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HP-35

HP-35

The HP-35 was Hewlett-Packard's first pocket calculator and the world's first scientific pocket calculator (a calculator with trigonometric and exponential functions). Like some of HP's desktop calculators it used reverse Polish notation. Introduced at US$395, the HP-35 was available from 1972 to 1975. Market studies at the time had shown no market for pocket sized calculators. However, HP co-founder Bill Hewlett began development of a "shirt-pocket sized HP-9100", and it turned out that the marketing studies were wrong. In the first months orders were exceeding HP's expectations as to the entire market size. Before the HP-35, the only practically portable device for performing trigonometric and exponential functions were slide rules. Existing pocket calculators at the time were only four-function; i.e., could only do addition, subtraction, multiplication, and division. It had been originally known simply as "The Calculator," but Hewlett suggested that it be called the HP-35 because it had 35 keys. The calculator used a traditional floating decimal display for numbers that could be displayed in that format, but automatically switched to scientific notation for other numbers. The fifteen digit LED display was capable of displaying a 10 digit mantissa plus its sign and a decimal point and a two digit exponent plus its sign. The display was unique in that the multiplexing was designed to illuminate a single LED segment at a time (rather than a single LED digit) because HP research had shown that this method was perceived by the human eye as brighter for equivalent power. The calculator used 3 "AA"-sized NiCd batteries assembled into a removable proprietary battery pack. Replacement battery packs are no longer available, leaving existing HP-35s to rely on AC power, or their users to rebuild the battery packs themselves using available cells. An external battery charger was available and the calculator could also run from the charger, with or without batteries installed. Internally, the calculator was organized around a serial (one-bit) processor chipset processing 56-bit floating-point numbers (representing 14-digit BCD numbers). The HP-35 was the start of a family of related calculators which all shared similar mechanical packaging:
- The HP-45 added many more features, including the ability to control the output format (rather than the purely automatic format of the HP-35). It also contained an undocumented timer feature. The timer worked, but was not accurate enough to use as a stopwatch due to lack of a crystal oscillator.
- The HP-55 provided storage for smaller programs (but didn't provide any external storage). The timer that was already present on the HP-45 was now crystal-controlled to achieve the needed accuracy and explicitly documented.
- The HP-65 added programmability, with program storage on magnetic cards.
- The HP-80 (and the less expensive HP-70) provided financial, rather than scientific functions, such as future value and net present value. Follow-on calculators used varying mechanical packaging but most were operationally similar. The HP-41C was a major advancement in programmability and offered CMOS memory so that programs were not lost when the calculator was switched off. It was the first calculator to offer alphanumeric capabilities for both the display and the keyboard. Four external ports below the display area allowed memory expansion (RAM modules), loading of additional programs (ROM modules) and interfacing a wide variety of peripherals including HP-IL ("HP Interface Loop"), a scaled-down version of the HPIB/GPIB/IEEE-488 instrument bus. The later HP-28C and HP-28S added much more memory and a substantially different, more powerful programming metaphor.

Calculator trivia

- The HP-35's 15-character display, when viewed upside down, could produce a limited number of alphabetic messages. For example, 710.77345 would read as "SHELL OIL". The most extravagant display was probably 57738.57734 E+40 which would read as "Oh hELLS BELLS".
- The HP-35 was exactly 5.8 inches long and 3.2 inches wide. This was the size of William Hewlett's pocket, hence "pocket calculator".
- The LED display power requirement was responsible for the HP-35's short battery life between charges — about three hours. To extend operating time and avoid wearing out the on/off slide switch, users would press the decimal point key to force the display to illuminate just a single LED junction.
- Introduction of the HP-35 and similarly capable scientific calculators by Texas Instruments soon thereafter signaled the demise of the slide rule as a status symbol among science and engineering students. Slide rule holsters began to rapidly give way to "electronic slide rule" holsters, and colleges began to drop slide rule classes from their curricula.

External links

- [http://www.hpmuseum.org/hp35.htm The Museum of HP Calculators' article on the HP-35]
- [http://www.hpl.hp.com/hpjournal/72jun/toc-06-72.htm HP Journal, June 1972]
- [http://www.hp.com/hpinfo/abouthp/histnfacts/museum/personalsystems/0023/index.html HP Virtual Museum:] HP-35 35

Hewlett-Packard

The Hewlett-Packard Company , commonly known as HP, is a very large global company headquartered in Palo Alto, California, United States. Its products are concentrated in the fields of computing, printing, and digital imaging. It also sells software and services.

Company history

From '39 until the seventies

HP was founded in 1939 by Bill Hewlett and Dave Packard, who had both graduated from Stanford University in 1934, as a manufacturer of test and measurement instruments. Their first product was a precision audio oscillator, the Model 200A. Their innovation was the use of a light bulb as a temperature stabilized resistor in a critical portion of the circuit. This allowed them to sell the Model 200A for $54.40 when competitors were selling less stable oscillators for over US$ 200. Their company's name, Hewlett-Packard, was derived by their last names and had Bill not won the coin toss, the company today may have been known as Packard-Hewlett. One of the company's earliest customers was Walt Disney Productions, who bought eight Model 200B oscillators (at $71.50 each) for use in testing the Fantasound stereophonic sound system for the movie Fantasia.

First Computers

Fantasia HP is [http://www.wired.com/wired/archive/8.12/mustread.html?pg=11 acknowledged by] Wired magazine as the producer of the world's first personal computer, in 1968, the Hewlett-Packard 9100A. HP called it a desktop calculator because, as Bill Hewlett said, "If we had called it a computer, it would have been rejected by our customers' computer gurus because it didn't look like an IBM. We therefore decided to call it a calculator, and all such nonsense disappeared". An engineering triumph at the time, the logic circuit was produced without any integrated circuits; the assembly of the CPU having been entirely executed in discrete components. The mathematical functions and programmability rival the most powerful scientific calculators of the present day. With CRT readout, magnetic card storage, and printer the price was around $5000. The company earned global respect for a variety of products. They introduced the world's first handheld scientific electronic calculator in 1972 (the HP-35), the first handheld programmable in 1974 (the HP-65), the first alphanumeric, programmable, expandable in 1979 (the HP-41C), and the first symbolic and graphing calculator HP-28C. Like their scientific and business calculators, their oscilloscopes, logic analyzers, and other measurement instruments have a reputation for sturdiness and usability (the latter products are now part of spin-off Agilent's product line). The company's design philosophy in this period was summarized as "design for the guy at the next bench". HP is recognized as the symbolic founder of Silicon Valley, although it did not actively investigate semiconductor devices until a few years after the "Traitorous Eight" had abandoned William Shockley to create Fairchild Semiconductor in 1957. Hewlett-Packard's HP Associates division, established around 1960, developed semiconductor devices primarily for internal use. Instruments and calculators were some of the products using these devices.

The eighties and beyond

In 1984, HP introduced both inkjet and laser printers for the desktop. Along with its scanner product line, these have later been developed into successful multifunction products, the most significant being single-unit printer/scanner/copier/fax machines. As of 2003, HP's major competitors in this growing part of the home office market are Brother, Canon, Epson, and Lexmark. Another vendor of note who rivals HP printers is Dell, who rebrands and repackages Lexmark products. In the 1990s, HP expanded their computer product line, which initially had been targeted at university, research, and business customers, to reach consumers. HP also grew through acquisitions, buying Apollo Computer in 1989, Convex Computer in 1995, and Compaq in 2002. Compaq itself had bought Tandem Computers in 1997 (which had been started by ex-HP employees), and Digital Equipment Corporation in 1998. Following this strategy HP became a major player in desktops, laptops, and servers for many different markets. The buyout made HP the world's largest manufacturer of personal computers. In 1987, the Palo Alto garage where Hewlett and Packard started their business was designated as a California State historical landmark. However, Agilent Technologies, not HP, bears the legacy of the original instrument company founded by Bill Hewlett and Dave Packard in 1939. Agilent was spun off from HP in 1999. HPshopping.com was launched in 1998 as HP's direct-to-consumer e-commerce store, and in 1999 became incorporated as a wholly owned subsidiary. In 2002, Compaq's direct-to-consumer e-commerce store, CompaqAtHome, joined hpshopping.com, creating a comprehensive, one-stop, on-line store for HP and Compaq home and home-office products. Many long-time HP calculator users were surprised and disappointed when HP announced in March 2002 that the company would no longer manufacture financial and scientific calculators – a product line and, indeed, a market, that HP had started thirty years before. The decision was especially hard to fathom in light of the HP-48 graphing calculator range's success. However, despite its spring 2002 press release stating the opposite, the company nevertheless returned to the market during the fall of 2003 with several new models (flagship: HP-49g+) competing against similar offerings from competitor Texas Instruments. The extremely popular HP 12c financial calculator, introduced in 1981, still remains in production today.

