Calculation error

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In mathematics or in everyday life, arithmetic errors are errors in arithmetic that arise during arithmetic operations due to a deviation from the calculation rules .

General

In addition to spelling errors , thinking errors , printing errors , fat finger errors , speech errors or typing errors , calculation errors are common everyday mistakes . All of them can become gross mistakes if they go undetected and cause damage.

Arithmetic errors in mental arithmetic represent a violation of arithmetic laws. If calculating machines (such as pocket calculators ) are used, a calculation error occurs due to incorrect input . The calculation error is shown in both cases by the deviation of the calculation result from the true mathematical value . In chain problems, calculation errors continue as error propagation . The calculation errors also include the rounding errors and - strictly speaking - the approximate value .

The errors that arise mainly through rounding (replacement of an irrational number (e.g. ) with a finite decimal fraction or, in general, replacement of a real number with a floating point number ) are also referred to as calculation errors . Individual local invoice errors can result in an accumulated invoice error. The greater the number of operations , the greater the risk of the result being falsified and even completely useless.

Legal questions about calculation errors

Calculation errors can lead to legal consequences in business dealings . The business partner wants to point out that his invoice may be incorrect with the note “arithmetic errors reserved” ( Latin salvo errōre calcŭli , abbreviation: sec). The most common calculation errors in accounting are addition errors , and less often subtraction errors , with which balance errors are closely related. There are also multiplication errors during invoicing ( ).

The calculation error is widespread. In the case of offers or orders , despite the calculation error, there is a binding contract offer within the meaning of Section 145 BGB , so that it cannot be withdrawn even after the calculation error has been discovered. Binding means that the offerer can not revoke his application ; it is in the power of the other party to bring about the contract through acceptance . Even with the tender , the bidder's offer represents a binding contract offer within the meaning of § 145 BGB in accordance with the respective terms and conditions accepted by him. When awarding, the offers of the bidders in the construction industry according to § 16c para. 1 VOB / A are based on compliance with requirements, in particular from a computational, technical and economic point of view. In the cases of the offer, the order and the invitation to tender, there is also no place for a contestation due to error: The offer amount was not due to a mere typing error that would have justified a successful contestation due to an error of explanation according to Section 119 (1) BGB. Rather, the calculation error always belongs to the area of ​​irrelevant motive errors . According to the case law of the Federal Court of Justice (BGH), this applies even if the recipient of the declaration has recognized this calculation error.

"Tactical" (that is, deliberate) calculation errors are also incorporated into tenders in order to be able to correct the offer after publication of the tender results. In the economy, one can create the impression of the cheapest bidder through a "built-in" calculation error and then try to correct the price upwards by uncovering the error, but this fails because of the error of motive. Intentional calculation errors of the bidder do not lead to the exclusion of the offer, but in the case of a large number of corresponding errors may give rise to doubts about the suitability of the bidder.

Spelling mistakes, calculation errors and similar obvious inaccuracies, including errors in administrative acts and judgments based on mechanical oversight, can be corrected ex officio by the authorities or the court at any time ( Section 118 VwGO , Section 107 FGO , Section 138 SGG ). Thus, for example, tax authorities in accordance with § 129 AO spelling errors, miscalculations and similar obvious mistakes that have undermined the adoption of an administrative act, be corrected. For a correction according to § 129 AO it is not necessary that the inaccuracy is also recognizable for the taxpayer; What is decisive is whether the error in the disclosure of the facts is clearly recognizable as an obvious inaccuracy for any impartial third party .

When operating expense reports or utility costs calculation errors, wrong square footage or incorrect allocation formula are considered to be material errors. An error-related claim or reimbursement of operating costs is permissible as long as the billing period is running. A correction after the deadline may no longer lead to an additional payment by the tenant ( Section 556 (3) sentence 3 BGB)

Famous calculation errors

There have been arithmetic errors since primeval times . The book of Ezra ( Ezra 1,8-11  EU ) lists silver and gold temple implements and at the end gives a total of 5400. However, if you add up the individual numbers, you only get 2499. The difference only disappears when you add the gifts of the Jews who remained in Babel mentioned in verse 6 to these temple objects. Joshua ( Jos 19,2-6  EU ) lists 14 cities, but speaks of 13: “And it became their inheritance Beersheba, Scheme, Molada, Hazar-Shual, Baala, Ezem, Eltolad, Betul, Horma, Ziklag, Bet -Markabot, Hazar-Susa, Bet-Lebaot, Sharuhe ”.

On January 31, 1918, Albert Einstein submitted an article to his academy under the title “About Gravitational Waves” because his earlier account was “defaced by an unfortunate calculation error”.

In the state election in Schleswig-Holstein in 1992 , the official result of the Greens on election day was given as 5.0%, calculated with rounding per constituency to one decimal place , in fact it was 4.97%, as it turned out the next day. The calculation error meant that this party unexpectedly did not come to the state parliament because of the threshold clause .

The Pentium FDIV bug became known in November 1994 about 1 ½ years after the market launch and led to inaccurate results for floating point divisions with certain relatively few value pairs.

Discovery and Avoidance

The handling of calculation errors is the responsibility of error management and error cause analysis . A distinction must be made as to whether the calculator controls his arithmetic task himself and / or another control instance takes over this control. Calculation errors that do not correspond to the result expectation of the calculator are noticed and corrected by him. On the other hand, we easily overlook calculation errors that conform to expectations; this is known as selective error correction . When self-control help recalculation (by recalculation with conflicting basic arithmetic) of Nine and penalty trial , plausibility checks , estimates or rough calculations to avoid miscalculations. Supervising bodies (such as teachers in schools or auditing / auditing offices in authorities or companies) compare the calculation results with the ideal solutions available to them.

See also

Individual evidence

  1. ^ Gisela Engeln-Müllges , Fritz Reutter: Collection of formulas for numerical mathematics with standard FORTRAN-77 programs . 5th edition. Bibliographisches Institut, Zurich 1986, ISBN 3-411-03125-5 , p. 9 .
  2. ↑ Pierer's Universal Lexicon of the Past and Present . 4th edition. Verlagbuchhandlung von HA Pierer , Altenburg 1865 ( zeno.org [accessed on August 29, 2019] lexicon entry “Salvo”).
  3. ^ Alfred Isaac, Revision und Wirtschaftsprüfung , 1951, p. 157
  4. ↑ BGH, judgment of July 7, 1998, Az .: X ZR 17/97
  5. ↑ Bundesfinanzhof, judgment of October 28, 1992, Az .: II R 111/89
  6. ^ BGH, judgment of January 12, 2011, Az .: VII ZR 296/09
  7. ↑ Rüdiger Vaas, Beyond Einstein's Universe , 2015, o. P.