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More general distributions of matter or charge are obtained from this by convolution , giving the Poisson equation. The fundamental concept of storing programs in the computer's memory alongside the data they operate on is the crux of the von Neumann, or stored program [ citation needed ] , architecture. Seshu Aiyar, and B. Special cases include Coulomb's law and Newton's law of universal gravitation. By remembering where it was executing prior to the interrupt, the computer can return to that task later. These programs enable computers to perform an extremely wide range of tasks. See also Fagan, Garrett G.

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It could add or subtract times a second, a thousand times faster than any other machine. It also had modules to multiply, divide, and square root. High speed memory was limited to 20 words about 80 bytes. Built under the direction of John Mauchly and J. The machine was huge, weighing 30 tons, using kilowatts of electric power and contained over 18, vacuum tubes, 1, relays, and hundreds of thousands of resistors, capacitors, and inductors. The principle of the modern computer was proposed by Alan Turing in his seminal paper, [35] On Computable Numbers.

Turing proposed a simple device that he called "Universal Computing machine" and that is now known as a universal Turing machine. He proved that such a machine is capable of computing anything that is computable by executing instructions program stored on tape, allowing the machine to be programmable.

The fundamental concept of Turing's design is the stored program , where all the instructions for computing are stored in memory. Von Neumann acknowledged that the central concept of the modern computer was due to this paper. Except for the limitations imposed by their finite memory stores, modern computers are said to be Turing-complete , which is to say, they have algorithm execution capability equivalent to a universal Turing machine.

Early computing machines had fixed programs. Changing its function required the re-wiring and re-structuring of the machine. A stored-program computer includes by design an instruction set and can store in memory a set of instructions a program that details the computation. The theoretical basis for the stored-program computer was laid by Alan Turing in his paper.

In , Turing joined the National Physical Laboratory and began work on developing an electronic stored-program digital computer. His report "Proposed Electronic Calculator" was the first specification for such a device. The Manchester Baby was the world's first stored-program computer. The Mark 1 in turn quickly became the prototype for the Ferranti Mark 1 , the world's first commercially available general-purpose computer.

At least seven of these later machines were delivered between and , one of them to Shell labs in Amsterdam. The LEO I computer became operational in April [42] and ran the world's first regular routine office computer job.

The bipolar transistor was invented in From onwards transistors replaced vacuum tubes in computer designs, giving rise to the "second generation" of computers.

Compared to vacuum tubes, transistors have many advantages: Silicon junction transistors were much more reliable than vacuum tubes and had longer, indefinite, service life.

Transistorized computers could contain tens of thousands of binary logic circuits in a relatively compact space. At the University of Manchester , a team under the leadership of Tom Kilburn designed and built a machine using the newly developed transistors instead of valves. The next great advance in computing power came with the advent of the integrated circuit. The idea of the integrated circuit was first conceived by a radar scientist working for the Royal Radar Establishment of the Ministry of Defence , Geoffrey W.

Produced at Fairchild Semiconductor, it was made of silicon , whereas Kilby's chip was made of germanium. This new development heralded an explosion in the commercial and personal use of computers and led to the invention of the microprocessor. While the subject of exactly which device was the first microprocessor is contentious, partly due to lack of agreement on the exact definition of the term "microprocessor", it is largely undisputed that the first single-chip microprocessor was the Intel , [52] designed and realized by Ted Hoff , Federico Faggin , and Stanley Mazor at Intel.

With the continued miniaturization of computing resources, and advancements in portable battery life, portable computers grew in popularity in the s. These so-called smartphones and tablets run on a variety of operating systems and have become the dominant computing device on the market, with manufacturers reporting having shipped an estimated million devices in 2Q The term hardware covers all of those parts of a computer that are tangible physical objects.

Circuits, computer chips, graphic cards, sound cards, memory RAM , motherboard, displays, power supplies, cables, keyboards, printers and "mice" input devices are all hardware. A general purpose computer has four main components: These parts are interconnected by buses , often made of groups of wires.