HP today

In 2002, Hewlett-Packard merged with Compaq, a highly controversial move intended to make the company the leader in the personal computing space. The merger opposition was led by Walter Hewlett, son of original founder William Hewlett. Hewlett-Packard is a supporter of Open Source and Linux. Some HP employees, such as Linux CTO and former Debian Project Leader Bdale Garbee actively contribute – a few have official Open Source job responsibilities. Many others participate in the Open Source community as unpaid volunteers. Hewlett-Packard also partners extensively with Microsoft. Given the size and scope of HP operations, HP leverages technology from most major software and hardware vendors. Microsoft Other HP products/technologies include:
- Inkjet and LaserJet printers, consumables and related products.
- the HP-UX and Tru64 operating systems (two UNIX implementations, the second from DEC)
- the OpenVMS large-scale, highly-available server operating system (from DEC)
- the NonStop high-reliability architecture and operating system (from Tandem Computers)
- the PA-RISC processor architecture
- the IA64 processor architecture (with Intel)
- the Alpha processor architecture (from DEC)
- the HP 9000 line of servers and workstations
- the ProLiant line of x86 based servers (from Compaq)
- the UDC (Utility Data Center)
- the OpenView family of management software
- the ProCurve [http://www.hp.com/rnd/index.htm] family of network switches, wireless access points, and routers.
- the Indigo Digital Press HP has a successful line of printers, scanners, calculators, PDAs, servers, workstations, and home-small business computers. Until recently HP even offered a re-branded version of Apple's famous iPod. HP today promotes itself as not just being a hardware and software company, but also one that offers a full range of services to architect, implement and support today's IT infrastructure. In 2003, HP had 140,000 employees world wide. From July 1999 to February 9, 2005, the chairman and CEO was Carly Fiorina, the first woman ever to serve as CEO of a company included in the Dow Jones Industrial Average. Amid controversy over her performance and threats of reductions in her responsibilities by the HP Board of Directors, Fiorina was eventually forced to resign. The current non-executive chairman is Patricia C. Dunn. She was given this responsibility in February 2005 after Ms. Fiorina left the company. Unlike previous CEOs, Mark Hurd the new CEO of HP does not hold the position of Chairman.

Management

- Founder and CEO: David Packard (CEO: 1964 - 1969)
- Founder and CEO: William Hewlett (CEO: 1969 - 1978)
- CEO: John A. Young (1978 - 1992)
- CEO: Lewis Platt (1992 - July 18, 1999)
- Chairman and CEO: Carly Fiorina (July 19, 1999 - February 9, 2005, Appointed chairman in 2000)
- Interim CEO: Robert P. Wayman (Feburary 10, 2005 - March 28, 2005)
- CEO: Mark Hurd (March 29, 2005 - current)

Diversity

Hewlett-Packard received a 100% rating on the Corporate Equality Index released by the Human Rights Campaign starting in 2003, the second year of the report. In addition, the company was named one of the 100 Best Companies for Working Mothers in 2004 by Working Mothers magazine. Hewlett-Packard is also involved in the NEPAD e-school programme to provide all schools in Africa with computers and internet access.

Ad campaigns

A television ad campaign for Hewlett-Packards digital photography (titled "You + HP: digital photography") has been noted for its simplistic special effects and choice of music. It won "Campaign of the Year" from Adweek magazine.

Songs used in "You + HP" Campaign

- "Picture Book" by The Kinks
- "Out of the Picture" by The Robins
- "Pictures of You" by The Cure
- "The Rainbow" by The Apples in Stereo
- "Across the Universe" by The Beatles

Competitors

Major competitors of HP in the PC business include:
- Dell, Inc
- Gateway, Inc.
- Lenovo (Purchased the ThinkPad notebook line from IBM)
- Sony
- Toshiba Major competitors of HP in the server business include:
- Sun Microsystems
- IBM
- Dell, Inc

External links

;HP
- [http://www.hp.com HP corporate homepage]
- [http://index.hpshopping.com HP online store]
- [http://www.hp.com/hpinfo/abouthp/histnfacts/ HP History and Facts]
- [http://opensource.hp.com/ HP Open Source portal]
- [http://www.hp.com/linux HP Linux portal]
- [http://www.hp.com/calculators/ HP Calculators] ;Data
- [http://biz.yahoo.com/ic/10/10723.html Yahoo! - Hewlett-Packard Company Company Profile] ;Third-party
- [http://www.siliconvalleyinfozone.com/companies/Hewlett-Packard Silicon Valley InfoZone - Hewlett-Packard]
- [http://www.paloaltoonline.com/weekly/morgue/2002/2002_04_10.hpway10.html The rise and fall of the HP Way] by Jocelyn Dong, with Pam Sturner (Online Edition of the Palo Alto Weekly, April 10, 2002)
- [http://www.hydrix.com/wiki/ The HP Calculator Wiki]
- [http://tuxmobil.org/hp.html Linux on Hewlett-Packard laptops]
- [http://www.openpa.net/ The OpenPA Project] - resource for information on PA-RISC based computers from HP and other vendors
- [http://www.hpmuseum.org The Museum of HP Calculators], [http://www.hpmuseum.net HP Computer Museum] Category:Electronics companies Category:Hewlett-Packard Category:Portable Audio Player Manufacturers Category:Fortune 500 companies Category:Companies based in California ja:ヒューレット・パッカード

Pocket calculator

A calculator is a device for performing numerical calculations. The type is considered distinct from both a calculating machine and a computer in that the calculator is a special-purpose device that may not qualify as a Turing machine. Although modern calculators often incorporate a general purpose computer, the device as a whole is designed for ease of use to perform specific operations, rather than for flexibility. The complexity of calculators varies with the intended purpose. A simple one with only four functions (addition, subtraction, multiplication and division and perhaps a single-number memory) may be useful for everyday activities such as shopping or checking a bill. More complex ones may include complex mathematical functions suitable to engineering or accounting as well as a substantial memory and the ability to execute moderately complex programs. Since the late-1980's, it has become common to incorporate simple calculators in other small devices, such as mobile phones, pagers or wrist watches. In most developed countries, students use calculators for schoolwork. There was some initial resistance to the idea out of fear that basic arithmetic skills would suffer. There remains disagreement about the importance of the ability to perform calculations by hand or "in the head", with some curricula restricting calculator use until a certain level of proficiency has been obtained, while others concentrate more on teaching estimation techniques and problem-solving.

Overview

Modern calculators are electrically powered, most often by battery, and are made by numerous manufacturers, in countless shapes and sizes varying from cheap, give-away, credit-card sized models to more sturdy adding machine-like models with built-in printers. Only a very few companies develop and make modern professional engineering and finance calculators: The most well-known are Casio, Sharp, Hewlett-Packard (HP) and Texas Instruments (TI). Such calculators are good examples of embedded systems. They are also often complex enough to be programmed; calculator applications include algebraic equation solvers, financial models and even games. In the near past, mechanical and clerical aids such as abacuses, comptometers, Napier's bones, books of mathematical tables, slide rules, adding machines, were used for serious numeric work, and the word "calculator" denoted a person (most often male) who did such work for a living using such aids as well as pen and paper. This semi-manual process of calculation was tedious and error-prone.

Electronic calculators

Today most calculators are handheld microelectronic devices, but in the past some calculators were as large as today's computers. The first mechanical calculators were mechanical desktop devices, which were soon replaced by electromechanical desktop calculators, and then by electronic devices using first thermionic valves, then transistors, then hard-wired integrated circuit logic. A pocket calculator is a small battery-powered or solar powered electronic digital computer made possible by integrated circuit and semiconductor technology. Typically they are limited to an 8–10 digit single-number display and a few basic functions of arithmetic, but some modern calculators have more of the features of a general-purpose computer. Pocket calculators rendered the slide rule obsolete. Calculators vary in their capabilities. Some are limited to only basic arithmetic; others support trigonometric, statistical and other mathematical functions. The most advanced modern calculators are programmable, can display graphics, and include features of computer algebra systems.

Personal computing

Personal computers and personal digital assistants can perform general calculations in a variety of ways:
- computers often have a separate calculator program, varying from one that just emulates a simple calculator, such as Microsoft Calculator, to advanced spreadsheet programs such as Excel or OpenOffice.org Calc
- for more advanced calculations one can use a computer algebra program, such as Mathematica, Maple or Matlab.
- browsers can perform calculations using client-side scripting, e.g. using Client-side JavaScript by entering "javascript:alert(12
- 13)" in the address bar (the answer 156 appears in a separate alert window) or "document.write (12
- 13)" in a HTML file, preceded with "<script type="text/javascript">" and followed by "</script>".
- an interpreter or compiler for a general programming language can be used
- calculations can also be performed server-side, e.g. with the calculator feature of the Google search engine

History

Origin: The Abacus

calculator feature of the Google search engine The first calculators were abacuses, and were often constructed as a wooden frame with beads sliding on wires. Abacuses were in use centuries before the adoption of the written Arabic numerals system and are still widely used by merchants and clerks in China and elsewhere.

