calculator

A calculator is a portable, handy electronic calculating machine , with the aid of numerical calculations can be performed. Some recent technical and scientific calculator also dominate symbolic mathematics using a computer algebra system (CAS), so they can about equations change or dissolve.

Virtually all of today's calculator use electronic integrated circuits and LC displays as a display and a battery or solar cell with power supplied.

1972 Casio Mini, one of the first affordable consumer calculators

history

Commodore SR36 from 1974

TI-89 graphing calculator

Even before the introduction of electronic pocket calculators, there was a need for portable computing aids. This was satisfied with mechanical pocket calculators and slide rules. Most of them were simple adding machines. Four-species machines - that is, calculating machines that could add, subtract, multiply and divide - were available in pocket-sized sizes. The best known example is the Curta .

The forerunners of the electronic pocket calculator were electronic desk calculators , in which the degree of integration of the circuit technology was even lower and therefore had larger dimensions.

The first electronic, actually palm-sized pocket calculator was developed by Texas Instruments in 1967 , with a patent from Jack Kilby representing the design extensively. A 1.5 kg prototype of this first pocket calculator is on display at the Smithsonian Institution today . This also ran on batteries; earlier computers needed a power connection. The first commercially sold pocket calculators were manufactured in 1969 and 1970 by the Californian company Compucorp , as well as the Japanese companies Sanyo , Sharp and Canon . Intel developed one of the first microprocessors for the Japanese company Busicom , the Intel 4004 , which came on the market in 1971 and was used in the Busicom 141-PF model. The Casio Mini , published in 1972, is considered to be the first pocket calculator that was affordable for the general public at a retail price of 10,000 yen . In 1972 Texas Instruments launched the SR 10 pocket calculator with its own TMS1000 microprocessor . These calculators did little more than the four basic arithmetic operations . In 1971 Bowmar produced the first pocket calculator available in the USA (Bowmar 901B / "Bowmar Brain", dimensions: 131 mm × 77 mm × 37 mm). It had four functions and an eight-digit red LED display. It was sold for US $ 240. Bowmar had to close in 1976.

In 1972, the HP-35 from Hewlett-Packard was the first technical and scientific pocket calculator with trigonometric , logarithmic and exponential calculation functions . It became a sales success and ushered in the end of the slide rule that was still widespread at the time. One of its developers was Steve Wozniak , who a few years later co-founded the Apple company and, as a computer engineer, significantly influenced the development of the personal computer .

Especially Hewlett-Packard and Texas Instruments developed programmable pocket calculators from 1974 onwards . The first graphical pocket calculators (GTR) came onto the market in the late 1980s .

Distinguishing features

keyboard

In most pocket calculators, data is entered with your finger using small pushbuttons . The keyboard layout depends on the variant of the computer. Some devices are equipped with an alphanumeric keyboard. The following keys can be found on almost every calculator in the order shown:

Basic keyboard layout

MC	MR	M−	M +
C.	±	%	√
7th	8th	9	÷
4th	5	6th	×
1	2	3	-
0	.	=	+

Explanation of the basic keys

MC	M emory C lear (clear memory)
MR	M emory R ecall (retrieve stored value)
M−	M emory subtraction (subtract from memory content)
M+	M emory addition (add to the memory contents)
C	C lear (erase everything)
±	Change of sign
%	percent
÷	division
×	multiplication
−	subtraction
+	addition
.	Decimal point
√	square root
=	Result

Often there is Calso a CEkey next to the key : C lear E ntry; (only delete last entry).

Input logic

Casio fx-991DE Plus

Depending on the type of calculator, different entries are required to calculate the same function:

• Sequential input: immediate execution of the operations:

3 × 8 + 2 =returns 26, but 2 + 8 × 3 =returns 30. The operations are evaluated directly in the order in which they are entered. Operations (a + b) × (c + d) with two intermediate results cannot be calculated directly.

• Algebraic Notation: When Algebraic operating system that is operator precedence considered:

Both 2 + 8 × 3 =and 8 × 3 + 2 =gives 26. Both entries give the result 26, since multiplication has priority over addition. But if (2 + 8) × 3 is asked, you have 2 + 8 = × 3 =to type. Operations (a + b) × (c + d) with two intermediate results cannot be calculated directly.

• Algebraic notation with brackets :

Both 2 + 8 × 3 =and 8 × 3 + 2 =results in 26. However, if (2 + 8) × 3 is asked, you must type ( 2 + 8 ) × 3 =. The additional bracket keys enable a more flexible input sequence. There is a maximum number of bracket levels (usually 8).

