1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
2.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
3.
Statistics
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Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e. g. a scientific, industrial, or social problem, populations can be diverse topics such as all people living in a country or every atom composing a crystal. Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys, statistician Sir Arthur Lyon Bowley defines statistics as Numerical statements of facts in any department of inquiry placed in relation to each other. When census data cannot be collected, statisticians collect data by developing specific experiment designs, representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. In contrast, an observational study does not involve experimental manipulation, inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. A standard statistical procedure involves the test of the relationship between two data sets, or a data set and a synthetic data drawn from idealized model. A hypothesis is proposed for the relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the hypothesis is done using statistical tests that quantify the sense in which the null can be proven false. Working from a hypothesis, two basic forms of error are recognized, Type I errors and Type II errors. Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis, measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random or systematic, the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics continues to be an area of research, for example on the problem of how to analyze Big data. Statistics is a body of science that pertains to the collection, analysis, interpretation or explanation. Some consider statistics to be a mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty, mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. In applying statistics to a problem, it is practice to start with a population or process to be studied. Populations can be diverse topics such as all living in a country or every atom composing a crystal. Ideally, statisticians compile data about the entire population and this may be organized by governmental statistical institutes
4.
Probability
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Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1, the higher the probability of an event, the more certain that the event will occur. A simple example is the tossing of a fair coin, since the coin is unbiased, the two outcomes are both equally probable, the probability of head equals the probability of tail. Since no other outcomes are possible, the probability is 1/2 and this type of probability is also called a priori probability. Probability theory is used to describe the underlying mechanics and regularities of complex systems. For example, tossing a coin twice will yield head-head, head-tail, tail-head. The probability of getting an outcome of head-head is 1 out of 4 outcomes or 1/4 or 0.25 and this interpretation considers probability to be the relative frequency in the long run of outcomes. A modification of this is propensity probability, which interprets probability as the tendency of some experiment to yield a certain outcome, subjectivists assign numbers per subjective probability, i. e. as a degree of belief. The degree of belief has been interpreted as, the price at which you would buy or sell a bet that pays 1 unit of utility if E,0 if not E. The most popular version of subjective probability is Bayesian probability, which includes expert knowledge as well as data to produce probabilities. The expert knowledge is represented by some prior probability distribution and these data are incorporated in a likelihood function. The product of the prior and the likelihood, normalized, results in a probability distribution that incorporates all the information known to date. The scientific study of probability is a development of mathematics. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, there are reasons of course, for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the study of probability. According to Richard Jeffrey, Before the middle of the century, the term probable meant approvable. A probable action or opinion was one such as people would undertake or hold. However, in legal contexts especially, probable could also apply to propositions for which there was good evidence, the sixteenth century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes
5.
Units of measurement
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A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same quantity. Any other value of quantity can be expressed as a simple multiple of the unit of measurement. For example, length is a physical quantity, the metre is a unit of length that represents a definite predetermined length. When we say 10 metres, we actually mean 10 times the definite predetermined length called metre, the definition, agreement, and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day. Different systems of units used to be very common, now there is a global standard, the International System of Units, the modern form of the metric system. In trade, weights and measures is often a subject of regulation, to ensure fairness. The International Bureau of Weights and Measures is tasked with ensuring worldwide uniformity of measurements, metrology is the science for developing nationally and internationally accepted units of weights and measures. In physics and metrology, units are standards for measurement of quantities that need clear definitions to be useful. Reproducibility of experimental results is central to the scientific method, a standard system of units facilitates this. Scientific systems of units are a refinement of the concept of weights, science, medicine, and engineering often use larger and smaller units of measurement than those used in everyday life and indicate them more precisely. The judicious selection of the units of measurement can aid researchers in problem solving, in the social sciences, there are no standard units of measurement and the theory and practice of measurement is studied in psychometrics and the theory of conjoint measurement. A unit of measurement is a quantity of a physical property. Units of measurement were among the earliest tools invented by humans, primitive societies needed rudimentary measures for many tasks, constructing dwellings of an appropriate size and shape, fashioning clothing, or bartering food or raw materials. Weights and measures are mentioned in the Bible and it is a commandment to be honest and have fair measures. As of the 21st Century, multiple unit systems are used all over the world such as the United States Customary System, the British Customary System, however, the United States is the only industrialized country that has not yet completely converted to the Metric System. The systematic effort to develop an acceptable system of units dates back to 1790 when the French National Assembly charged the French Academy of Sciences to come up such a unit system. After this treaty was signed, a General Conference of Weights, the CGPM produced the current SI system which was adopted in 1954 at the 10th conference of weights and measures. Currently, the United States is a society which uses both the SI system and the US Customary system
6.