Inside each of these parts are thousands to trillions of small electrical circuits which can be turned off or on by means of an electronic switch. Each circuit represents a bit binary digit of information so that when the circuit is on it represents a "1", and when off it represents a "0" in positive logic representation.

The circuits are arranged in logic gates so that one or more of the circuits may control the state of one or more of the other circuits. When unprocessed data is sent to the computer with the help of input devices, the data is processed and sent to output devices. The input devices may be hand-operated or automated.

The act of processing is mainly regulated by the CPU. Some examples of input devices are:. The means through which computer gives output are known as output devices.

Some examples of output devices are:. The control unit often called a control system or central controller manages the computer's various components; it reads and interprets decodes the program instructions, transforming them into control signals that activate other parts of the computer.

A key component common to all CPUs is the program counter , a special memory cell a register that keeps track of which location in memory the next instruction is to be read from. The control system's function is as follows—note that this is a simplified description, and some of these steps may be performed concurrently or in a different order depending on the type of CPU:. Since the program counter is conceptually just another set of memory cells, it can be changed by calculations done in the ALU.

Adding to the program counter would cause the next instruction to be read from a place locations further down the program. Instructions that modify the program counter are often known as "jumps" and allow for loops instructions that are repeated by the computer and often conditional instruction execution both examples of control flow.

The sequence of operations that the control unit goes through to process an instruction is in itself like a short computer program, and indeed, in some more complex CPU designs, there is another yet smaller computer called a microsequencer , which runs a microcode program that causes all of these events to happen. Early CPUs were composed of many separate components but since the mids CPUs have typically been constructed on a single integrated circuit called a microprocessor.

The ALU is capable of performing two classes of operations: Some can only operate on whole numbers integers while others use floating point to represent real numbers , albeit with limited precision.

However, any computer that is capable of performing just the simplest operations can be programmed to break down the more complex operations into simple steps that it can perform. Therefore, any computer can be programmed to perform any arithmetic operation—although it will take more time to do so if its ALU does not directly support the operation. An ALU may also compare numbers and return boolean truth values true or false depending on whether one is equal to, greater than or less than the other "is 64 greater than 65?

Logic operations involve Boolean logic: These can be useful for creating complicated conditional statements and processing boolean logic.

Superscalar computers may contain multiple ALUs, allowing them to process several instructions simultaneously. A computer's memory can be viewed as a list of cells into which numbers can be placed or read. Each cell has a numbered "address" and can store a single number. The computer can be instructed to "put the number into the cell numbered " or to "add the number that is in cell to the number that is in cell and put the answer into cell Letters, numbers, even computer instructions can be placed into memory with equal ease.

Since the CPU does not differentiate between different types of information, it is the software's responsibility to give significance to what the memory sees as nothing but a series of numbers. In almost all modern computers, each memory cell is set up to store binary numbers in groups of eight bits called a byte. To store larger numbers, several consecutive bytes may be used typically, two, four or eight. When negative numbers are required, they are usually stored in two's complement notation.

Other arrangements are possible, but are usually not seen outside of specialized applications or historical contexts. A computer can store any kind of information in memory if it can be represented numerically.

Modern computers have billions or even trillions of bytes of memory. The CPU contains a special set of memory cells called registers that can be read and written to much more rapidly than the main memory area.

There are typically between two and one hundred registers depending on the type of CPU. Registers are used for the most frequently needed data items to avoid having to access main memory every time data is needed. As data is constantly being worked on, reducing the need to access main memory which is often slow compared to the ALU and control units greatly increases the computer's speed. ROM is typically used to store the computer's initial start-up instructions.

In general, the contents of RAM are erased when the power to the computer is turned off, but ROM retains its data indefinitely. In embedded computers , which frequently do not have disk drives, all of the required software may be stored in ROM. Software stored in ROM is often called firmware , because it is notionally more like hardware than software.

