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Complex number

Complex number
A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i is the imaginary unit which satisfies i2 = −1. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation x2 = −1, that is, i2 = −1.[1] In this expression, a is the real part and b is the imaginary part of the complex number. Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane (also called Argand plane) by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. Overview[edit] Complex numbers allow for solutions to certain equations that have no solutions in real numbers. Definition[edit] . or or z*. and .

Related:  Mathematics

Fractional quantum Hall effect The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2D electrons shows precisely quantised plateaus at fractional values of . It is a property of a collective state in which electrons bind magnetic flux lines to make new quasiparticles, and excitations have a fractional elementary charge and possibly also fractional statistics. The 1998 Nobel Prize in Physics was awarded to Robert Laughlin, Horst Störmer and Daniel Tsui for the discovery and explanation of the fractional Hall effect.[1] However, Laughlin's explanation was a phenomenological guess and only applies to fillings where is an odd integer. Fundamental theorem of algebra The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with zero imaginary part. History[edit]

Circuit Theory/Phasor Analysis Phasor Analysis[edit] The mathematical representations of individual circuit elements can be converted into phasor notation, and then the circuit can be solved using phasors. Resistance, Impedance and Admittance[edit] In phasor notation, resistance, capacitance, and inductance can all be lumped together into a single term called "impedance". The phasor used for impedance is

Mathematical induction Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. Mathematical induction is a method of mathematical proof typically used to establish a given statement for all natural numbers. It is done in two steps. The first step, known as the base case, is to prove the given statement for the first natural number.

Cosine The cosine function is one of the basic functions encountered in trigonometry (the others being the cosecant, cotangent, secant, sine, and tangent). Let Polar coordinate system Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 degrees, or (3,60°). In blue, the point (4,210°). History[edit]

Phasor An example of series RLC circuit and respective phasor diagram for a specific Glossing over some mathematical details, the phasor transform can also be seen as a particular case of the Laplace transform, which additionally can be used to (simultaneously) derive the transient response of an RLC circuit.[10][8] However, the Laplace transform is mathematically more difficult to apply and the effort may be unjustified if only steady state analysis is required.[10] Definition[edit] Matrix (mathematics) Each element of a matrix is often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. Applications of matrices are found in most scientific fields.

Number A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (telephone numbers), for ordering (serial numbers), and for codes (e.g., ISBNs). Classification of numbers[edit] Sine For the angle α, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse. The sine function graphed on the Cartesian plane. In this graph, the angle x is given in radians (π = 180°). The sine and cosine functions are related in multiple ways. The derivative of Mathematics Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers),[2] structure,[3] space,[2] and change.[4][5][6] There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.[7][8] Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.[11]

Related:  ALGEBRA