Artifact (error)
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In natural science and signal processing, an artifact is any error in the perception or representation of any information introduced by the involved equipment or technique. In computer science, digital artifacts are introduced into digital signals as a result of digital processing. In microscopy, artifacts are introduced during the processing of samples into slide form. Such an artifact may be called a statistical artifact, physicians typically learn to recognize some of these artifacts to avoid mistaking them for actual pathology. In medical electrophysiological monitoring, artifacts are anomalous signals that originate from some other than the electrophysiological structure being studied. Offending artifacts may obscure, distort, or completely misrepresent the true underlying electrophysiological signal sought, in radar signal processing, some echoes can be related to fix targets, multipath returns, jamming, atmospheric effect, anomalous propagation, and many other effects. Visual artifact, in imaging, any unwanted visual alteration introduced by the imaging equipment, compression artifact, in computer graphics, distortion of media by the data compression
7.
Maxima and minima
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Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. The value of the function at a point is called the maximum value of the function. Similarly, the function has a minimum point at x∗ if f ≤ f for all x in X within distance ε of x∗. A similar definition can be used when X is a space, since the definition just given can be rephrased in terms of neighbourhoods. Note that a maximum point is always a local maximum point. In both the global and local cases, the concept of a strict extremum can be defined, note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points. A continuous real-valued function with a compact domain always has a maximum point, an important example is a function whose domain is a closed interval of real numbers. Finding global maxima and minima is the goal of mathematical optimization, if a function is continuous on a closed interval, then by the extreme value theorem global maxima and minima exist. Furthermore, a global maximum either must be a maximum in the interior of the domain. So a method of finding a maximum is to look at all the local maxima in the interior, and also look at the maxima of the points on the boundary. Local extrema of functions can be found by Fermats theorem. For any function that is defined piecewise, one finds a maximum by finding the maximum of each piece separately, the function x2 has a unique global minimum at x =0. The function x3 has no global minima or maxima, although the first derivative is 0 at x =0, this is an inflection point. The function x x has a global maximum at x = e. The function x−x has a global maximum over the positive real numbers at x = 1/e. The function x3/3 − x has first derivative x2 −1, setting the first derivative to 0 and solving for x gives stationary points at −1 and +1. From the sign of the second derivative we can see that −1 is a local maximum, note that this function has no global maximum or minimum
8.
Meanness
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Meanness is a personal quality whose classical form, discussed by many from Aristotle to Thomas Aquinas, characterizes it as a vice of lowness, but whose modern form deals more with cruelty. These, in particular the one, largely summarize the aspects of the classical definition of meanness that have been propounded by philosophers, Aristotelian and otherwise. It is the deficiency of giving to or the excess of taking from others, the other vice is prodigality, which Aristotle describes as both less common than meanness and less of a vice. Meanness can take many forms, as there are ways in which one can deviate from the liberal/generous virtue. Aristotle said that meanness we always impute to those who care more than they ought for wealth, the translation of what Aristotle meant is not without problems. Aristotles actual word ἐλευθερία corresponds with both liberality and freedom and its opposite, ανἐλευθερία, is both meanness and servility. Philosophers such as Susanne E. Foster have attempted to reconcile these by explaining that the connotation is that meanness is enslavement by ones material possessions. A slave has no possessions to give and is not the recipient of gifts from others, whereas a person, in a gift-exchanging culture, can give. Meanness, therefore, is the vice of a person behaving, in such a culture. Another problem that has taxed philosophers analysing Aristotle is that the continuum is from excess to deficiency. Philosophers have attempted to explain this, and characterize what Aristotle was trying to say as an excess or a deficiency in some other form, howard J. Exactly which vice of the two that brackets a virtue is excess and which is deficiency is entirely arbitrary. Curzer also contends that Aristotles statement that meanness is worse than prodigality is a mistake, in his Summa Theologica, Thomas Aquinas explicitly cited Aristotle and held meanness — parvificentia in his original Latin — to be the opposite of magnificence. Whilst a magnificent person is willing to pay for projects and good causes. Theodore Parker, like Aristotle, opposed meanness and generosity, declaring meanness as an unmanly and unwomanly vice. He differentiated two forms of meanness, pecuniary meanness, which is meanness in things measurable with money, and meanness of behaviour, which is meanness in things not thus measurable. George Crabb described meanness as hatever a man does in common with those below him and he considered meanness to be only relatively bad, as what may constitute meanness for one person may be generosity or prudence for another. Seeking to save or to gain for oneself at the expense of others, henry Sidgwick in The Methods of Ethics held meanness to be both the opposite of liberality and generosity. A mean person chooses a trifling gain to himself rather than the avoidance of disappointment to others, samuel Johnson wrote in The False Alarm that n infallible characteristic of meanness is cruelty
9.