• Conventional algebraic notation :

While operations with two operands (+, -, *, /) are entered, however they are written, and are only executed when you press "=", functions (one-digit operations) are executed immediately when you press the corresponding key, because there is no need to wait for a second operand. This means that you have to enter the argument before the function, e.g. B. 4 sin 30 ° is entered as 4 × 3 0 sin =.

• Direct algebraic logic - depending on the manufacturer with "DAL" (Sharp), "VPAM" - engl. for Visually Perfect Algebraic Method (Casio) or "AOS" ( Algebraic Operating System , Texas Instruments) - and is usually printed on the housing:

The difference is important for teaching in schools, because there it regularly happens that key sequences are announced and several students are also typed. If the students have computers with different input logic, misunderstandings arise.

• Reverse Polish Notation (RPN), based on a stack - Computer Architecture :

With this input logic, the operator is always entered after the operands. Occasionally the ENTER key must be used to separate operands. Computers of this type can usually be recognized by the ENTER key, while the "=" key is missing

3 ENTER 8 × 2 +, unusual but possible 2 ENTER 3 ENTER 8 × +.

Some pocket calculators like the HP-49G + and the HP 35s can also be switched between the reverse Polish notation and the algebraic notation.

• Two-dimensional input editor :

More and more newer models like Casio fx-991ES or TI-30X Plus MultiView also have a two-dimensional input editor like the HP-48, which was built in 1989 . This means that the input and typically also the output takes place as you write or print.

variants

• Easy ( basic arithmetic , percentage calculation )
• Finance ( interest calculation , ...), for example the "classic" HP-12C (manufactured since 1981), HP 17 B (II) (manufactured since 1988)
• Boolean calculations (calculating with dual , octal and hexadecimal numbers , e.g. TI Programmer )
• Scientific ( trigonometric functions , logarithm , mathematical statistics , ...), for example TI-30 (since 1976 the same type designation for completely different internal devices)
• Programmable calculator
• Graphical pocket calculator (function / curve display) - from 1985 devices from Casio (fx series, current model fx 9860G SD), 1989 until today the HP-48-49 series, from 1993 the TI-82 and its successors, currently the TI-84 Plus and TI-Nspire. Graphics calculators are typically also programmable.
• Computer algebra calculators (graphical pocket calculators with built-in computer algebra core ); The first popular device was the HP-48 from Hewlett-Packard (from 1989), later devices for example TI-92 (+) (from 1995), TI-89 (from 1998) and Voyage 200 (from 2002), TI-Nspire CAS (from 2007) from Texas Instruments , ClassPad 300 from Casio , HP 49g + from Hewlett-Packard.

Most current models contain several of the above function groups, in some cases even with a simple spreadsheet .

Numerical accuracy

Even if today's pocket calculators usually have hardly any program errors in simple calculations, different accuracies and resolutions can be determined for numerical calculations between different pocket calculator models. The reasons are the numerical approximation methods (for example Horner's scheme and CORDIC ), with which, for example, transcendent functions such as the sine function are calculated. More precisely, it depends on the number of stored coefficients for the function approximations: the memory space required for this was an extreme bottleneck, especially in the early days. These small differences in the procedures and different levels of accuracy can also be used as identifiers for a specific firmware .

For example, the numerical calculation of sin (22) in radians on different pocket calculators gives the following results that differ from one another:

computer	Value for sin (22)
The first 40 significant digits :	−0.008851309290403875921690256815772332463289 ...
Casio FX-3900Pv	−0.008851309 4194
Casio fx-991D, Casio FX-82SX, Casio FX-702P, Casio FX-603P, Casio fx -5000F	−0.0088513092 19
Casio FX-992S	−0.008851309290 957
Casio fx-7400GII, Casio fx-CG 20	−0.008851309290 35653
Casio FX-850P, Casio FX-880P 20	−0.0088513092 1901
Casio ALGEBRA FX 2.0 PLUS, Casio FX-85ES, Casio CFX-9850G, Casio fx-991DE PLUS, Casio fx-82DE PLUS, Casio fx-991DE X	−0.008851309290 35655
Casio ClassPad 330 (Ver. 3.03)	−0.008851309290 35651226567489 ...
Casio fx-991ES	−0.008851309290 21092
Casio fx-180P	−0.00885130 78196
HP-10s	−0.008851309290 389
HP-11C, HP15C , HP-34C, HP-41 , Casio FX-85MS, Casio FX-115MS, Casio fx-991WA	−0.0088513092 89
HP-25 , HP 45, HP-65	−0.00885130 6326
HP-48 S / X, HP 48G / X, HP 49G, HP 49G +, HP 50, HP-33s, HP 35s , HP-71B , HP Prime	−0.0088513092904
Logitech LC-605	−0.00885130 4
Sharp EL-506 P, Sharp EL-5020, Sharp EL-5120, TI-35x, TI-52, Sharp PC-1401	−0.008851309
Sharp EL-W506, EL-W531	−0.008851309290 2112
Sharp EL-520R	−0.008851309 15412
Sharp EL-9900	−0.008851309290 2122
Sharp PC-E500 (S) (after switching to DEFDBL)	−0.008851309290403875921 7
Simvalley Instruments GRC-1000	−0.0088513092 88957
Texas Instruments TI-25, TI-30-SLX , S chul- R echner 1	−0.0088 487
Texas Instruments TI-30 (Red LEDs), TI-45, CASIO fx-3600P	−0.00885130 7832
Texas Instruments TI-30 eco RS	−0.008851309 3286
Texas Instruments TI-30X IIS, TI-36X II	−0.0088513092 88956
Texas Instruments TI-35 II	−0.0088513
Texas Instruments SR-51-II	−0.00885130929 151
Texas Instruments TI-51-III	−0.008851309 7488
Texas Instruments TI-59	−0.0088513092 85516
Texas Instruments TI-66	−0.00885130929040 8
Texas Instruments TI-89	−0.0088513092904
Texas Instruments TI-200 , TI-89 Titanium , TI-83 Plus	−0.008851309290 3565
Texas Instruments TI-Nspire CAS (early version)	−0.008851309290 1566
Texas Instruments TI-Nspire CAS (current version)	−0.008851309290 16

Even if only the basic arithmetic operations are used, errors can occur, since every simple pocket calculator works with fixed-point numbers and every scientific pocket calculator works with floating-point numbers . Is z. B. on a simple 8-digit calculator

1 2 3 4 5 6 7 8 + 0 . 1 - 1 2 3 4 5 6 7 8 =

the result is 0 instead of the correct 0.1. Similarly, on a scientific pocket calculator with a 12-digit mantissa, entering

1 + 1 ×10^ - 1 3 - 1 =

to the wrong result 0 (instead of correct ), because the mean summand is smaller than the machine accuracy. ${\textstyle 10^{-13}}$

Both examples also show that laws of mathematics such as the commutative law are generally no longer valid on pocket calculators; If you swap the second with the third addend when entering the data, the calculations are correct in both cases.

Especially when performing calculations one after the other, the errors can accumulate to a completely unusable end result .

Recent developments

• Pocket calculator with exact arithmetic and natural representation of terms : B. rationalizing the denominator. This allows them to advance into areas that were previously reserved for computer algebra computers. Examples are Casio FX-85ES , Casio fx-991ES or Texas Instruments TI-30X Plus MultiView .
• Integrated / interactive pocket calculators or computers : pocket calculators that integrate the basic mathematical software types ( computer algebra , dynamic geometry , spreadsheets ) into a coherent system and thus go well beyond the previously known computer algebra calculators. The first representatives have been the Casio ClassPad 300 models since 2002 and Texas Instruments TI-Nspire and TI-Nspire CAS since 2007 .

Admission requirements in schools

Various abbreviations for the respective device classes have emerged in schools:

• WTR: Scientific pocket calculator, scientific school calculator
• GTR: graphical calculator, numerical graphing calculator
• CAS: graphical calculator with computer algebra system

Situation in Germany

With a resolution dated October 18, 2012, the Standing Conference (KMK) introduced educational standards for the general higher education entrance qualification in various subjects, including mathematics, thereby replacing the uniform examination requirements for the Abitur examination (EPA) for these subjects. The Institute for Quality Development in Education (IQB) is putting together a pool of tasks on behalf of the Conference of Ministers of Education , from which future Abitur exams are to be fed. In this context, requirements for the use of digital tools were defined. A “simple scientific pocket calculator” or a computer algebra system (CAS) are permitted as digital aids. For each of the two digital tools it is assumed that it does not allow access to networks of any kind when used.

The information on the “simple scientific pocket calculator” corresponds to the requirements of the federal states of Baden-Württemberg and Bavaria. The use of programmable pocket calculators is not intended. A calculator is considered to be programmable if additional routines can be stored that are not part of the original functionality. With the exception of Bavaria and Baden-Württemberg as well as Berlin and Brandenburg, the current examination conditions of the other federal states allow scientific pocket calculators that contradict the IQB's requirements in all points, unless GTR or CAS are required as digital aids in the Abitur examination.