The 17th century

Wilhelm Schickard built the first automatic calculator called the "Calculating Clock" in 1623. Some 20 years later, in 1645, French philosopher Blaise Pascal invented the calculation device later known as Pascal's calculator, which was used for taxes in France until 1799. The German philosopher G.W.v.Leibniz also produced a calculating machine.

1930s to 1960s

calculating machine From approximately the 1930s through the 1960s, mechanical calculators were often used (see Mechanical Calculator under History of computing hardware). These desktop devices were motor-driven and had multiple columns of keys for each digit. Addition and subtraction were performed in a single operation, as on a conventional adding machine, but multiplication and division were accomplished by repeated mechanical additions and subtractions. Handheld mechanical calculators such as the Curta continued to be used until they were displaced by electronic calculators in the 1970s. In 1954, IBM demonstrated a large all-transistor calculator and, in 1957, they released the first commercial all-transistor calculator (the IBM 608). In October 1961, the world's first all-electronic desktop calculator, the Bell Punch/Sumlock Comptometer ANITA Mk.VII was released. This British designed-and-built machine used vacuum tubes in its circuits and cold-cathode nixie tubes for its display. It was superseded, technologically, in 1964 when Sharp introduced the CS-10A—the world's first all-transistor desktop calculator—which weighed 25 kg (55 lb) and cost 500,000 yen (~US$2500). The first handheld electronic calculators went on sale in 1970 with models from Japanese manufacturers Sharp and Canon weighing around 770 g (1.7 lb).

1970s to mid-1980s

In the early 1970s, the Monroe EPIC programmable calculator came on the market. A large desk-top unit, with an attached floor-standing logic tower, it was capable of being programmed to perform many computer-like functions. However, the only branch instruction was an implied unconditional branch (GOTO) at the end of the operation stack, returning the program to its starting instruction. Thus, it was not possible to include any conditional branch (IF-THEN-ELSE) logic. During this era, the absence of the conditional branch was sometimes used to distinguish a programmable calculator from a computer. The first pocket-sized calculator, the Bowmar 901B (popularly referred to as The Bowmar Brain), measuring 5.2×3.0×1.5 in (131×77×37 mm), came out in the fall of 1971, with four functions and an eight-digit red LED display, for $240, while in August 1972 the four-function Sinclair Executive became the first slimline pocket calculator measuring 5.4×2.2×0.35 in (138×56×9 mm) and weighing 2.5 oz (70g). It retailed for around $150 (GB£79). By the end of the decade, similar calculators were priced less than $10 (GB£5). The first pocket calculator with scientific functions, i.e. the first slide rule-replacing model, was the 1972 HP-35 from Hewlett Packard (HP); it, along with all later HP engineering calculators, used reverse Polish notation (RPN) (where a calculation like "6 – 2" is performed by pressing "6", "Enter↑", "2", and "–"; instead of algebraically: "6", "–", "2", "="). In 1973, Texas Instruments (TI) introduced the SR-10, (SR signifying slide rule) a hand-held algebraic notation calculator, which was later followed by the SR-11 and eventually the TI-30. The first programmable hand-held calculator was the HP-65, in 1974; it had a capacity of 100 instructions, and could store and retrieve programs with a built-in magnetic card reader. A year later the HP-25C introduced continuous memory, i.e. programs and data were retained in memory during power-off. In 1979, HP released the first alphanumeric, programmable, expandable calculator, the HP-41C. It could be expanded with RAM (memory) and ROM (software) modules, as well as peripherals like bar code readers, microcassette and floppy disk drives, paper-roll thermal printers, and miscellaneous communication interfaces (RS-232, HP-IL, HP-IB).

Mid-1980s to present

HP-IB HP-IB calculator.]] The two leading manufacturers, HP and TI, released steadily more feature-laden calculators during the 1980s and 90s. At the turn of the millennium, the line between a graphing calculator and a PDA/ handheld computer was not always clear (forgetting the keyboard for the sake of the argument), as some very advanced calculators such as the TI-89 and HP-49G could differentiate and integrate functions, run word processing and PIM software, and connect by wire or IR to other calculators/computers. In March 2002, HP announced that the company would no longer produce calculators, which was hard to fathom for some fans of the company's products; the HP-48 range in particular had an extremely loyal customer base. Nevertheless, HP restarted their production of calculators in late 2003. The new models, however, reportedly didn't have the mechanical quality and sober design HP's earlier calculators were famous for (instead featuring the more "youthful" look and feel of contemporary competing designs from TI). The business calculator HP-12C is still produced. It was introduced in 1981 and is still being made with nearly no changes. In 2003 several new models were released, including an improved version of the HP-12C, the "HP-12C platinum edition".

Drawbacks

- Built-in inaccuracy commonly due to arithmetic underflow is a drawback occurring in many ordinary digital calculators. To obtain an example of this potential problem, the following exercise may be performed: enter the number one, divide by three, to reach 0.333 (recurring, i.e. followed by a theoretically infinite number of 3s), and then multiply by three to get back to one. On some calculators this operation will not work correctly, in that the result is given as 0.999 (recurring)—roughly speaking, this anomaly happens because the calculator works with only a finite number of decimals. It is important to note, however, that with infinite precision, .999... repeating is equal to one.
- Another kind of "drawback" resulting from the use, rather than the construction, of calculators, is the tendency of users to carelessly rely on the calculator's output without double-checking the magnitude (in practice, the placement of the decimal separator) of the result. This problem was all but nonexistent in the era of slide rules and pencil-and-paper calculations, when the task of establishing the magnitudes of results had to be done by the (sufficiently meticulous) user.

Trivia

- The word "calculator" is occasionally used as a pejorative term to describe an inadequately capable general-purpose microcomputer. The synonym of this meaning is "bitty box", as discussed in the Jargon file.
- A curious episode of the mid 1970s involved the Melcor 635, a scientific calculator with a bug in its trigonometric functions. Because the CORDIC algorithms used in most calculators cannot compute the inverse trigonometric functions of zero, these need to be hardcoded — and some engineer at Melcor got it wrong. For any input other than exactly zero, even for instance 1.0E-99, the calculator worked correctly; the user simply had to remember not to compute the arc-cosine of zero. The company discovered this after making 50,000 calculators. The upshot was an advertisement in Scientific American headlined 'Somebody Goofed', offering these calculators for sale at half-price.
- As many schoolchildren and students know, some words and simple phrases can be written using an ordinary seven-segment display calculator; this involves entering certain numbers and then viewing the resulting words by turning the calculator display upside-down.

Patents

- – Complex computer – G. R. Stibitz (electromechanic device that would calculate, record, and print results)
- – Miniature electronic calculator – J. S. Kilby (TI electromechanic device)

External links

- [http://www.ti.com/corp/docs/company/history/calc.shtml On TI's US Patent No. 3819921] – From TI's own website
- [http://sharp-world.com/corporate/info/his/h_company/1994/ 30th Anniversary of the Calculator] – From Sharp's web presentation of its history; including a picture of the CS-10A desktop calculator
- [http://www.maths.hscripts.com/ Online Calculators and Converters]
- [http://web.peoriadesignweb.com/calculator Online Calculator Software]
- [http://www.satsig.net/seticalc.htm Online deep space SETI range calculator]
- [http://ostermiller.org/calc/calculator.html JavaScript Scientific Calculator] – Scientific notation, hex, octal, decimal, binary, and math functions; requires JavaScript (from ostermiller.org)
- [http://www.oldcalculatormuseum.com The Old Calculator Web Museum]
- [http://www.calculators.de Calculator Museum]
- [http://www.taswegian.com/MOSCOW/soviet.html Museum of Soviet Calculators]
- [http://www.rk86.com/frolov/calcolle.htm Soviet Calculators Collection]
- [http://www.vintagecalculators.com/index.html Vintage Calculators]
- [http://www.lendingok.com various calculators]
- [http://www.cut-the-knot.org/Curriculum/Arithmetic/BrokenCalculator.shtml Broken Calculator]
- [http://www.graphcalc.com GraphCalc – an Open Source graphing calculator program]
- [http://www.binarythings.com/hidigit/ HiDigit scientific calculator]
- [http://www.hpmuseum.org The Museum of HP Calculators] ([http://www.hpmuseum.org/prehp.htm slide rules/mech. section])
- [http://www.hydrix.com/wiki/ HP Calculator Wiki]
- [http://www.typeonline.co.uk/number_pad_lesson1.html Number pad typing tutorial]
- [http://www.casiocalc.org International Casio Calculator Community]
- [http://www.graph100.com French Casio Calculator Community]
- Calculator Category:Mathematical tools Category:Office equipment ja:電卓 th:เครื่องคิดเลข

Trigonometric functions

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to positive and negative values and even to complex numbers. All of these approaches will be presented below. In modern usage, there are six basic trigonometric functions, which are tabulated below along with equations relating them to one another. (Especially in the case of the last four, these relations are often taken as the definitions of those functions, but one can equally define them geometrically or by other means and derive the relations.) A few other functions were common historically (and appeared in the earliest tables), but are now little-used, such as:
- versed sine (versin = 1 − cos)
- exsecant (exsec = sec − 1). Many more relations between these functions are listed in the article about trigonometric identities.