Speed
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In everyday use and in kinematics, the speed of an object is the magnitude of its velocity, it is thus a scalar quantity. Speed has the dimensions of distance divided by time, the SI unit of speed is the metre per second, but the most common unit of speed in everyday usage is the kilometre per hour or, in the US and the UK, miles per hour. For air and marine travel the knot is commonly used, the fastest possible speed at which energy or information can travel, according to special relativity, is the speed of light in a vacuum c =299792458 metres per second. Matter cannot quite reach the speed of light, as this would require an amount of energy. In relativity physics, the concept of rapidity replaces the classical idea of speed, italian physicist Galileo Galilei is usually credited with being the first to measure speed by considering the distance covered and the time it takes. Galileo defined speed as the distance covered per unit of time, in equation form, this is v = d t, where v is speed, d is distance, and t is time. A cyclist who covers 30 metres in a time of 2 seconds, objects in motion often have variations in speed. If s is the length of the path travelled until time t, in the special case where the velocity is constant, this can be simplified to v = s / t. The average speed over a time interval is the total distance travelled divided by the time duration. Speed at some instant, or assumed constant during a short period of time, is called instantaneous speed. By looking at a speedometer, one can read the speed of a car at any instant. A car travelling at 50 km/h generally goes for less than one hour at a constant speed, if the vehicle continued at that speed for half an hour, it would cover half that distance. If it continued for one minute, it would cover about 833 m. Different from instantaneous speed, average speed is defined as the distance covered divided by the time interval. For example, if a distance of 80 kilometres is driven in 1 hour, likewise, if 320 kilometres are travelled in 4 hours, the average speed is also 80 kilometres per hour. When a distance in kilometres is divided by a time in hours, average speed does not describe the speed variations that may have taken place during shorter time intervals, and so average speed is often quite different from a value of instantaneous speed. If the average speed and the time of travel are known, using this equation for an average speed of 80 kilometres per hour on a 4-hour trip, the distance covered is found to be 320 kilometres. Linear speed is the distance travelled per unit of time, while speed is the linear speed of something moving along a circular path
10.
Sampling (statistics)
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Two advantages of sampling are that the cost is lower and data collection is faster than measuring the entire population. Each observation measures one or more properties of observable bodies distinguished as independent objects or individuals, in survey sampling, weights can be applied to the data to adjust for the sample design, particularly stratified sampling. Results from probability theory and statistical theory are employed to guide the practice, in business and medical research, sampling is widely used for gathering information about a population. Acceptance sampling is used to determine if a lot of material meets the governing specifications. In sampling, this includes defining the population from which our sample is drawn, a population can be defined as including all people or items with the characteristic one wishes to understand. Because there is rarely enough time or money to gather information from everyone or everything in a population. Sometimes what defines a population is obvious, in this case, the batch is the population. Although the population of interest often consists of objects, sometimes we need to sample over time, space. For the time dimension, the focus may be on periods or discrete occasions, in other cases, our population may be even less tangible. For example, Joseph Jagger studied the behaviour of roulette wheels at a casino in Monte Carlo, in this case, the population Jagger wanted to investigate was the overall behaviour of the wheel, while his sample was formed from observed results from that wheel. Similar considerations arise when taking repeated measurements of some characteristic such as the electrical conductivity of copper. This situation often arises when we seek knowledge about the system of which the observed population is an outcome. In such cases, sampling theory may treat the population as a sample from a larger superpopulation. Here the superpopulation is everybody in the country, given access to this treatment – a group which does not yet exist, note also that the population from which the sample is drawn may not be the same as the population about which we actually want information. Often there is large but not complete overlap between two groups due to frame issues etc. However, in the general case this is not usually possible or practical. There is no way to identify all rats in the set of all rats, where voting is not compulsory, there is no way to identify which people will actually vote at a forthcoming election. These imprecise populations are not amenable to sampling in any of the ways below, as a remedy, we seek a sampling frame which has the property that we can identify every single element and include any in our sample
11.