Flash memory blurs the distinction between ROM and RAM, as it retains its data when turned off but is also rewritable. It is typically much slower than conventional ROM and RAM however, so its use is restricted to applications where high speed is unnecessary.

In more sophisticated computers there may be one or more RAM cache memories , which are slower than registers but faster than main memory. Generally computers with this sort of cache are designed to move frequently needed data into the cache automatically, often without the need for any intervention on the programmer's part.

Hard disk drives , floppy disk drives and optical disc drives serve as both input and output devices. A graphics processing unit might contain fifty or more tiny computers that perform the calculations necessary to display 3D graphics.

A era flat screen display contains its own computer circuitry. While a computer may be viewed as running one gigantic program stored in its main memory, in some systems it is necessary to give the appearance of running several programs simultaneously.

This is achieved by multitasking i. By remembering where it was executing prior to the interrupt, the computer can return to that task later. If several programs are running "at the same time".

Since modern computers typically execute instructions several orders of magnitude faster than human perception, it may appear that many programs are running at the same time even though only one is ever executing in any given instant.

This method of multitasking is sometimes termed "time-sharing" since each program is allocated a "slice" of time in turn. Before the era of inexpensive computers, the principal use for multitasking was to allow many people to share the same computer. If a program is waiting for the user to click on the mouse or press a key on the keyboard, then it will not take a "time slice" until the event it is waiting for has occurred. This frees up time for other programs to execute so that many programs may be run simultaneously without unacceptable speed loss.

Some computers are designed to distribute their work across several CPUs in a multiprocessing configuration, a technique once employed only in large and powerful machines such as supercomputers , mainframe computers and servers.

Multiprocessor and multi-core multiple CPUs on a single integrated circuit personal and laptop computers are now widely available, and are being increasingly used in lower-end markets as a result. Supercomputers in particular often have highly unique architectures that differ significantly from the basic stored-program architecture and from general purpose computers.

Such designs tend to be useful only for specialized tasks due to the large scale of program organization required to successfully utilize most of the available resources at once. Supercomputers usually see usage in large-scale simulation , graphics rendering , and cryptography applications, as well as with other so-called " embarrassingly parallel " tasks.

Software refers to parts of the computer which do not have a material form, such as programs, data, protocols, etc. Software is that part of a computer system that consists of encoded information or computer instructions, in contrast to the physical hardware from which the system is built. Computer software includes computer programs , libraries and related non-executable data , such as online documentation or digital media.

It is often divided into system software and application software]] Computer hardware and software require each other and neither can be realistically used on its own. There are thousands of different programming languages—some intended to be general purpose, others useful only for highly specialized applications. The defining feature of modern computers which distinguishes them from all other machines is that they can be programmed.

That is to say that some type of instructions the program can be given to the computer, and it will process them. Modern computers based on the von Neumann architecture often have machine code in the form of an imperative programming language. In practical terms, a computer program may be just a few instructions or extend to many millions of instructions, as do the programs for word processors and web browsers for example. A typical modern computer can execute billions of instructions per second gigaflops and rarely makes a mistake over many years of operation.

Large computer programs consisting of several million instructions may take teams of programmers years to write, and due to the complexity of the task almost certainly contain errors. This section applies to most common RAM machine —based computers.

In most cases, computer instructions are simple: These instructions are read from the computer's memory and are generally carried out executed in the order they were given. However, there are usually specialized instructions to tell the computer to jump ahead or backwards to some other place in the program and to carry on executing from there.

These are called "jump" instructions or branches. Furthermore, jump instructions may be made to happen conditionally so that different sequences of instructions may be used depending on the result of some previous calculation or some external event.

Many computers directly support subroutines by providing a type of jump that "remembers" the location it jumped from and another instruction to return to the instruction following that jump instruction. Program execution might be likened to reading a book. While a person will normally read each word and line in sequence, they may at times jump back to an earlier place in the text or skip sections that are not of interest.

Similarly, a computer may sometimes go back and repeat the instructions in some section of the program over and over again until some internal condition is met. This is called the flow of control within the program and it is what allows the computer to perform tasks repeatedly without human intervention.

