Mental calculation

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Mental calculation has long been a component of mathematical education. Mental calculation at primary school.jpg
Mental calculation has long been a component of mathematical education.

Mental calculation (also known as mental computation [1] ) consists of arithmetical calculations made by the mind, within the brain, with no help from any supplies (such as pencil and paper) or devices such as a calculator. People may use mental calculation when computing tools are not available, when it is faster than other means of calculation (such as conventional educational institution methods), or even in a competitive context. Mental calculation often involves the use of specific techniques devised for specific types of problems. Many of these techniques take advantage of or rely on the decimal numeral system.

Contents

Capacity of short-term memory is a necessary factor for the successful acquisition of a calculation, [2] specifically perhaps, the phonological loop, in the context of addition calculations (only). [3] Mental flexibleness contributes to the probability of successful completion of mental effort - which is a concept representing adaptive use of knowledge of rules or ways any number associates with any other and how multitudes of numbers are meaningfully associative, and certain (any) number patterns, combined with algorithms process. [4]

It was found during the eighteenth century that children with powerful mental capacities for calculations developed either into very capable and successful scientists and or mathematicians or instead became a counter example having experienced personal retardation. [5] People with an unusual fastness with reliably correct performance of mental calculations of sufficient relevant complexity are prodigies or savants. [6] By the same token, in some contexts and at some time, such an exceptional individual would be known as a: lightning calculator, or a genius. [7]

In a survey of children in England it was found that mental imagery was used for mental calculation. [8] By neuro-imaging, brain activity in the parietal lobes of the right hemisphere was found to be associated with mental imaging. [9]

The teaching of mental calculation as an element of schooling, with a focus in some teaching contexts on mental strategies [10]

Exceptional calculational ability

An exceptional ability is mental calculation such as adding, subtracting, multiplying or dividing large numbers.

Skilled calculators were necessary in research centers such as CERN before the advent of modern electronic calculators and computers. See, for instance, Steven B. Smith's 1983 book The Great Mental Calculators, or the 2016 book Hidden Figures [11] and the film adapted from it.

Competitions

World Cup

The Mental Calculation World Cup is an international competition that attempts to find the world's best mental calculator, and also the best at specific types of mental calculation, such as addition, multiplication, square root or calendar reckoning. The first Mental Calculation World Cup [12] took place in 2004. It is an in-person competition that occurs every other year in Germany. It consists of four different standard tasks --- addition of ten ten-digit numbers, multiplication of two eight-digit numbers, calculation of square roots, and calculation of weekdays for given dates --- in addition to a variety of "surprise" tasks. [12] The last edition was organized in September 2024 and won by Aaryan Nitin Shukla, who successfully defended his title to become two time World Champion.

Mental Calculation & Speed Reading Olympiad

The Mind Sports Olympiad has staged annual world championships since 1998.

Memoriad – World Memory

The first international Memoriad [13] was held in Istanbul, Turkey, in 2008. The second Memoriad took place in Antalya, Turkey, on 24–25 November 2012. 89 competitors from 20 countries participated. Awards and money prizes were given for 10 categories in total; of which 5 categories had to do about Mental Calculation (Mental addition, Mental Multiplication, Mental Square Roots (non-integer), Mental Calendar Dates calculation and Flash Anzan). The third Memoriad was held in Las Vegas, USA, from November 8, 2016 through November 10, 2016.

Mind Sports Organisation

The Mind Sports Organisation recognizes six grandmasters of mental calculation: Robert Fountain (1999), George Lane (2001), Gert Mittring (2005), Chris Bryant (2017), Wenzel Grüß (2019), and Kaloyan Geshev (2022), and one international master, Andy Robertshaw (2008). In 2021, Aaryan Nitin Shukla became the youngest champion ever at an age of just 11 years.

World records

Shakuntala Devi from India has been often mentioned on the Guinness World Records. Neelakantha Bhanu Prakash from India has been often mentioned on the Limca Book of Records for racing past the speed of a calculator in addition. [14] Sri Lankan-Malaysian performer Yaashwin Sarawanan was the runner-up in 2019 Asia's Got Talent .

13th root

Extracting the 13th root of a number is a famous category for the mental calculation world records. The challenge consists of being given a large perfect 13th power (possibly over 100 digits) and asked to return the number that, when taken to the 13th power, equals the given number. For example, the 13th root of 8,192 is 2 and the 13th root of 96,889,010,407 is 7.