Exponential distribution
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It is a particular case of the gamma distribution. It is the analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson processes, the probability density function of an exponential distribution is f = { λ e − λ x x ≥0,0 x <0. Alternatively, this can be defined using the right-continuous Heaviside step function, H where H=1, f = λ e − λ x H Here λ >0 is the parameter of the distribution, the distribution is supported on the interval [0, ∞). If a random variable X has this distribution, we write X ~ Exp, the exponential distribution exhibits infinite divisibility. The cumulative distribution function is given by F = {1 − e − λ x x ≥0,0 x <0. Where β >0 is mean, standard deviation, and scale parameter of the distribution and that is to say, the expected duration of survival of the system is β units of time. The parametrization involving the rate parameter arises in the context of events arriving at a rate λ, the alternative specification is sometimes more convenient than the one given above, and some authors will use it as a standard definition. This alternative specification is not used here, unfortunately this gives rise to a notational ambiguity. An example of this switch, reference uses λ for β. The mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by E =1 λ = β, see above. In light of the examples given above, this sense, if you receive phone calls at an average rate of 2 per hour. The variance of X is given by Var =1 λ2 = β2, the moments of X, for n =1,2. are given by E = n. The median of X is given by m = ln λ < E , where ln refers to the natural logarithm. Thus the absolute difference between the mean and median is | E − m | =1 − ln λ <1 λ = standard deviation, an exponentially distributed random variable T obeys the relation Pr = Pr, ∀ s, t ≥0. The exponential distribution and the distribution are the only memoryless probability distributions. The exponential distribution is also necessarily the only continuous probability distribution that has a constant Failure rate. The quantile function for Exp is F −1 = − ln λ,0 ≤ p <1 The quartiles are therefore, first quartile, ln/λ median, ln/λ third quartile, ln/λ And as a consequence the interquartile range is ln/λ
12.
Median
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The median is the value separating the higher half of a data sample, a population, or a probability distribution, from the lower half. In simple terms, it may be thought of as the value of a data set. For example, in the set, the median is 6. The median is a commonly used measure of the properties of a set in statistics. The basic advantage of the median in describing data compared to the mean is that it is not skewed so much by extremely large or small values, and so it may give a better idea of a typical value. For example, in understanding statistics like household income or assets which vary greatly, Median income, for example, may be a better way to suggest what a typical income is. The median of a finite list of numbers can be found by arranging all the numbers from smallest to greatest, if there is an odd number of numbers, the middle one is picked. For example, consider the set of numbers,1,3,3,6,7,8,9 This set contains seven numbers, the median is the fourth of them, which is 6. If there are a number of observations, then there is no single middle value. For example, in the set,1,2,3,4,5,6,8,9 The median is the mean of the middle two numbers, this is ÷2, which is 4.5. The formula used to find the number of a data set of n numbers is ÷2. This either gives the number or the halfway point between the two middle values. For example, with 14 values, the formula will give 7.5, and you will also be able to find the median using the Stem-and-Leaf Plot. There is no accepted standard notation for the median. In any of these cases, the use of these or other symbols for the needs to be explicitly defined when they are introduced. The median is used primarily for skewed distributions, which it summarizes differently from the arithmetic mean, the median is 2 in this case, and it might be seen as a better indication of central tendency than the arithmetic mean of 4. The widely cited empirical relationship between the locations of the mean and the median for skewed distributions is, however. There are, however, various relationships for the difference between them, see below
![{\displaystyle (4\times 36\times 45\times 50\times 75)^{\frac {1}{5}}={\sqrt[{5}]{24\;300\;000}}=30.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b516046ef2a7b8b23301b7ab228cec73f38e062)