Comparatively, a person using a pocket calculator can perform a basic arithmetic operation such as adding two numbers with just a few button presses. But to add together all of the numbers from 1 to 1, would take thousands of button presses and a lot of time, with a near certainty of making a mistake.

On the other hand, a computer may be programmed to do this with just a few simple instructions. The following example is written in the MIPS assembly language:. Once told to run this program, the computer will perform the repetitive addition task without further human intervention. It will almost never make a mistake and a modern PC can complete the task in a fraction of a second. In most computers, individual instructions are stored as machine code with each instruction being given a unique number its operation code or opcode for short.

The command to add two numbers together would have one opcode; the command to multiply them would have a different opcode, and so on. The simplest computers are able to perform any of a handful of different instructions; the more complex computers have several hundred to choose from, each with a unique numerical code. Since the computer's memory is able to store numbers, it can also store the instruction codes. This leads to the important fact that entire programs which are just lists of these instructions can be represented as lists of numbers and can themselves be manipulated inside the computer in the same way as numeric data.

The fundamental concept of storing programs in the computer's memory alongside the data they operate on is the crux of the von Neumann, or stored program [ citation needed ] , architecture. In some cases, a computer might store some or all of its program in memory that is kept separate from the data it operates on. This is called the Harvard architecture after the Harvard Mark I computer. Modern von Neumann computers display some traits of the Harvard architecture in their designs, such as in CPU caches.

While it is possible to write computer programs as long lists of numbers machine language and while this technique was used with many early computers, [67] it is extremely tedious and potentially error-prone to do so in practice, especially for complicated programs.

These mnemonics are collectively known as a computer's assembly language. Converting programs written in assembly language into something the computer can actually understand machine language is usually done by a computer program called an assembler.

Programming languages provide various ways of specifying programs for computers to run. Unlike natural languages , programming languages are designed to permit no ambiguity and to be concise.

They are purely written languages and are often difficult to read aloud. They are generally either translated into machine code by a compiler or an assembler before being run, or translated directly at run time by an interpreter.

Sometimes programs are executed by a hybrid method of the two techniques. Machine languages and the assembly languages that represent them collectively termed low-level programming languages tend to be unique to a particular type of computer. For instance, an ARM architecture computer such as may be found in a smartphone or a hand-held videogame cannot understand the machine language of an x86 CPU that might be in a PC.

Although considerably easier than in machine language, writing long programs in assembly language is often difficult and is also error prone. Therefore, most practical programs are written in more abstract high-level programming languages that are able to express the needs of the programmer more conveniently and thereby help reduce programmer error. High level languages are usually "compiled" into machine language or sometimes into assembly language and then into machine language using another computer program called a compiler.

It is therefore often possible to use different compilers to translate the same high level language program into the machine language of many different types of computer. This is part of the means by which software like video games may be made available for different computer architectures such as personal computers and various video game consoles.

Fourth-generation languages 4GL are less procedural than 3G languages. The benefit of 4GL is that they provide ways to obtain information without requiring the direct help of a programmer. Program design of small programs is relatively simple and involves the analysis of the problem, collection of inputs, using the programming constructs within languages, devising or using established procedures and algorithms, providing data for output devices and solutions to the problem as applicable.

As problems become larger and more complex, features such as subprograms, modules, formal documentation, and new paradigms such as object-oriented programming are encountered. Large programs involving thousands of line of code and more require formal software methodologies.

The task of developing large software systems presents a significant intellectual challenge. Producing software with an acceptably high reliability within a predictable schedule and budget has historically been difficult; the academic and professional discipline of software engineering concentrates specifically on this challenge.

Errors in computer programs are called " bugs ". They may be benign and not affect the usefulness of the program, or have only subtle effects. But in some cases, they may cause the program or the entire system to " hang ", becoming unresponsive to input such as mouse clicks or keystrokes, to completely fail, or to crash.