In the case of computer algebra systems, it is assumed that the CAS has typical functions such as the algebraic solving of equations and systems of equations, differentiation and integration, calculation with vectors and matrices and the like. It is also assumed that the CAS is put into a state before it can be used in the exam in which access to files and programs that are not part of the scope of delivery or a system update is prevented.

The following table was developed on the basis of the information provided by the education ministries of the federal states. If these could not be found, the information from the various pocket calculator manufacturers was used. It reflects the conditions at grammar schools with regard to admission to examinations, since use in the classroom is possible anywhere due to the educational freedom of the teacher.

state	academic school computer (WTR) z. B. Casio fx-991DE PLUS , TI-30X Plus MultiView	Numerical graphic calculator (GTR) z. B. Casio FX-CG20 , TI-84 Plus	Computer Algebra Pocket Computer (CAS) e.g. B. Casio ClassPad 300 , TI-Nspire CAS , HP Prime
Baden-Württemberg	Yes	No	No
Bavaria	Yes	No	Yes
Berlin	Yes	Yes	Yes
Brandenburg	Yes	No	Yes
Bremen	Yes	Yes	Yes
Hamburg	Yes	No	Yes
Hesse	Yes	Yes	Yes
Mecklenburg-Western Pomerania	Yes	No	Yes
Lower Saxony	No	Yes	Yes
North Rhine-Westphalia	Yes	Yes	Yes
Rhineland-Palatinate	Yes	Yes	Yes
Saarland	Yes	Yes	No
Saxony	No	Yes	Yes
Saxony-Anhalt	Yes	No	No
Schleswig-Holstein	Yes	Yes	Yes
Thuringia	No	No	Yes

Situation in Austria

Until the introduction of the central matriculation examination in the middle of the 2010s, there were no nationwide uniform regulations for the admission of certain aids for the Matura, as the Matura itself was drawn up decentrally, i.e. by the teachers on site. The decision as to whether a certain aid was approved or not was up to the respective teacher.

Since 2018, the minimum requirements for technical aids (such as pocket calculators; computers are also permitted) are as follows:

• Grammar schools: "basic functions for displaying function graphs, for numerically solving equations and systems of equations, for determining derivative or master functions, for numerical integration and for supporting methods and procedures in stochastics" (§ 18 Paragraph 3 AHS examination regulations)
• Higher vocational schools: "Basic functions for displaying function graphs, for numerical solving of equations and systems of equations, for matrix calculation, for numerical integration as well as for support with methods and procedures in stochastics." (§ 17 Paragraph 3 BMHS Examination Regulations)

Situation in Switzerland

At present there are neither Switzerland-wide nor canton-wide uniform regulations for the admission of certain aids for the Matura examination, as the examination is decentralized, i.e. drawn up by the teachers themselves. The decision as to whether a certain aid is approved or not is up to the respective teacher. The usual Matura exams have two parts, one (several hours, written) with a calculator, the other (short, oral) part without a calculator. The federal Matura examination is only taken orally, without a calculator.

Picture gallery

• 

  Forerunner of the electronic pocket calculator: Mechanical pocket calculator from the sixties

• 

  Sharp EL-8 , the first portable calculator (1971)

• 

  HP-35 , first scientific calculator (1972), reverse Polish notation and LED display

• 

  MBO pocket calculator with LED display (approx. 1972), one of the first pocket calculators made in Germany

• 

  TI pocket calculator SR 10 with scientific notation (1973)

• 

  Source Privilege Scientific Calculator (1974)

• 

  Pocket calculator (from 1975): LED display with exponential notation , one of the first with gon functions

• 

  Pocket calculator with vacuum fluorescence display (around 1975)

• 

  Pocket calculator from 1976 (price at that time 49 DM, which corresponds to approx. 63 EUR today)

• 

  Interior view of a Casio FX 20 (approx. 1976)

• 

  Brown ET 23 (1977)

• 

  Braun ETS 77 (1987)

• 

  HP-15C with numerical integration, zero point and matrix calculation

• 

  DDR pocket calculator MR 609 , identical to the school calculator SR1

• 

  Front view of the Sharp PC-1403 , a pocket computer

• 

  TI programmer

• 

  Calculator scientific dual power

• 

  TI-Nspire (CAS) from Texas Instruments

• 

  Casio JS-20WK (2018)