History

The earliest systematic study of trigonometric functions and tabulation of their values was performed by Hipparchus of Nicaea (180-125 BC), who tabulated the lengths of circle arcs (angle A times radius r) with the lengths of the subtending chords (2r sin(A/2)). Later, Ptolemy (2nd century) expanded upon this work in his Almagest, deriving addition/subtraction formulas for the equivalent of sin(A + B) and cos(A + B). Ptolemy also derived the equivalent of the half-angle formula sin²(A/2) = (1 − cos(A))/2, allowing him to create tables with any desired accuracy. Neither the tables of Hipparchus nor of Ptolemy have survived to the present day. The next significant development of trigonometry was in India, in the works known as the Siddhantas (4th–5th century), which first defined the sine as the modern relationship between half an angle and half a chord. The Siddhantas also contained the earliest surviving tables of sine values (along with 1 − cos values), in 3.75-degree intervals from 0 to 90 degrees. The Hindu works were later translated and expanded by the Arabs, who by the 10th century (in the work of Abu'l-Wefa) were using all six trigonometric functions, and had sine tables in 0.25-degree increments, to 8 decimal places of accuracy, as well as tables of tangent values. Our modern word sine comes, via sinus ("bay" or "fold") in Latin, from a mistranslation of the Sanskrit jiva (or jya). jiva (originally called ardha-jiva, "half-chord", in the 6th century Aryabhata) was transliterated by the Arabs as jiba (جب), but was confused for another word, jaib (جب) ("bay"), by European translators such as Robert of Chester and Gherardo of Cremona in Toledo in the 12th century, probably because jiba (جب) and jaib (جب) are written the same in Arabic (many vowels are excluded from words written in the Arabic alphabet). All of these earlier works on trigonometry treated it mainly as an adjunct to astronomy; perhaps the first treatment as a subject in its own right was by the De triangulis omnimodus (1464) of Regiomontanus (1436–1476), as well as his later Tabulae directionum (which included the tangent function, unnamed). The Opus palatinum de triangulis of Rheticus, a student of Copernicus, was the first to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596. The Introductio in analysin infinitorum (1748) of Euler was primarily responsible for establishing the analytic treatment of trigonometric functions, defining them as infinite series and presenting "Euler's formula" e^ix = cos(x) + i sin(x). Euler used the near-modern abbreviations sin., cos., tang., cot., sec., and cosec..

Right triangle definitions

Euler's formula In order to define the trigonometric functions for the angle A, start with an arbitrary right triangle that contains the angle A: We use the following names for the sides of the triangle:
- The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h.
- The opposite side is the side opposite to the angle we are interested in, in this case a.
- The adjacent side is the side that is in contact with the angle we are interested in and the right angle, hence its name. In this case the adjacent side is b. All triangles are taken to exist in the Euclidean plane so that the inside angles of each triangle sum to π radians (or 180°). Then, 1) The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case :

\sin A = \frac   = \frac

. Note that this ratio does not depend on the particular right triangle chosen, as long as it contains the angle A, since all those triangles are similar. The set of zeroes of sine is

\left\

. 2) The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case :

\cos A = \frac   = \frac

. The set of zeroes of cosine is

\left\

. 3) The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case :

\tan A = \frac   = \frac

. The set of zeroes of tangent is

\left\

. The remaining three functions are best defined using the above three functions. 4) The cosecant csc(A) is the multiplicative inverse of sin(A), i.e. the ratio of the length of the hypotenuse to the length of the opposite side: :

\csc A = \frac   = \frac

. 5) The secant sec(A) is the multiplicative inverse of cos(A), i.e. the ratio of the length of the hypotenuse to the length of the adjacent side: :

\sec A = \frac   = \frac

. 6) The cotangent cot(A) is the multiplicative inverse of tan(A), i.e. the ratio of the length of the adjacent side to the length of the opposite side: :

\cot A = \frac   = \frac

Mnemonics

There are a number of mnemonics for the above definitions, for example SOHCAHTOA (sounds like "soak a toe-a" or "sock-a toe-a" depending upon which side of the Atlantic you hail from. Can also be read as "soccer tour"). It means:
- SOH ... sin = opposite/hypotenuse
- CAH ... cos = adjacent/hypotenuse
- TOA ... tan = opposite/adjacent. Many other such words and phrases have been contrived. For more see: trigonometry mnemonics.

Slope definitions

Equivalent to the right-triangle definitions, the trigonometric functions can be defined in terms of the rise, run, and slope of a line segment relative to some horizontal line. The slope is commonly taught as "rise over run" or rise/run. The three main trigonometric functions are commonly taught in the order sine, cosine, tangent. With a unit circle, this gives rise to the following matchings: # Sine is first, rise is first. Sine takes an angle and tells the rise. # Cosine is second, run is second. Cosine takes an angle and tells the run. # Tangent is the slope formula that combines the rise and run. Tangent takes an angle and tells the slope. This shows the main use of tangent and arctangent, which is converting between the two ways of telling how slanted a line is: angles and slopes. While the radius of the circle makes no difference for the slope (the slope doesn't depend on the length of the slanted line), it does affect rise and run. To adjust and find the actual rise and run, just multiply the sine and cosine by the radius. For instance, if the circle has radius 5, the run at an angle of 1 is 5 cos(1).

Unit-circle definitions

unit circle] The six trigonometric functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin. The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles. The unit circle definition does, however, permit the definition of the trig functions for all positive and negative arguments, not just for angles between 0 and π/2 radians. It also provides a single visual picture that encapsulates at once all the important triangles used so far. The equation for the unit circle is: :

x^2 + y^2 = 1 \,

In the picture, some common angles, measured in radians, are given. Measurements in the counter clockwise direction are positive angles and measurements in the clockwise direction are negative angles. Let a line making an angle of θ with the positive half of the x-axis intersect the unit circle. The x- and y-coordinates of this point of intersection are equal to cos θ and sin θ, respectively. The triangle in the graphic enforces the formula; the radius is equal to the hypotenuse and has length 1, so we have sin θ = y/1 and cos θ = x/1. The unit circle can be thought of as a way of looking at an infinite number of triangles by varying the lengths of their legs but keeping the lengths of their hypotenuses equal to 1. circle For angles greater than 2π or less than −2π, simply continue to rotate around the circle. In this way, sine and cosine become periodic functions with period 2π: :

\sin\theta = \sin\left(\theta + 2\pi k \right)

\cos\theta = \cos\left(\theta + 2\pi k \right)

for any angle θ and any integer k. The smallest positive period of a periodic function is called the primitive period of the function. The primitive period of the sine, cosine, secant, or cosecant is a full circle, i.e. 2π radians or 360 degrees; the primitive period of the tangent or cotangent is only a half-circle, i.e. π radians or 180 degrees. Above, only sine and cosine were defined directly by the unit circle, but the other four trig functions can be defined by: :

\tan\theta = \frac \quad \sec\theta = \frac

\csc\theta = \frac \quad \cot\theta = \frac

integer Alternatively, all of the basic trigonometric functions can be defined in terms of a unit circle centered at O (shown at right), and similar such geometric definitions were used historically. In particular, for a chord AB of the circle, where θ is half of the subtended angle, sin(θ) is AC (half of the chord), a definition introduced in India (see below). cos(θ) is the horizontal distance OC, and versin(θ) = 1 − cos(θ) is CD. tan(θ) is the length of the segment AE of the tangent line through A, hence the word tangent for this function. cot(θ) is another tangent segment, AF. sec(θ) = OE and csc(θ) = OF are segments of secant lines (intersecting the circle at two points), and can also be viewed as projections of OA along the tangent at A to the horizontal and vertical axes, respectively. DE is exsec(θ) = sec(θ) − 1 (the portion of the secant outside, or ex, the circle). From these constructions, it is easy to see that the secant and tangent functions diverge as θ approaches π/2 (90 degrees) and that the cosecant and cotangent diverge as θ approaches zero. (Many similar constructions are possible, and the basic trigonometric identities can also be proven graphically.)

Series definitions

exsec Please note: Here, and generally in calculus, all angles are measured in radians. (See also below). Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine and the derivative of cosine is the opposite of sine. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x: :

\sin x = x - \frac + \frac - \frac + \cdots = \sum_^\infty \frac

\cos x = 1 - \frac + \frac - \frac + \cdots = \sum_^\infty \frac

These identities are often taken as the definitions of the sine and cosine function. They are often used as the starting point in a rigorous treatment of trigonometric functions and their applications (e.g. in Fourier series), since the theory of infinite series can be developed from the foundations of the real number system, independent of any geometric considerations. The differentiability and continuity of these functions are then established from the series definitions alone. Other series can be found: :

\tan x = x + \frac + \frac + \frac + \cdots = \sum_^\infty \frac, \quad \left | x \right | < \frac

\csc x = \frac   + \frac   + \frac   + \frac   + \cdots = \frac   + \sum_^\infty \frac, \quad 0 < \left | x \right | < \frac

\sec x = 1 + \frac   + \frac   + \frac   + \cdots = 1+ \sum_^\infty \frac, \quad \left | x \right | < \frac

\cot x = \frac   - \frac  - \frac   - \frac   - \cdots = \frac   - \sum_^\infty \frac, \quad 0 < \left | x \right | < \frac

where :

E_n \,

is the nth Euler number, and :

U_n \,

is the nth up/down number.