Otherwise benign bugs may sometimes be harnessed for malicious intent by an unscrupulous user writing an exploit , code designed to take advantage of a bug and disrupt a computer's proper execution. Bugs are usually not the fault of the computer. Since computers merely execute the instructions they are given, bugs are nearly always the result of programmer error or an oversight made in the program's design.

Firmware is the technology which has the combination of both hardware and software such as BIOS chip inside a computer. This chip hardware is located on the motherboard and has the BIOS set up software stored in it. Computers have been used to coordinate information between multiple locations since the s. In time, the network spread beyond academic and military institutions and became known as the Internet.

The emergence of networking involved a redefinition of the nature and boundaries of the computer. Computer operating systems and applications were modified to include the ability to define and access the resources of other computers on the network, such as peripheral devices, stored information, and the like, as extensions of the resources of an individual computer.

Initially these facilities were available primarily to people working in high-tech environments, but in the s the spread of applications like e-mail and the World Wide Web , combined with the development of cheap, fast networking technologies like Ethernet and ADSL saw computer networking become almost ubiquitous. In fact, the number of computers that are networked is growing phenomenally. A very large proportion of personal computers regularly connect to the Internet to communicate and receive information.

A computer does not need to be electronic , nor even have a processor , nor RAM , nor even a hard disk. While popular usage of the word "computer" is synonymous with a personal electronic computer, the modern [74] definition of a computer is literally: Historically, computers evolved from mechanical computers and eventually from vacuum tubes to transistors. However, conceptually computational systems as flexible as a personal computer can be built out of almost anything.

For example, a computer can be made out of billiard balls billiard ball computer ; an often quoted example. There is active research to make computers out of many promising new types of technology, such as optical computers , DNA computers , neural computers , and quantum computers.

Most computers are universal, and are able to calculate any computable function , and are limited only by their memory capacity and operating speed. However different designs of computers can give very different performance for particular problems; for example quantum computers can potentially break some modern encryption algorithms by quantum factoring very quickly.

There are many types of computer architectures:. Of all these abstract machines , a quantum computer holds the most promise for revolutionizing computing. The ability to store and execute lists of instructions called programs makes computers extremely versatile, distinguishing them from calculators. The Church—Turing thesis is a mathematical statement of this versatility: Therefore, any type of computer netbook , supercomputer , cellular automaton , etc.

A computer will solve problems in exactly the way it is programmed to, without regard to efficiency, alternative solutions, possible shortcuts, or possible errors in the code. Computer programs that learn and adapt are part of the emerging field of artificial intelligence and machine learning. Artificial intelligence based products generally fall into two major categories: Rule based systems attempt to represent the rules used by human experts and tend to be expensive to develop.

Pattern based systems use data about a problem to generate conclusions. Examples of pattern based systems include voice recognition, font recognition, translation and the emerging field of on-line marketing. As the use of computers has spread throughout society, there are an increasing number of careers involving computers.

The need for computers to work well together and to be able to exchange information has spawned the need for many standards organizations, clubs and societies of both a formal and informal nature. From Wikipedia, the free encyclopedia. For other uses, see Computer disambiguation and Computer system disambiguation. History of computing hardware.

It has been suggested that this section be split out into another article titled Digital computer. Computer hardware , Personal computer hardware , Central processing unit , and Microprocessor.

CPU design and Control unit. Computer program and Computer programming. This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed.

July Learn how and when to remove this template message. Like the cosine, the complex exponential can be defined in one of several ways. The set of complex numbers at which exp z is equal to one is then an imaginary arithmetic progression of the form:.

A circle encloses the largest area that can be attained within a given perimeter. Second, since no transcendental number can be constructed with compass and straightedge , it is not possible to " square the circle ". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is exactly equal to the area of a given circle.

These numbers are among the most well-known and widely used historical approximations of the constant. Some approximations of pi include:. Any complex number , say z , can be expressed using a pair of real numbers. This formula establishes a correspondence between imaginary powers of e and points on the unit circle centered at the origin of the complex plane.