Calculator as a program

After pocket calculators had become a widespread tool in (professional) life, their functionality was simulated in computer programs . These soon became part of the basic equipment of operating systems , for example in personal computers and cell phones . There is also a large selection of programs that offer complex functionalities such as programmability or the conversion of physical quantities.

literature

• Mathias Gerlach: Milestone: Pocket calculator. In: Chip No. 3/2016, p. 86

Web links

Wiktionary: Pocket calculator  - explanations of meanings, word origins, synonyms, translations

Commons : Calculator  - collection of pictures, videos and audio files

• Pocket calculator from Robotron at www.robotrontechnik.de
• MyCalcDB: Pocket Calculators - Database / Museum - Database /
• www.calculators.de - Pocket calculator museum by category with a short data sheet for each device
• Texas Instruments and Hewlett-Packard Calculator Internet Museum - lots of text /
• www.datamath.org - Datamath Calculator Museum
• www.hpmuseum.com - private calculator museum specializing in HP models
• Programmable Calculators at www.rskey.org

Individual evidence

1. ↑ 40 years of electric adder: The first pocket calculator weighed 1.5 kg , spiegel.de.
2. ↑ Patent US3819921A : Miniature Electronic Calculator. Applied December 21, 1972 , published June 25, 1974 , assignee: Texas Instruments Inc., inventors: Jack S. Kilby, Jerry D. Merryman, James H. van Tassel (Based on a dropped patent US 671777 dated 09/29/1967 ). ‌
3. ^ Casio History (1970-1979) .
4. ↑ Definition of the approved and non-approved functions of a pocket calculator according to the IQB .
5. ↑ Casio's approval guidelines ( Memento of the original dated August 28, 2011 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. . 
6. ↑ Texas Instruments approval guidelines (PDF; 178 kB).
7. ↑ The pocket calculator model fx-991DE plus is not permitted in some federal states due to the possibility of solving equations. Under CASIO school computer - approval guidelines, the manufacturer offers the option of checking the models for approval by federal state. Accessed November 3, 2015.
8. ↑ Requirements for the functional scope of scientific pocket calculators in final exams BW Ministry for Culture, Youth and Sport Baden-Württemberg. Accessed September 14, 2018.
9. ↑ Main framework conditions Abitur examination from 2014 website of the ISB. Accessed April 4, 2013.
10. ↑ Implementation Regulations for School Examinations, p. 108 ( Memento of the original from March 19, 2013 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 1.4 MB) Website of the Senate Department for Education, Science and Research. Accessed April 4, 2013. 
11. ↑ Examination tasks Abitur Bildungsserver Berlin-Brandenburg. Accessed April 4, 2013.
12. ↑ Regulations for the Abitur examination 2013 (PDF; 197 kB) Education server Bremen. Accessed April 4, 2013.
13. ↑ Regulations for the Abitur examination 2013 ( Memento of the original from January 23, 2013 in the Internet Archive ) Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 888 kB) Education server Hamburg. Accessed April 4, 2013 
14. ↑ Regulations for the Abitur examination 2013 ( Memento of the original dated May 12, 2013 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 341 kB) Website of the State Parents' Council. Accessed April 4, 2013. 
15. ↑ Preliminary information for the tasks for the central written Abitur examinations in the general subjects school year 2012/13 ( Memento of the original from December 24, 2012 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. Education server Mecklenburg-Western Pomerania. Accessed April 4, 2013. 
16. ↑ Notes on the 2013 written Abitur examination in mathematics ( Memento of the original from October 21, 2012 in the Internet Archive ) Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 35 kB) Lower Saxony education server. Accessed April 4, 2013. 
17. ↑ Requirements for the teaching requirements for the written examinations in the Abitur in the gymnasiale Oberstufe in 2013 ( Memento of the original from March 26, 2013 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. Ministry of Education website. Accessed April 4, 2013. 
18. ↑ Administrative regulation of the Saxon State Ministry for Culture to prepare for the Abitur examination and the supplementary exams 2016 at general high schools, evening grammar schools and colleges in the Free State of Saxony (VwV Abitur examination 2016) from April 28, 2014, MBl. SMK 6/2014, p. 100 ( PDF ; 281K).
19. ↑ Administrative regulation of the Saxon State Ministry for Culture for the special performance assessment in grade 10 at the grammar school in the school year 2014/15 from April 28, 2014, MBl. SMK 6/2014, p. 99 ( PDF ; 281K).
20. ↑ Orientation tasks for the Abitur from 2014 school portal Thuringia. Accessed April 4, 2013.