Summation in the context of Operation (mathematics)

The national average salary (or national average wage) is the mean salary for the working population of a nation. It is calculated by summing all the annual salaries of all persons in work (surveyed) and dividing the total by the number of workers (surveyed). It is not the same as the Gross domestic product (GDP) per capita, which is calculated by dividing the GDP by the total population of a country, including the unemployed and those not in the workforce (e.g. retired people, children, students, etc.). It can be useful in understanding economic conditions, and to employers and employees in negotiating salaries. The national median salary is usually significantly less than the national average salary because the distribution of workers by salary is skewed.

Addition, usually denoted with the plus sign +, is one of the four basic operations of arithmetic, the other three being subtraction, multiplication, and division. The addition of two whole numbers results in the total or sum of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples. This observation is expressed as "3 + 2 = 5", which is read as "three plus two equals five".

Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers, and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, and elements of additive groups.

In mathematics, an integral is the continuous analog of a sum, and is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter.

A definite integral computes the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function; in this case, they are also called indefinite integrals. The fundamental theorem of calculus relates definite integration to differentiation and provides a method to compute the definite integral of a function when its antiderivative is known; differentiation and integration are inverse operations.

In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance.

Among the Ancient Greeks, the idea that a potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes. Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the quadrature of the parabola. The mathematical side of Zeno's paradoxes was resolved using the concept of a limit during the 17th century, especially through the early calculus of Isaac Newton. The resolution was made more rigorous and further improved in the 19th century through the work of Carl Friedrich Gauss and Augustin-Louis Cauchy, among others, answering questions about which of these sums exist via the completeness of the real numbers and whether series terms can be rearranged or not without changing their sums using absolute convergence and conditional convergence of series.

In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence ${\displaystyle (a_{1},a_{2},a_{3},\ldots )}$ defines a series S that is denoted

The nth partial sum S_n is the sum of the first n terms of the sequence; that is,

Leonhard Euler (/ˈɔɪlər/ OY-lər; 15 April 1707 – 18 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia.

Euler is credited for popularizing the Greek letter ${\displaystyle \pi }$ (lowercase pi) to denote the ratio of a circle's circumference to its diameter, as well as first using the notation ${\displaystyle f(x)}$ for the value of a function, the letter ${\displaystyle i}$ to express the imaginary unit ${\displaystyle {\sqrt {-1}}}$, the Greek letter ${\displaystyle \Sigma }$ (capital sigma) to express summations, the Greek letter ${\displaystyle \Delta }$ (capital delta) for finite differences, and lowercase letters to represent the sides of a triangle while representing the angles as capital letters. He gave the current definition of the constant ${\displaystyle e}$, the base of the natural logarithm, now known as Euler's number. Euler made contributions to applied mathematics and engineering, such as his study of ships, which helped navigation; his three volumes on optics, which contributed to the design of microscopes and telescopes; and his studies of beam bending and column critical loads.

In a Euclidean space, the sum of angles of a triangle equals a straight angle (180 degrees, π radians, two right angles, or a half-turn). A triangle has three angles, and has one at each vertex, bounded by a pair of adjacent sides.

The sum can be computed directly using the definition of angle based on the dot product and trigonometric identities, or more quickly by reducing to the two-dimensional case and using Euler's identity.

Sigma (/ˈsɪɡmə/ SIG-mə; uppercase Σ, lowercase σ, lowercase in word-final position ς; Ancient Greek: σίγμα) is the eighteenth letter of the Greek alphabet. When used at the end of a letter-case word (one that does not use all caps), the final form (ς) is used. In Ὀδυσσεύς (Odysseus), for example, the two lowercase sigmas (σ) in the center of the name are distinct from the word-final sigma (ς) at the end.

In the system of Greek numerals, sigma has a value of 200. In general mathematics, uppercase Σ is used as an operator for summation. The Latin letter S derives from sigma while the Cyrillic letter Es derives from a lunate form of this letter.

In signal processing, the power spectrum ${\displaystyle S_{xx}(f)}$ of a continuous time signal ${\displaystyle x(t)}$ describes the distribution of power into frequency components ${\displaystyle f}$ composing that signal. Fourier analysis shows that any physical signal can be decomposed into a distribution of frequencies over a continuous range, where some of the power may be concentrated at discrete frequencies. The statistical average of the energy or power of any type of signal (including noise) as analyzed in terms of its frequency content, is called its spectral density.

When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the energy spectral density. More commonly used is the power spectral density (PSD, or simply power spectrum), which applies to signals existing over all time, or over a time period large enough (especially in relation to the duration of a measurement) that it could as well have been over an infinite time interval. The PSD then refers to the spectral power distribution that would be found, since the total energy of such a signal over all time would generally be infinite. Summation or integration of the spectral components yields the total power (for a physical process) or variance (in a statistical process), identical to what would be obtained by integrating ${\displaystyle x^{2}(t)}$ over the time domain, as dictated by Parseval's theorem.

A mixing console or mixing desk is an electronic device for mixing audio signals, used in sound recording and reproduction and sound reinforcement systems. Inputs to the console include microphones, signals from electric or electronic instruments, or recorded sounds. Mixers may control analog or digital signals. The modified signals are summed to produce the combined output signals, which can then be broadcast, amplified through a sound reinforcement system or recorded.

Mixing consoles are used for applications including recording studios, public address systems, sound reinforcement systems, nightclubs, broadcasting, and post-production. A typical, simple application combines signals from microphones on stage into an amplifier that drives one set of loudspeakers for the audience. A DJ mixer may have only two channels, for mixing two record players. A coffeehouse's small stage might only have a six-channel mixer, enough for two singer-guitarists and a percussionist. A nightclub stage's mixer for rock music shows may have 24 channels for mixing the signals from a rhythm section, lead guitar and several vocalists. A mixing console in a professional recording studio may have as many as 96 channels. Consoles used for live sound can go even higher, with some having up to 384 input channels.

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ with ⁠${\displaystyle a>b>0}$⁠, ⁠${\displaystyle a}$⁠ is in a golden ratio to ⁠${\displaystyle b}$⁠ if $${\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi ,}$$ where the Greek letter phi (⁠${\displaystyle \varphi }$⁠ or ⁠${\displaystyle \phi }$⁠) denotes the golden ratio. The constant ⁠${\displaystyle \varphi }$⁠ satisfies the quadratic equation ⁠${\displaystyle \textstyle \varphi ^{2}=\varphi +1}$⁠ and is an irrational number with a value of${\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}={}}$1.618033988749....The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli; it also goes by other names.

Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of ⁠${\displaystyle \varphi }$⁠—may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as financial markets, in some cases based on dubious fits to data. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.