After this, no further progress was made until the late medieval period. Astronomical calculations in the Shatapatha Brahmana ca. With a correct value for its seven first decimal digits, this value of 3. The Indian astronomer Aryabhata used a value of 3. An infinite series is the sum of the terms of an infinite sequence.

Nilakantha attributes the series to an earlier Indian mathematician, Madhava of Sangamagrama , who lived c. The second infinite sequence found in Europe , by John Wallis in , was also an infinite product: In Europe, Madhava's formula was rediscovered by Scottish mathematician James Gregory in , and by Leibniz in In John Machin used the Gregory—Leibniz series to produce an algorithm that converged much faster: After five terms, the sum of the Gregory—Leibniz series is within 0.

Series that converge even faster include Machin's series and Chudnovsky's series , the latter producing 14 correct decimal digits per term. John Machin ", leading to speculation that Machin may have employed the Greek letter before Jones. American mathematicians John Wrench and Levi Smith reached 1, digits in using a desk calculator. The iterative algorithms were independently published in — by American physicist Eugene Salamin and Australian scientist Richard Brent.

An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step that converges to the desired value. The approach was actually invented over years earlier by Carl Friedrich Gauss , in what is now termed the arithmetic—geometric mean method AGM method or Gauss—Legendre algorithm. The iterative algorithms were widely used after because they are faster than infinite series algorithms: For example, the Brent-Salamin algorithm doubles the number of digits in each iteration.

In , the Canadian brothers John and Peter Borwein produced an iterative algorithm that quadruples the number of digits in each step; and in , one that increases the number of digits five times in each step. New infinite series were discovered in the s and s that are as fast as iterative algorithms, yet are simpler and less memory intensive. This series converges much more rapidly than most arctan series, including Machin's formula. The associated random walk is. As n varies W n defines a discrete stochastic process.

This Monte Carlo method is independent of any relation to circles, and is a consequence of the central limit theorem , discussed above. American mathematicians Stan Wagon and Stanley Rabinowitz produced a simple spigot algorithm in Another spigot algorithm, the BBP digit extraction algorithm , was discovered in by Simon Plouffe: Variations of the algorithm have been discovered, but no digit extraction algorithm has yet been found that rapidly produces decimal digits.

After a new record is claimed, the decimal result is converted to hexadecimal, and then a digit extraction algorithm is used to calculate several random hexadecimal digits near the end; if they match, this provides a measure of confidence that the entire computation is correct. For example, an integral that specifies half the area of a circle of radius one is given by: The trigonometric functions rely on angles, and mathematicians generally use radians as units of measurement.

In many applications, it plays a distinguished role as an eigenvalue. One way to obtain this is by estimating the energy. The energy satisfies an inequality, Wirtinger's inequality for functions , [] which states that if a function f: As mentioned above , it can be characterized via its role as the best constant in the isoperimetric inequality: The Sobolev inequality is equivalent to the isoperimetric inequality in any dimension , with the same best constants.

This is the integral transform , that takes a complex-valued integrable function f on the real line to the function defined as:. The uncertainty principle gives a sharp lower bound on the extent to which it is possible to localize a function both in space and in frequency: The physical consequence, about the uncertainty in simultaneous position and momentum observations of a quantum mechanical system, is discussed below.

The fields of probability and statistics frequently use the normal distribution as a simple model for complex phenomena; for example, scientists generally assume that the observational error in most experiments follows a normal distribution.

For this to be a probability density, the area under the graph of f needs to be equal to one. This follows from a change of variables in the Gaussian integral: Let V be the set of all twice differentiable real functions f: Then V is a two-dimensional real vector space , with two parameters corresponding to a pair of initial conditions for the differential equation.

The Euler characteristic of a sphere can be computed from its homology groups and is found to be equal to two. The constant appears in many other integral formulae in topology, in particular, those involving characteristic classes via the Chern—Weil homomorphism.

