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Imaginary Number
An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . For example, is an imaginary number, and its square is . By definition, zero is considered to be both real and imaginary. Originally coined in the 17th century by René Descartes as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler (in the 18th century) and Augustin-Louis Cauchy and Carl Friedrich Gauss (in the early 19th century). An imaginary number can be added to a real number to form a complex number of the form , where the real numbers and are called, respectively, the ''real part'' and the ''imaginary part'' of the complex number. History Although the Greek mathematician and engineer Hero of Alexandria is noted as the first to present a calculati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Of Unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly th roots of unity, except when is a multiple of the (positive) characteristic of the field. General definition An ''th root of unity'', where is a positive integer, is a number satisfying the equation :z^n = 1. Unless otherwise specified, the roots of unity may be taken to be complex number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Rowan Hamilton
Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Ireland, living at Dunsink Observatory. Hamilton's scientific career included the study of geometrical optics, ideas from Fourier analysis, and his work on quaternions which made him one of the founders of modern linear algebra. He made major contributions in optics, classical mechanics and abstract algebra. His work was fundamental to modern theoretical physics, particularly his reformulation of Newtonian mechanics, now called Hamiltonian mechanics. It is now central both to electromagnetism and to quantum mechanics. Early life Hamilton was the fourth of nine children born to Sarah Hutton (1780–1817) and Archibald Hamilton (1778–1819),Bruno (2003) who lived in Dublin at 29 Dominick Street, later renumbered to 36. Hamilton's father, who ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negative Number
In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset. If a quantity, such as the charge on an electron, may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as ''positive'' and ''negative''. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic. For example, −(−3) = 3 because the opposite of an opposite is the original value. Negative numbers are usually written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of three, and is pronounced "min ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because . Every nonnegative real number has a unique nonnegative square root, called the ''principal square root'', which is denoted by \sqrt, where the symbol \sqrt is called the '' radical sign'' or ''radix''. For example, to express the fact that the principal square root of 9 is 3, we write \sqrt = 3. The term (or number) whose square root is being considered is known as the ''radicand''. The radicand is the number or expression underneath the radical sign, in this case 9. For nonnegative , the principal square root can also be written in exponent notation, as . Every positive number has two square roots: \sqrt, which is positive, and -\sqrt, which is negative. The two roots can be written more concisely using the ± sign as \plusmn\sq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positive real number. For example, 4 has two square roots: 2 and −2; of these the positive root, 2, is considered the principal root and is denoted as \sqrt. Motivation Consider the complex logarithm function log ''z''. It is defined as the complex number ''w'' such that :e^w = z. Now, for example, say we wish to find log ''i''. This means we want to solve :e^w = i for ''w''. Clearly ''i''π/2 is a solution. But is it the only solution? Of course, there are other solutions, which is evidenced by considering the position of ''i'' in the complex plane and in particular its argument arg ''i''. We can rotate counterclockwise π/2 radians from 1 to reach ''i'' initially, but if we rotate further another 2π we reach ''i'' again. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton. Its first book was a new 1912 edition of John Witherspoon's ''Lectures on Moral Philosophy.'' History Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two already existing local publishers, that of the ''Princeton Alumni Weekly'' and the Princeton Press. The new press printed both local newspapers, university documents, ''The Daily Princetonian'', and later added book publishing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Argument (complex Analysis)
In mathematics (particularly in complex analysis), the argument of a complex number ''z'', denoted arg(''z''), is the angle between the positive real axis and the line joining the origin and ''z'', represented as a point in the complex plane, shown as \varphi in Figure 1. It is a multi-valued function operating on the nonzero complex numbers. To define a single-valued function, the principal value of the argument (sometimes denoted Arg ''z'') is used. It is often chosen to be the unique value of the argument that lies within the interval . Definition An argument of the complex number , denoted , is defined in two equivalent ways: #Geometrically, in the complex plane, as the 2D polar angle \varphi from the positive real axis to the vector representing . The numeric value is given by the angle in radians, and is positive if measured counterclockwise. #Algebraically, as any real quantity \varphi such that z = r (\cos \varphi + i \sin \varphi) = r e^ for some positive r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional object has an infinite number of possible central axes and rotational directions. If the rotation axis passes internally through the body's own center of mass, then the body is said to be ''autorotating'' or '' spinning'', and the surface intersection of the axis can be called a '' pole''. A rotation around a completely external axis, e.g. the planet Earth around the Sun, is called ''revolving'' or ''orbiting'', typically when it is produced by gravity, and the ends of the rotation axis can be called the '' orbital poles''. Mathematics Mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed. This definition applies to rotations within both two and three dimensions (in a plane and in spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Wiley & Sons
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in print and electronically, as well as online products and services, training materials, and educational materials for undergraduate, graduate, and continuing education students. History The company was established in 1807 when Charles Wiley opened a print shop in Manhattan. The company was the publisher of 19th century American literary figures like James Fenimore Cooper, Washington Irving, Herman Melville, and Edgar Allan Poe, as well as of legal, religious, and other non-fiction titles. The firm took its current name in 1865. Wiley later shifted its focus to scientific, technical, and engineering subject areas, abandoning its literary interests. Wiley's son John (born in Flatbush, New York, October 4, 1808; died in East Orang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a point. The integers are often shown as specially-marked points evenly spaced on the line. Although the image only shows the integers from –3 to 3, the line includes all real numbers, continuing forever in each direction, and also numbers that are between the integers. It is often used as an aid in teaching simple addition and subtraction, especially involving negative numbers. In advanced mathematics, the number line can be called as a real line or real number line, formally defined as the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space (or affine space), a metric space, a topological space, a measure space, or a linear continuum. Just li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can be defined between two lines (or two line segments), between a line and a plane, and between two planes. Perpendicularity is one particular instance of the more general mathematical concept of '' orthogonality''; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its '' normal vector''. Definitions A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one sid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
