branches of modern mathematics set theory occupies a unique place: with a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects."
14. ^ Waltershausen
15. ^ Einstein, p. 28. The quote is Einstein's answer to the question: "how can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?" He, too, is concerned with The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
16. ^ Popper 1995, p. 56
17. ^ Ziman
18. ^ "The Fields Medal is now indisputably the best known and most influential award in mathematics." Monastyrsky
19. ^ Riehm
20. ^ Clay Mathematics Institute P=NP
References
* Benson, Donald C., The Moment of Proof: Mathematical Epiphanies, Oxford University Press, USA; New Ed edition (December 14, 2000). ISBN 0-19-513919-4.
* Boyer, Carl B., A History of Mathematics, Wiley; 2 edition (March 6, 1991). ISBN 0-471-54397-7. — A concise history of mathematics from the Concept of Number to contemporary Mathematics.
* Courant, R. and H. Robbins, What Is Mathematics? : An Elementary Approach to Ideas and Methods, Oxford University Press, USA; 2 edition (July 18, 1996). ISBN 0-19-510519-2.
* Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Mariner Books; Reprint edition (January 14, 1999). ISBN 0-395-92968-7.— A gentle introduction to the world of mathematics.
* Einstein, Albert (1923). "Sidelights on Relativity (Geometry and Experience)". P. Dutton., Co.
* Eves, Howard, An Introduction to the History of Mathematics, Sixth Edition, Saunders, 1990, ISBN 0-03-029558-0.
* Gullberg, Jan, Mathematics—From the Birth of Numbers. W. W. Norton & Company; 1st edition (October 1997). ISBN 0-393-04002-X. — An encyclopedic overview of mathematics presented in clear, simple language.
* Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. — A translated and expanded version of a Soviet mathematics encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM, and online [1].
* Jourdain, Philip E. B., The Nature of Mathematics, in The World of Mathematics, James R. Newman, editor, Dover, 2003, ISBN 0-486-43268-8.
* Kline, Morris, Mathematical Thought from Ancient to Modern Times, Oxford University Press, USA; Paperback edition (March 1, 1990). ISBN 0-19-506135-7.
* Monastyrsky, Michael (2001). "Some Trends in Modern Mathematics and the Fields Medal". Canadian Mathematical Society. Retrieved on 2006-07-28.
* Oxford English Dictionary, second edition, ed. John Simpson and Edmund Weiner, Clarendon Press, 1989, ISBN 0-19-861186-2.
* The Oxford Dictionary of English Etymology, 1983 reprint. ISBN 0-19-861112-9.
* Pappas, Theoni, The Joy Of Mathematics, Wide World Publishing; Revised edition (June 1989). ISBN 0-933174-65-9.
* Peirce, Benjamin. "Linear Associative Algebra". American Journal of Mathematics (Vol. 4, No. 1/4. (1881). JSTOR.
* Peterson, Ivars, Mathematical Tourist, New and Updated Snapshots of Modern Mathematics, Owl Books, 2001, ISBN 0-8050-7159-8.
* Paulos, John Allen (1996). A Mathematician Reads the Newspaper. Anchor. ISBN 0-385-48254-X.
* Popper, Karl R. (1995). "On knowledge", In Search of a Better World: Lectures and Essays from Thirty Years. Routledge. ISBN 0-415-13548-6.
* Riehm, Carl (August 2002). "The Early History of the Fields Medal". Notices of the AMS 49 (7): 778-782. AMS.
* Sevryuk, Mikhail B. (January 2006). "Book Reviews" (PDF). Bulletin of the American Mathematical Society 43 (1): 101-109. Retrieved on 2006-06-24.
* Waltershausen, Wolfgang Sartorius von (1856, repr. 1965). Gauss zum Gedächtniss. Sändig Reprint Verlag H. R. Wohlwend. ISBN 3-253-01702-8.
* Ziman, J.M., F.R.S. (1968). "Public Knowledge:An essay concerning the social dimension of science".
External links
Find more information on Mathematics by searching Wikipedia's sister projects
Dictionary definitions from Wiktionary
Textbooks from Wikibooks
Quotations from Wikiquote
Source texts from Wikisource
Images and media from Commons
News stories from Wikinews
Learning resources from Wikiversity
Wikiversity
At Wikiversity you can learn more and teach others about Mathematics at:
The School of Mathematics
* Online Encyclopaedia of Mathematics [2] from Springer. Graduate-level reference work with over 8,000 entries, illuminating nearly 50,000 notions in mathematics.
* Some mathematics applets, at MIT
* Rusin, Dave: The Mathematical Atlas. A guided tour through the various branches of modern mathematics. (Can also be found here.)
* Stefanov, Alexandre: Textbooks in Mathematics. A list of free online textbooks and lecture notes in mathematics.
* Weisstein, Eric et al.: MathWorld: World of Mathematics. An online encyclopedia of mathematics.
* Polyanin, Andrei: EqWorld: The World of Mathematical Equations. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
* Planet Math. An online mathematics encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
* Metamath. A site and a language, that formalize mathematics from its foundations.
* Mathematician Biographies. The MacTutor History of Mathematics archive Extensive history and quotes from all famous mathematicians.
* Cain, George: Online Mathematics Textbooks available free online.
* Math & Logic: The history of formal mathematical, logical, linguistic and methodological ideas. In The Dictionary of the History of Ideas.
* Nrich, a prize-winning site for students from age five from Cambridge University
* 'FreeScience Library->Mathematics ' The mathematics section of FreeScience library
* Open Problem Garden, a wiki of open problems in mathematics
* Applications of High School Algebra
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