Relationship to exponential function

It can be shown from the series definitions that the sine and cosine functions are the imaginary and real parts, respectively, of the complex exponential function when its argument is purely imaginary: :

e^ = \cos\theta + i\sin\theta \,.

This relationship was first noted by Euler and the identity is called Euler's formula. In this way, trigonometric functions become essential in the geometric interpretation of complex analysis. For example, with the above identity, if one considers the unit circle in the complex plane, defined by e^ix, and as above, we can parametrize this circle in terms of cosines and sines, the relationship between the complex exponential and the trigonometric functions becomes more apparent. Furthermore, this allows for the definition of the trigonometric functions for complex arguments z: :

\sin z \, = \, \sum_^\fracz^ \, = \,  = -\imath \sinh \left( \imath z\right)

\cos z \, = \, \sum_^\fracz^ \, = \,  = \cosh \left(\imath z\right)

where i^{2 = −1. Also, for purely real x,

: $\cos x \, = \, \mbox (e^)$

: $\sin x \, = \, \mbox (e^)$

It is also shown that exponential processes are intimately linked to periodic behavior.

Definitions via differential equations

Both the sine and cosine functions satisfy the differential equation

: $y\,$ =-y

i.e. each is the additive inverse of its own second derivative. Within the 2-dimensional vector space V consisting of all solutions of this equation, the sine function is the unique solution satisfying the initial conditions y(0) = 0 and y′(0) = 1, and the cosine function is the unique solution satisfying the initial conditions y(0) = 1 and y′(0) = 0. Since the sine and cosine functions are linearly independent, together they form a basis of V. This method of defining the sine and cosine functions is essentially equivalent to using Euler's formula. (See linear differential equation.) It turns out that this differential equation can be used not only to define the sine and cosine functions but also to prove the trigonometric identities for the sine and cosine functions. See the trigonometric identity article for this technique.

The tangent function is the unique solution of the nonlinear differential equation

: $y\,'=1+y^2$

satisfying the initial condition y(0) = 0. There is a very interesting visual proof that the tangent function satisfies this differential equation; see [http://www.usfca.edu/vca/PDF/vca-preface.pdf].

The significance of radians
Radians constitute a special argument to the sine and cosine functions. In particular, only those sines and cosines which map radians to ratios satisfy the differential equations which classically describe them. If an argument to sine or cosine in radians is scaled by frequency,
: $f(x) = \sin(kx); k \ne 0, k \ne 1 \,$
then the derivatives will scale by amplitude.
: $f'(x) = k\cos(kx) \,$ .

Here, k is a constant that represents a mapping between units. If x is in degrees, then
: $k = \frac.$

This means that the second derivative of a sine in degrees satisfies not the differential equation
: $y$ = -y \,,
but
: $y$ = -k^2y \,;
similarly for cosine.

This means that these sines and cosines are different functions, and that the fourth derivative of sine will be sine again only if the argument is in radians.

Other definitions

Theorem: There exists exactly one pair of real functions
s, c with the following properties:

For any $x, y \in\mathbb$ :

: $s(x)^2 + c(x)^2 = 1,\,$

: $s(x+y) = s(x)c(y) + c(x)s(y),\,$

: $c(x+y) = c(x)c(y) - s(x)s(y),\,$

: $0 < xc(x) < s(x) < x\ \mathrm\ 0 < x < 1.$

Computation

The computation of trigonometric functions is a complicated subject, which can today be avoided by most people because of the widespread availability of computers and scientific calculators that provide built-in trigonometric functions for any angle. In this section, however, we describe more details of their computation in three important contexts: the historical use of trigonometric tables, the modern techniques used by computers, and a few "important" angles where simple exact values are easily found. (Below, it suffices to consider a small range of angles, say 0 to π/2, since all other angles can be reduced to this range by the periodicity and symmetries of the trigonometric functions.)

Prior to computers, people typically evaluated trigonometric functions by interpolating from a detailed table of their values, calculated to many significant figures. Such tables have been available for as long as trigonometric functions have been described (see History, above), and were typically generated by repeated application of the half-angle and angle-addition identities starting from a known value (such as sin(π/2)=1). See also: Generating trigonometric tables.

Modern computers use a variety of techniques (Kantabutra, 1996). One common method, especially on higher-end processors with floating point units, is to combine a polynomial approximation (such as a Taylor series or a rational function) with a table lookup — they first look up the closest angle in a small table, and then use the polynomial to compute the correction. On simpler devices that lack hardware multipliers, there is an algorithm called CORDIC (as well as related techniques) that is more efficient, since it uses only shifts and additions. All of these methods are commonly implemented in hardware for performance reasons.

Finally, for some simple angles, the values can be easily computed by hand using the Pythagorean theorem, as in the following examples. In fact, the sine, cosine and tangent of any integer multiple of π/60 radians (three degrees) can be found exactly by hand.

Consider a right triangle where the two other angles are equal, and therefore are both π/4 radians (45 degrees). Then the length of side b and the length of side a are equal; we can choose a = b = 1. The values of sine, cosine and tangent of an angle of π/4 radians (45 degrees) can then be found using the Pythagorean theorem:

: $c = \sqrt = \sqrt2$

Therefore:

: $\sin \left(\pi / 4 \right) = \sin \left(45^\circ\right) = \cos \left(\pi / 4 \right) = \cos \left(45^\circ\right) =$
: $\tan \left(\pi / 4 \right) = \tan \left(45^\circ\right) = = 1$

To determine the trigonometric functions for angles of π/3 radians (60 degrees) and π/6 radians (30 degrees), we start with an equilateral triangle of side length 1. All its angles are π/3 radians (60 degrees). By dividing it into two, we obtain a right triangle with π/6 radians (30 degrees) and π/3 radians (60 degrees) angles. For this triangle, the shortest side = 1/2, the next largest side =(√3)/2 and the hypotenuse = 1. This yields:

: $\sin \left(\pi / 6 \right) = \sin \left(30^\circ\right) = \cos \left(\pi / 3 \right) = \cos \left(60^\circ\right) =$
: $\cos \left(\pi / 6 \right) = \cos \left(30^\circ\right) = \sin \left(\pi / 3 \right) = \sin \left(60^\circ\right) =$
: $\tan \left(\pi / 6 \right) = \tan \left(30^\circ\right) = \cot \left(\pi / 3 \right) = \cot \left(60^\circ\right) =$

See also: Exact trigonometric constants

Inverse functions

The trigonometric functions are periodic, so we must restrict their domains before we are able to define a unique inverse. In the following, the functions on the left are defined by the equation on the right; these are not proved identities. The principal inverses are usually defined as:

: $\begin

\mbox & -\frac \le y \le \frac,
& y = \arcsin(x) & \mbox & x = \sin(y) \\
\mbox & 0 \le y \le \pi,
& y = \arccos(x) & \mbox & x = \cos(y) \\
\mbox & -\frac < y < \frac,
& y = \arctan(x) & \mbox & x = \tan(y) \\
\mbox & -\frac \le y \le \frac, y \ne 0,
& y = \arccsc(x) & \mbox & x = \csc(y) \\
\mbox & 0 \le y \le \pi, y \ne \frac,
& y = \arcsec(x) & \mbox & x = \sec(y) \\
\mbox & -\frac < y < \frac, y \ne 0,
& y = \arccot(x) & \mbox & x = \cot(y) \\

\end$

For inverse trigonometric functions, the notations sin⁻¹ and cos⁻¹ are often used for arcsin and arccos, etc. When this notation is used, the inverse functions are sometimes confused with the multiplicative inverses of the functions. Our notation avoids such confusion.