Vector calculus is a branch of calculus that is concerned with the properties of vector fields , and has many physical applications such as to electricity and magnetism. The Newtonian potential for a point source Q situated at the origin of a three-dimensional Cartesian coordinate system is []. The field, denoted here by E , which may be the Newtonian gravitational field or the Coulomb electric field , is the negative gradient of the potential:. Special cases include Coulomb's law and Newton's law of universal gravitation.

More general distributions of matter or charge are obtained from this by convolution , giving the Poisson equation. The factorial function n! The gamma function extends the concept of factorial normally defined only for non-negative integers to all complex numbers, except the negative real integers. The gamma function is defined by its Weierstrass product development: Further, it follows from the functional equation that.

The gamma function can be used to create a simple approximation to the factorial function n! Ehrhart's volume conjecture is that this is the optimal upper bound on the volume of a convex body containing only one lattice point. Finding a simple solution for this infinite series was a famous problem in mathematics called the Basel problem. For distinct primes, these divisibility events are mutually independent; so the probability that two numbers are relatively prime is given by a product over all primes: This is a special case of Weil's conjecture on Tamagawa numbers , which asserts the equality of similar such infinite products of arithmetic quantities, localized at each prime p , and a geometrical quantity: This functional determinant can be computed via a product expansion, and is equivalent to the Wallis product formula.

The Fourier decomposition shows that a complex-valued function f on T can be written as an infinite linear superposition of unitary characters of T. That is, continuous group homomorphisms from T to the circle group U 1 of unit modulus complex numbers. There is a unique character on T , up to complex conjugation, that is a group isomorphism. For example, the Chudnovsky algorithm involves in an essential way the j-invariant of an elliptic curve. An example is the Jacobi theta function.

Certain identities hold for all automorphic forms. The total probability is equal to one, owing to the integral:.

The Cauchy distribution plays an important role in potential theory because it is the simplest Furstenberg measure , the classical Poisson kernel associated with a Brownian motion in a half-plane.

The Hilbert transform H is the integral transform given by the Cauchy principal value of the singular integral. A simple formula from the field of classical mechanics gives the approximate period T of a simple pendulum of length L , swinging with a small amplitude g is the earth's gravitational acceleration: It is defined as exactly.

The sinuosity is the ratio between the actual length and the straight-line distance from source to mouth. Faster currents along the outside edges of a river's bends cause more erosion than along the inside edges, thus pushing the bends even farther out, and increasing the overall loopiness of the river. However, that loopiness eventually causes the river to double back on itself in places and "short-circuit", creating an ox-bow lake in the process.

The first word has three letters, the second word has one, the third has four, the fourth has one, the fifth has five, and so on. An early example of a memorization aid, originally devised by English scientist James Jeans , is "How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.

The digits are large wooden characters attached to the dome-like ceiling. The digits were based on an calculation by English mathematician William Shanks , which included an error beginning at the th digit. The error was detected in and corrected in Several college cheers at the Massachusetts Institute of Technology include "3.

The bill is notorious as an attempt to establish a value of scientific constant by legislative fiat. The bill was passed by the Indiana House of Representatives, but rejected by the Senate, meaning it did not become a law. The versions are 3, 3. From Wikipedia, the free encyclopedia. This article is about the mathematical constant. For the Greek letter, see Pi letter. For other uses, see Pi disambiguation.

The Gauss—Legendre iterative algorithm: Needles a and b are dropped randomly. Random dots are placed on the quadrant of a square with a circle inscribed in it. Zeit Online in German. Archived from the original on 17 March Die Ludolphsche Zahl oder Kreiszahl erhielt nun auch das Symbol, unter dem wir es heute kennen: Archived from the original on 28 July Retrieved 18 June Calculus, volume 1 2nd ed.

Principles of Mathematical Analysis. Real and complex analysis. Lawrence Berkeley National Laboratory. Archived from the original on 20 October Retrieved 10 November Archived from the original on Retrieved 4 November Science and Its Times: Understanding the Social Significance of Scientific Discovery. Retrieved 12 April The American Mathematical Monthly.

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