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Plus–minus Sign
The plus–minus sign or plus-or-minus sign () and the complementary minus-or-plus sign () are symbols with broadly similar multiple meanings. *In mathematics, the sign generally indicates a choice of exactly two possible values, one of which is obtained through addition and the other through subtraction. *In statistics and experimental sciences, the sign commonly indicates the confidence interval or uncertainty bounding a range of possible errors in a measurement, often the standard deviation or standard error. The sign may also represent an inclusive range of values that a reading might have. *In chess, the sign indicates a clear advantage for the white player; the complementary minus-plus sign () indicates a clear advantage for the black player. Other meanings occur in other fields, including medicine, engineering, chemistry, electronics, linguistics, and philosophy. History A version of the sign, including also the French word ''ou'' ("or"), was used in its mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Albert Girard
Albert Girard () (11 October 1595 in Saint-Mihiel, France − 8 December 1632 in Leiden, Dutch Republic) was a French-born mathematician. He studied at the University of Leiden. He "had early thoughts on the fundamental theorem of algebra" and gave the inductive definition for the Fibonacci numbers. He was the first to use the abbreviations 'sin', 'cos' and 'tan' for the trigonometric functions in a treatise. Girard was the first to state, in 1625, that each prime of the form 1 mod 4 is the sum of two squares. (See Fermat's theorem on sums of two squares.) It was said that he was quiet-natured and, unlike most mathematicians, did not keep a journal for his personal life. In the opinion of Charles Hutton, Girard was ...the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation. This had previously b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Normal Distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac e^\,. The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma^2 is the variance. The standard deviation of the distribution is (sigma). A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Margin Of Error
The margin of error is a statistic expressing the amount of random sampling error in the results of a Statistical survey, survey. The larger the margin of error, the less confidence one should have that a poll result would reflect the result of a simultaneous census of the entire Statistical population, population. The margin of error will be positive whenever a population is incompletely sampled and the outcome measure has positive variance, which is to say, whenever the measure ''varies''. The term ''margin of error'' is often used in non-survey contexts to indicate observational error in reporting measured quantities. Concept Consider a simple ''yes/no'' poll P as a sample of n respondents drawn from a population N \text(n \ll N) reporting the percentage p of ''yes'' responses. We would like to know how close p is to the true result of a survey of the entire population N, without having to conduct one. If, hypothetically, we were to conduct a poll P over subsequent sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Engineering Tolerance
Engineering tolerance is the permissible limit or limits of variation in: # a physical dimension; # a measured value or physical property of a material, manufactured object, system, or service; # other measured values (such as temperature, humidity, etc.); # in engineering and safety, a physical distance or space (tolerance), as in a truck (lorry), train or boat under a bridge as well as a train in a tunnel (see structure gauge and loading gauge); # in mechanical engineering, the space between a bolt and a nut or a hole, etc. Dimensions, properties, or conditions may have some variation without significantly affecting functioning of systems, machines, structures, etc. A variation beyond the tolerance (for example, a temperature that is too hot or too cold) is said to be noncompliant, rejected, or exceeding the tolerance. Considerations when setting tolerances A primary concern is to determine how wide the tolerances may be without affecting other factors or the outcome of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Square Number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of Figurate number, figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). In the Real number, real number system, square numbers are non-negative. A non-negative integer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Conjugate (square Roots)
In mathematics, the conjugate of an expression of the form a + b \sqrt d is a - b \sqrt d, provided that \sqrt d does not appear in and . One says also that the two expressions are conjugate. In particular, the two solutions of a quadratic equation are conjugate, as per the \pm in the quadratic formula x = \frac. Complex conjugation is the special case where the square root is i = \sqrt, the imaginary unit. Properties As (a + b \sqrt d)(a - b \sqrt d) = a^2 - b^2 d and (a + b \sqrt d) + (a - b \sqrt d) = 2a, the sum and the product of conjugate expressions do not involve the square root anymore. This property is used for removing a square root from a denominator, by multiplying the numerator and the denominator of a fraction by the conjugate of the denominator (see Rationalisation). An example of this usage is: \frac = \frac = \frac. Hence: \frac = \frac{a^2 - db^2}. A corollary property is that the subtraction: :(a+b\sqrt d) - (a-b\sqrt d)= 2b\sqrt d, leaves only a term c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Trigonometric Identity
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle. These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity. Pythagorean identities The basic relationship between the sine and cosine is given by the Pythagorean identity: \sin^2\theta + \cos^2\theta = 1, where \sin^2 \theta means ^2 and \cos^2 \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Quadratic Equation
In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and then the equation is linear equation, linear, not quadratic.) The numbers , , and are the ''coefficients'' of the equation and may be distinguished by respectively calling them, the ''quadratic coefficient'', the ''linear coefficient'' and the ''constant coefficient'' or ''free term''. The values of that satisfy the equation are called ''solution (mathematics), solutions'' of the equation, and ''zero of a function, roots'' or ''zero of a function, zeros'' of the quadratic function on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a double root. If all the coefficients are real numbers, there are either two real solutions, or a single real double root, or two comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Quadratic Formula
In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic equations, such as completing the square, yield the same solutions. Given a general quadratic equation of the form , with representing an unknown, and coefficients , , and representing known real number, real or complex number, complex numbers with , the values of satisfying the equation, called the Zero of a function, ''roots'' or ''zeros'', can be found using the quadratic formula, x = \frac, where the plus–minus sign, plus–minus symbol "" indicates that the equation has two roots. Written separately, these are: x_1 = \frac, \qquad x_2 = \frac. The quantity is known as the discriminant of the quadratic equation. If the coefficients , , and are real numbers then when , the equation has two distinct real number, real roots; when , the equation has one repeated root, repeated real root; and when , the equation h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Plus And Minus Signs
The plus sign () and the minus sign () are Glossary of mathematical symbols, mathematical symbols used to denote sign (mathematics), positive and sign (mathematics), negative functions, respectively. In addition, the symbol represents the operation of addition, which results in a Sum (mathematics), sum, while the symbol represents subtraction, resulting in a difference (mathematics), difference. Their use has been extended to many other meanings, more or less analogous. and are Latin terms meaning 'more' and 'less', respectively. The forms and are used in many countries around the world. Other designs include for plus and for minus. History Though the signs now seem as familiar as the alphabet or the Arabic numerals, they are not of great antiquity. The Egyptian hieroglyphic sign for addition, for example, resembles a pair of legs walking in the direction in which the text was written (Egyptian language, Egyptian could be written either from right to left or left to r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