The following series definition may be obtained:

: $\begin
\arcsin z & = & z + \left( \frac \right) \frac + \left( \frac \right) \frac + \left( \frac \right) \frac + \cdots\\
& = & \sum_^\infty \left( \frac \right) \frac
\end
\ , \quad \left| z \right| < 1$

: $\begin
\arccos z & = & \frac - \arcsin z \\
& = & \frac - (z + \left( \frac \right) \frac + \left( \frac \right) \frac + \left( \frac \right) \frac + \cdots ) \\
& = & \frac - \sum_^\infty \left( \frac \right) \frac
\end
\ , \quad \left| z \right| < 1$

: $\begin
\arctan z & = & z - \frac +\frac -\frac +\cdots \\
& = & \sum_^\infty \frac
\end
\ , \quad \left| z \right| < 1$

: $\begin
\arccsc z & = & \arcsin\left(z^\right) \\
& = & z^ + \left( \frac \right) \frac + \left( \frac \right) \frac + \left( \frac \right) \frac +\cdots \\
& = & \sum_^\infty \left( \frac \right) \frac \end
\ , \quad \left| z \right| > 1$

: $\begin
\arcsec z & = & \arccos\left(z^\right) \\
& = & \frac - (z^ + \left( \frac \right) \frac + \left( \frac \right) \frac + \left( \frac \right) \frac + \cdots ) \\
& = & \frac - \sum_^\infty \left( \frac \right) \frac
\end
\ , \quad \left| z \right| > 1$

: $\begin
\arccot z & = & \frac - \arctan z \\
& = & \frac - ( z - \frac +\frac -\frac +\cdots ) \\
& = & \frac - \sum_^\infty \frac
\end
\ , \quad \left| z \right| < 1$

These functions may also be defined by proving that they are antiderivatives of other functions.

: $\arcsin\left(x\right) =
\int_0^x \frac 1 \,\mathrmz, \quad |x| < 1$
: $\arccos\left(x\right) =
\int_x^1 \frac \,\mathrmz,\quad |x| < 1$
: $\arctan\left(x\right) =
\int_0^x \frac 1 \,\mathrmz,
\quad \forall x \in \mathbb$
: $\arccot\left(x\right) =
\int_x^\infty \frac \,\mathrmz,
\quad z > 0$
: $\arcsec\left(x\right) =
\int_x^1 \frac 1 \,\mathrmz, \quad x > 1$
: $\arccsc\left(x\right) =
\int_x^\infty \frac \,\mathrmz, \quad x > 1$

Inverse trigonometric functions can be generalized to complex arguments using the complex logarithm.

: $\arcsin (z) = -i \log \left( i \left( z + \sqrt\right) \right)$
: $\arccos (z) = -i \log \left( z + \sqrt\right)$
: $\arctan (z) = \frac \log\left(\frac\right)$

Note: arcsec can also mean arcsecond.

Identities

: $\sin \left(x+y\right)=\sin x \cos y + \cos x \sin y$

: $\sin \left(x-y\right)=\sin x \cos y - \cos x \sin y$

: $\cos \left(x+y\right)=\cos x \cos y - \sin x \sin y$

: $\cos \left(x-y\right)=\cos x \cos y + \sin x \sin y$

: $\sin x+\sin y=2\sin \left( \frac \right) \cos \left( \frac \right)$

: $\sin x-\sin y=2\cos \left( \frac \right) \sin \left( \frac \right)$

: $\cos x+\cos y=2\cos \left( \frac \right) \cos \left( \frac \right)$

: $\cos x-\cos y=-2\sin \left( \frac \right)\sin \left( \frac \right)$

: $\tan x+\tan y=\frac$

: $\tan x-\tan y=\frac$

: $\cot x+\cot y=\frac$

: $\cot x-\cot y=\frac$

See also trigonometric identity.

Properties and applications

The trigonometric functions, as the name suggests, are of crucial importance in trigonometry, mainly because of the following two results:

Law of sines

The law of sines for an arbitrary triangle states:

: $\frac = \frac = \frac$

It can be proven by dividing the triangle into two right ones and using the above definition of sine. The common number (sinA)/a occurring in the theorem is the reciprocal of the diameter of the circle through the three points A, B and C. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

Law of cosines

The law of cosines (also known as the cosine formula) is an extension of the Pythagorean theorem:

: $c^2=a^2+b^2-2ab\cos C \,$

Again, this theorem can be proven by dividing the triangle into two right ones. The law of cosines is useful to determine the unknown data of a triangle if two sides and an angle are known.

If the angle is not contained between the two sides, the triangle may not be unique. Be aware of this ambiguous case of the Cosine law.

Law of tangents

There is also a law of tangents:

: $\frac = \frac$

law of tangents The trigonometric functions are also important outside of the study of triangles. They are periodic functions with characteristic wave patterns as graphs, useful for modelling recurring phenomena such as sound or light waves. Every signal can be written as a (typically infinite) sum of sine and cosine functions of different frequencies; this is the basic idea of Fourier analysis, where trigonometric series are used to solve a variety of boundary-value problems in partial differential equations.

The image on the right displays a two-dimensional graph based on such a summation of sines and cosines, illustrating the fact that arbitrarily complicated closed curves can be described by a Fourier series. Its equation is:
: $(x(\theta),\,y(\theta)) = \sum_^\infty \frac (\sin(\theta\cdot F(n)),\, \cos(\theta\cdot F(n)))$
where F(n) is the nth Fibonacci number.

For a compilation of many relations between the trigonometric functions, see trigonometric identities.

References

- Carl B. Boyer, A History of Mathematics, 2nd ed. (Wiley, New York, 1991).

- Eli Maor, [http://www.pupress.princeton.edu/books/maor/ Trigonometric Delights] (Princeton Univ. Press, 1998).

- "[http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Trigonometric_functions.html Trigonometric functions]", MacTutor History of Mathematics Archive.

- Tristan Needham, Visual Complex Analysis, (Oxford University Press, 2000), ISBN 0198534469 [http://www.usfca.edu/vca Book website]

- Vitit Kantabutra, "On hardware for computing exponential and trigonometric functions," IEEE Trans. Computers 45 (3), 328-339 (1996).

See also

- Generating trigonometric tables

- Hyperbolic function

- Pythagoras

- Pythagorean theorem

- Trigonometric identity

- Uses of trigonometry

- Direction cosines

- [http://wikisource.org/wiki/Trigonometric_functions_of_angles_0%C2%B0_to_90%C2%B0_by_degree Trigonometric functions of angles 0° to 90° by degree]

External links

- [http://www.walterzorn.com/grapher/grapher_e.htm Javascript function grapher] uses a javascript library to display functions. Works in nearly every modern browser.

- [http://www.geocities.com/SiliconValley/Garage/3323/aat/a_sin.html Sine and cosine function ] with an implementation in Rexx.
Category:Trigonometry
Category:Special functions

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Exponential function
The exponential function is one of the most important functions in mathematics. It is written as exp(x) or e^x, where e is the base of the natural logarithm.

base of the natural logarithm

As a function of the real variable x, the graph of e^x is always positive (above the x axis) and increasing (viewed left-to-right). It never touches the x axis, although it gets arbitrarily close to it (thus, the x axis is a horizontal asymptote to the graph). Its inverse function, the natural logarithm, ln(x), is defined for all positive x.

Sometimes, especially in the sciences, the term exponential function is reserved for functions of the form ka^x,
where a, called the base, is any positive real number. This article will focus initially on the exponential function with base e.

In general, the variable x can be any real or complex number, or even an entirely different kind of mathematical object; see the formal definition below.

Properties
Using the natural logarithm, one can define more general exponential functions. The function
: $\!\, a^x=e^$
defined for all a > 0, and all real numbers x, is called the exponential function with base a.

Note that the equation above holds for a = e, since
: $\!\, e^=e^=e^x.$

Exponential functions "translate between addition and multiplication" as is expressed in the following exponential laws:
: $\!\, a^0 = 1$
: $\!\, a^1 = a$
: $\!\, a^ = a^x a^y$
: $\!\, a^ = \left( a^x \right)^y$
: $\!\, = \left(\right)^x = a^$
: $\!\, a^x b^x = (a b)^x$

These are valid for all positive real numbers a and b and all real numbers x and y. Expressions involving fractions and roots can often be simplified using exponential notation because:
: $= a^$
and, for any a > 0, real number b, and integer n > 1:
: $\sqrt[n] = \left(\sqrt[n]\right)^b = a^$

For any real constant c holds:
: $f'(0)=\lim_\frac=c$
for $f(x)=e^$

Derivatives and differential equations

The importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives. In particular,

: $e^x = e^x$

That is, e^x is its own derivative, a property unique among real-valued functions of a real variable. Other ways of saying the same thing include:

- The slope of the graph at any point is the height of the function at that point.

- The rate of increase of the function at x is equal to the value of the function at x.

- The function solves the differential equation $y'=y$ .

- exp is a fixed point of derivative as a functional

In fact, many differential equations give rise to exponential functions, including the Schrödinger equation and the Laplace's equation as well as the equations for simple harmonic motion.

For exponential functions with other bases:

: $a^x = (\ln a) a^x$

Thus any exponential function is a constant multiple of its own derivative.

If a variable's growth or decay rate is proportional to its size — as is the case in unlimited population growth (see Malthusian catastrophe), continuously compounded interest, or radioactive decay — then the variable can be written as a constant times an exponential function of time.

Furthermore for any differentiable function f(x) holds:

: $e^ = f'(x)e^$

Formal definition

The exponential function e^x can be defined in two equivalent ways, as an infinite series:
: $e^x = \sum_^ = 1 + x + + + + \cdots$
or as the limit of a sequence:
: $e^x = \lim_ \left( 1 + \right)^n$ .

In these definitions, $n!$ stands for the factorial of n, and x can be any real number, complex number, element of a Banach algebra (for example, a square matrix), or member of the field of p-adic numbers.

For further explanation of these definitions and a proof of their equivalence, see the article Definitions of the exponential function.

Numerical value

To obtain the numerical value of the exponential function. The infinite series can be rewritten as :

: $e^x = + x \, \left( + x \, \left( + x \, \left( + \cdots \right)\right)\right)$
: $= 1 + \left(1 + \left(1 + \left(1 + \cdots \right)\right)\right)$

This expression will converge quickly if we can ensure that x is less than one.

To ensure this, we can use the following identity.

:

- Where $z$ is the integer part of $x$

- Where $f$ is the fractional part of $x$

- Hence, $f$ is a always less than 1 and $f$ and $z$ add up to $x$ .

The value of the constant e^z can be calculated beforehand by multiplying e with itself z times.

On the complex plane

When considered as a function defined on the complex plane, the exponential function retains the important properties
: $\!\, e^ = e^z e^w$
: $\!\, e^0 = 1$
: $\!\, e^z \ne 0$
: $\!\, e^z = e^z$
for all z and w.

It is a holomorphic function which is periodic with imaginary period $2 \pi i$ and can be written as
: $\!\, e^ = e^a (\cos b + i \sin b)$
where a and b are real values. This formula connects the exponential function with the trigonometric functions and to the hyperbolic functions. Thus we see that all elementary functions except for the polynomials spring from the exponential function in one way or another.

See also Euler's formula.

Extending the natural logarithm to complex arguments yields a multi-valued function, ln(z). We can then define a more general exponentiation:
: $\!\, z^w = e^$
for all complex numbers z and w. This is also a multi-valued function. The above stated exponential laws remain true if interpreted properly as statements about multi-valued functions.

The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin. This can be seen by noting that the case of a line parallel with the real or imaginary axis maps to a line or circle.

Matrices and Banach algebras

The definition of the exponential function given above can be used verbatim for every Banach algebra, and in particular for square matrices (in which case the function is called the matrix exponential). In this case we have
: $\ e^ = e^x e^y \mbox xy = yx$
: $\ e^0 = 1$
: $\ e^x$ is invertible with inverse $\ e^$
: the derivative of $\ e^x$ at the point $\ x$ is that linear map which sends $\ u$ to $\ ue^x$ .

In the context of non-commutative Banach algebras, such as algebras of matrices or operators on Banach or Hilbert spaces, the exponential function is often considered as a function of a real argument:
: $\ f(t) = e^$
where A is a fixed element of the algebra and t is any real number. This function has the important properties
: $\ f(s + t) = f(s) f(t)$
: $\ f(0) = 1$
: $\ f'(t) = A f(t)$

On Lie algebras
The "exponential map" sending a Lie algebra to the Lie group that gave rise to it shares the above properties, which explains the terminology. In fact, since R is the Lie algebra of the Lie group of all positive real numbers with multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie algebra M(n, R) of all square real matrices belongs to the Lie group of all invertible square matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.

Double exponential function
The term double exponential function can have two meanings:

- a function with two exponential terms, with different exponents

- a function f(x)=a^a^x; this grows even faster than an exponential function; for example, if a=10: f(-1)=1.26, f(0)=10, f(1)=1e10, f(2)=1e100=googol, f(3)=1e1000, ..., f(100)=googolplex.

Compare the super-exponential function, which grows even faster.

See also

- exponential growth

External links

-

Category:Special functions
Category:Complex analysis
Category:Exponentials
Category:Special hypergeometric functions

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Reverse Polish notation
Reverse Polish notation (RPN), also known as postfix notation, was invented by Australian philosopher and computer scientist Charles Hamblin in the mid-1950s, to enable zero-address memory stores. It is derived from the Polish notation, which was introduced in 1920 by the Polish mathematician Jan Łukasiewicz. Hamblin presented his work at a conference in June 1957, and published it in 1957 and 1962.

The first computers to implement architectures enabling RPN were the English Electric Company's KDF9 machine, which was announced in 1960 and delivered (i.e. made available commercially) in 1963, and the American Burroughs B5000, announced in 1961 and also delivered in 1963. One of the designers of the B5000, R. S. Barton, later wrote that he developed RPN independently of Hamblin, sometime in 1958 while reading a textbook on symbolic logic, and before he was aware of Hamblin's work.

A decade after Hamblin first published his ideas, engineers at Hewlett-Packard (HP) developed a personal calculator, the 9100A Desktop Calculator, which used RPN. This calculator, the first in a long line by HP, was released in 1968, and it popularized RPN among the scientific and engineering communities; note, however, that early advertisements for the 9100A did not mention RPN. As a user interface for calculation the notation was first used in Hewlett-Packard's desktop calculators from the late 1960s and then in the HP-35 handheld scientific calculator launched in 1972. In RPN the operands precede the operator, thus dispensing with the need for parentheses. For example, the expression 3
- ( 4 + 7) would be written as 3 4 7 +
- , and done on an RPN calculator as "3", "Enter", "4", "Enter", "7", "+", "
- ". (Alternatively, and more-compactly, it could also be re-ordered and written as 4 7 + 3
- , and done on an RPN calculator as "4", "Enter", "7", "+", "3", "
- ".)

Implementations of RPN are stack-based; that is, operands are popped from a stack, and calculation results are pushed back onto it. Although this concept may seem obscure at first, RPN has the advantage of being extremely easy, and therefore fast, for a computer to analyze.

Practical implications

- Calculations proceed from left to right

- There are no brackets or parentheses, as they are unnecessary.

- Operands precede operator. They are removed as the operation is evaluated.

- When an operation is made, the result becomes an operand itself (for later operators)

- There is no hidden state. No need to wonder if you hit an operator or not.

- RPN calculators have no parentheses or brackets keys, nor an equals key.

- Fewer keystrokes are needed on an RPN calculator than on an algebraic notation calculator for most computations.

Example

The calculation: ((1 + 2)
- 4) + 3 can be written down like this in RPN:
1 2 + 4
- 3 +
The expression is evaluated in the following way (the Stack is displayed after Operation has taken place):

Input Stack Operation
1 1 Push operand
2 1, 2 Push operand
+ 3 Addition
4 3, 4 Push operand

- 12 Multiplication
3 12, 3 Push operand
+ 15 Addition

The final result, 15, lies on the top of the stack at the end of the calculation.

An alternate way of viewing the stack during the above operation is
shown below (as seen on HP48S calculator).

+---------------+
| |
| |
| 1 | 1 enter
+---------------+

+---------------+
| |
| 1 |
| 2 | 2 [enter]
+---------------+

+---------------+
| |
| |
| 3 | +
+---------------+

+---------------+
| |
| 3 |
| 4 | 4 [enter]
+---------------+

+---------------+
| |
| |
| 12 |
-
+---------------+

+---------------+
| |
| 12 |
| 3 | 3 [enter]
+---------------+

+---------------+
| |
| |
| 15 | +
+---------------+

The enters are in brackets because they are optional when followed by
an operator press. An enter is only needed to clear the insertion
mark from the line. Thus, RPN allows the expression to be entered and
evaluated in eight rather than eleven or twelve steps.

Converting from infix notation

Edsger Dijkstra invented an algorithm, named the "shunting yard" algorithm because its operation resembles that of a railroad shunting yard, which converts from infix notation to RPN. Like the evaluation of RPN, the shunting yard algorithm is stack-based. Infix expressions are the form of math most people are used to, for instance 3+4 or 3+4
- (2-1). For the conversion there are 2 text variables (strings), the input and the output. There is also a stack holding operators not yet added to the output stack. To convert, the program reads each letter in order and does something based on that letter.

A simple conversion

Input: 3+4
#Add 3 to the output queue (whenever a number is read it is added to the output)
#Push + (or its ID) onto the operator stack
#Add 4 to the output queue
#After reading expression pop the operators off the stack and add them to the output.
# In this case there is only one, "+".
#Output 3 4 +

This already shows a couple of rules:

- All numbers are added to the output when they are read.

- At the end of reading the expression, pop all operators off the stack and onto the output.

The algorithm in detail

- While there are tokens to be read:
:: Read a token.
::
- If the token is a number, then add it to the output queue.
::
- If the token is a function token, then push it onto the stack.
::
- If the token is a function argument separator (e.g., a comma):
::::
- Until the topmost element of the stack is a left parenthesis, pop the element onto the output queue. If no left parentheses are encountered, either the separator was misplaced or parentheses were mismatched.
::
- If the token is an operator, o₁, then:
:::1) while there is an operator, o₂, at the top of the stack, and either
:::::: o₁ is left-associative and its precedence is less than or equal to that of o₂, or
:::::: o₁ is right-associative and its precedence is less than that of o₂,

:::: pop o₂ off the stack, onto the output queue;
:::2) push o₁ onto the operator stack.
::
- If the token is a left parenthesis, then push it onto the stack.
::
- If the token is a right parenthesis, then pop operators off the stack, onto the output queue, until the token at the top of the stack is a left parenthesis, at which point it is popped off the stack but not added to the output queue. At this point, if the token at the top of the stack is a function token, pop it too onto the output queue. If the stack runs out without finding a left parenthesis, then there are mismatched parentheses.

- When there are no more tokens to read, pop all the tokens, if any, off the stack, add each to the output as it is popped out and exit. (These must only be operators; if a left parenthesis is popped, then there are mismatched parentheses.)

Complex example

Input 3+4
- 2/(1-5)^2
Read "3"
Add "3" to the output
Output: 3

Read "+"
Push "+" onto the stack
Output: 3
Stack: +

Read "4"
Add "4" to the output
Output: 3 4
Stack: +

Read "
- "
Push "
- " onto the stack
Output: 3 4
Stack: +
-

Read "2"
Add "2" to the output
Output: 3 4 2
Stack: +
-

Read "/"
Pop "
- " off stack and add it to output, push "/" onto the stack
Output: 3 4 2
-
Stack: + /

Read "("
Push "(" onto the stack
Output: 3 4 2
-
Stack: + / (

Read "1"
Add "1" to output
Output: 3 4 2
- 1
Stack: + / (

Read "-"
Push "-" onto the stack
Output: 3 4 2
- 1
Stack: + / ( -

Read "5"
Add "5" to output
Output: 3 4 2
- 1 5
Stack: + / ( -

Read ")"
Pop "-" off stack and add it to the output, pop (
Output: 3 4 2
- 1 5 -
Stack: + /

Read "^"
Push "^" onto stack
Output: 3 4 2
- 1 5 -
Stack: + / ^

Read "2"
Add "2" to output
Output: 3 4 2
- 1 5 - 2
Stack: + / ^

End of Expression
Pop stack to output
Output: 3 4 2
- 1 5 - 2 ^ / +

If you were writing an interpreter, this output would be tokenized and written to a compiled file to be later interpreted. Conversion from Infix to RPN can also allow for easier computer simplification of expressions. To do this, act like you are solving the RPN expression, however, whenever you come to a variable its value is null, and whenever an operator has a null value, it and its parameters are written to the output (this is a simplification, problems arise when the parameters are operators). When an operator has no null parameters its value can simply be written to the output. This method obviously doesn't include all the simplifications possible.

Real-world RPN use

- Forth programming language

- Hewlett-Packard science/engineering calculators

- PostScript page description language

- TI-68k (TI-89) [http://www.paxm.org/symbulator/download/rpn.html implementation]

- Unix system calculator program dc

- Writing an Interpreter

- Interactive JavaScript calculator [http://main.linuxfocus.org/~guido/javascript/rpnjcalc.html with RPN]

- [http://www.lowth.com/rope/LanguageReference Linux IpTables "Rope" programming language]

- Wikibooks:Ada Programming/Mathematical calculations (RPN calculator implemented in Ada)

See also

- HP calculators

- Infix notation

- LIFO

- Polish Notation

- Stack machine

- Subject Object Verb

- Subject Verb Object

External links

- [http://www.xnumber.com/xnumber/rpn_or_adl.htm RPN or DAL? A brief analysis of Reverse Polish Notation against Direct Algebraic Logic] – By James Redin

- [http://www.spsu.edu/cs/faculty/bbrown/web_lectures/postfix/ Postfix Notation Mini-Lecture] – By Bob Brown

- [http://www.langmaker.com/shallowfith.htm Fith: An Alien Conlang With A LIFO Grammar] – By Jeffrey Henning

- RPN
Category:Mathematical notation
Category:Science and technology in Poland

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ja:逆ポーランド記法

Scientific notation

Scientific notation (also known as Standard index notation) is a convenient way to write very small or large numbers. In this notation numbers are separated into two parts, a real number with an absolute value between 1 and 10 and an order of magnitude value written as a power of 10. Formally, scientific notation is floating-point notation with base 10.

Description

In scientific notation numbers are written as $a\times10^b$ where the exponent b is an integer and a is any real number. The number represented by a is called the significand or the mantissa, but the latter may cause confusion as it can also refer to the fractional part of the common logarithm. Usually a is chosen in the range of 1 to 10, excluding 10. Such a fixed range allows easy comparison of two numbers since the one with the larger exponent is larger. In that case b is the number's order of magnitude.

Engineering notation

Restricting the exponent b to multiples of 3 results in what is called engineering notation.

Exponential notation

Most calculators and many computer programs present very large and very small results in scientific notation. Usually the '10' is omitted and replaced by the letter E or, confusingly, e—which is short for exponent. Note that this is not related to the mathematical constant e. For example 1.56234 E+29 is the same as 1.56234×10²⁹. This is commonly called exponential notation.

Motivation
Scientific notation is a very convenient way to write large or small numbers. It also quickly conveys two properties of a measurement that are useful to scientists—significant figures and order of magnitude.

Examples

- An electron's mass is 0.00000000000000000000000000000091093826 Kg. In scientific notation, it is written 9.1093826×10^-31 Kg.

- The Earth's mass is 5,973,600,000,000,000,000,000,000 Kg. In scientific notation, it is written 5.9736×10²⁴ Kg.

Significant digits
Scientific notation is useful for indicating the precision with which the quantity was measured. Including only the significant figures, the digits that are known to be reliable, in the mantissa implicitly conveys value's precision. Any physical quantity in scientific notation is assumed to be precise to no fewer than the quoted number of digits of precision. However, where precision in such measurements is crucial, more sophisticated expressions of measurement error must be used.

As an example, consider the Earth's mass as presented above in conventional notation. Since the representation gives no indication of the accuracy of the reported value, a reader could incorrectly assume that it is known down to the last digit displayed. The scientific notation implicitly shows it is known with a precision of 0.00005×10²⁴ Kg, or 5×10¹⁹ Kg.

Order of magnitude
Scientific notation also enables simple order of magnitude comparisons. A proton's mass is 0.0000000000000000000000000016726 Kg. If it is written as 1.6726×10^-27 Kg, it is easy to compare this mass with that of the electron above. The difference in order of magnitude is obtained simply by comparing the exponents rather than counting all those zeroes. In this case, '-27' is larger than '-31' and therefore the proton is four orders of magnitude more massive than the electron.

Scientific notation also avoids regional differences in certain quantifiers, such as billion, which may be either 10⁹ or 10¹², thus avoiding misunderstanding.

Using scientific notation
Converting
Multiplication and division by 10 are easy to perform both with the mantissa and with the exponential part of a number represented in scientific notation.

At the mantissa, multiplication by 10 may be seen as shifting the decimal point one position to the right (adding a zero if needed): 12.34×10=123.4. Division may be seen as shifting it to the left: 12.34/10=1.234

In the exponential part multiplication by 10 results in adding 1 to the exponent: 10²×10=10³. Division by 10 results in subtracting 1 from the exponent: 10²/10=10¹.

Also notice that 1 is multiplication's neutral element and that 10⁰=1.

To convert between different representations of the same number, all that is needed is to perform the opposite operations to each part. Thus multiplying the mantissa by 10, n times is done by shifting the decimal point n times to the right. Dividing by 10 the same number of times is done by adding -n to the exponent. Some examples:

$123.4 = 123.4\times10^0 = (123.4/10^2) \times (10^0\times10^2) = 1.234\times10^2$

$.001234 = .001234\times10^0 = (.001234\times 10^3) \times (10^0 / 10^3) = 1.234\times10^$

Basic operations
Given two numbers in scientific notation,

: $x_0=a_0\times10^$

: $x_1=a_0\times10^$

Multiplication and division are performed using the rules for operation with exponential functions:

: $x_0 x_1=a_0 a_1\times10^$

: $\frac=\frac\times10^$

some examples are:
: $5.67\times10^ \times 2.34\times10^2 \approx 13.3\times10^ = 1.33\times10^$

: $\frac \approx 0.413\times10^ = 4.13\times10^6$

Addition and subtraction require the numbers to be represented using the same exponential part, in order to simply add, or subtract, the mantissas, so it may take two steps to perform. First, if needed, convert one number to a representation with the same exponential part as the other. This is usually done with the one with the smaller exponent. Second, add or subtract the mantissas.

: $x_1^\star = a_1^\star \times10^$

: $x_0 \pm x_1=x_0 \pm x_1^\star=(a_0\pm a_1^\star)\times10^$

an example:

: $2.34\times10^ + 5.67\times10^ = 2.34\times10^ + 0.567\times10^ \approx 2.91\times10^$

See also

- SI prefixes

- International standard ISO 31-0

External links

- [http://members.aol.com/profchm/sci_not.html What Is Scientific Notation And How Is It Used?]

- [http://www.math.toronto.edu/mathnet/plain/questionCorner/scinot.html Scientific Notation in Everyday Life]

- [http://science.widener.edu/svb/tutorial/scinot.html An exercise in converting to and from scientific notation]

Notation

Light-emitting diode

A light-emitting diode (LED) is a semiconductor device that emits}