Extracting the 13th root has certain properties. One is that the 13th root of a number is much smaller: a 13th root will have approximately 1/13th the number of digits. Thus, the 13th root of a 100-digit number only has 8 digits [15] and the 13th root of a 200-digit number will have 16 digits. Furthermore, the last digit of the 13th root is always the same as the last digit of the power. [15] For the 13th root of a 100-digit number there are 7,992,563 possibilities, in the range 41,246,264 – 49,238,826. This is considered a relatively easy calculation. There are 393,544,396,177,593 possibilities, in the range 2,030,917,620,904,736 – 2,424,462,017,082,328, for the 13th root of a 200-digit number. This is considered a difficult calculation.

100-digit numbers

The Guinness Book of World Records has published records for extracting the 13th root of a 100-digit number. [15] All world records for mentally extracting a 13th root have been for numbers with an integer root:

  • The first record was 23 minutes by Herbert B. de Grote (Mexico) on October 5, 1970. [15]
  • In 1981, Wim Klein (Netherlands) set a record of 88.8 seconds at the National Laboratory for High Energy Physics in Tsukuba, Japan. [16] : 356 
  • Gert Mittring calculated it in 39 seconds.
  • Alexis Lemaire has broken this record with 13.55 seconds in 2002. [17] This is the last official world record for extracting the 13th root of a 100-digit number. [18]
  • Mittring attempted to break this record with 11.8 seconds in November 2004. However, this was not counted as official as the Guinness Book of World Records had stopped recognizing records for root extraction of random numbers, due to the difficulty of standardizing the challenge. [17]
  • Lemaire broke this record unofficially 6 times, twice within 4 seconds: the best was 3.625 seconds in December 2004. [19]

200-digit numbers

Lemaire has also set the first world record for the 13th root of a 200-digit number: 513.55 seconds and 742 attempts on April 6, 2005, and broken it with 267.77 seconds and 577 attempts on June 3, 2005. [19] [20]

  • The same day, Lemaire has also set in front of official witnesses an unofficial record of 113 seconds and 40 attempts.
  • On February 27, 2007, he set a world record of 1 minute and 47 seconds
  • He broke this record on July 24, 2007, with a time of 1 minute and 17 seconds (77.99 seconds) at the Museum of History of Science, University of Oxford, UK. [21]
  • Lemaire broke his record on November 15, 2007, with a time of 72.4 seconds
  • Lemaire broke his record on December 10, 2007, with a time of 70.2 seconds

Media

Literary

Film

Fiction

Twentieth century
1980s
  • In the 1988 movie Rain Man , Raymond Babbitt, an autistic man with savant syndrome, can mentally calculate large numbers, amongst other abilities.
1990s
  • In the 1991 movie Little Man Tate , Fred Tate in the audience blurts out the answer during a mental calculation contest.
  • In the 1997 sci-fi thriller Cube , one of the prisoners, Kazan, is mentally disabled and is later revealed to be a savant who is able to calculate prime factors in his head.
  • In 1998 Darren Aronofsky's film Pi , Maximillian Cohen is asked a few times by a young child with a calculator to do large multiplications and divisions in his head, which he promptly does, correctly.
  • In 1998 film Mercury Rising , a 9-year-old autistic savant with prodigious math abilities cracks a top secret government code.
Twenty-first century
2000s
  • In the 2006 film Stranger than Fiction , the main character, Harold Crick, is able to perform rapid arithmetic at the request of his co-workers.
2010s
  • In the 2012 film Safe , a female child math genius is kidnapped to be used by the Chinese Triad.
  • In the 2016 film The Accountant , Christian Wolff is a high-functioning autistic man who tracks insider financial deceptions for numerous criminal organizations.
  • In the 2017 film Gifted , an intellectually gifted seven-year-old, Mary Adler, becomes the subject of a custody battle between her uncle and grandmother.

Bio

Television

Animation

In the 2009 Japanese animated film Summer Wars , the main character, mathematical genius Kenji Koiso, is able to mentally break purely mathematical encryption codes generated by the OZ virtual world's security system. He can also mentally calculate the day of the week a person was born, based on their birthday.

See also

Notes

    References

    1. ↑ Alistair McIntosh (2004). Mental Computation: A strategies approach (PDF). University of Tasmania: Department of Education. ISBN   1920865209.{{cite book}}: CS1 maint: publisher location (link)
    2. ↑ Hope, John Alfred (1984). Characteristics of unskilled, skilled and highly skilled mental calculators (Thesis). University of British Columbia. doi:10.14288/1.0096641.
    3. ↑ Noël, Marie-Pascale; Désert, Michel; Aubrun, Anne; Seron, Xavier (January 2001). "Involvement of short-term memory in complex mental calculation". Memory & Cognition . 29 (1): 34–42. doi:10.3758/BF03195738. PMID   11277462.
    4. ↑ Timo Flückiger; Elisabeth Rathgeb-Schnierer (February 2022). "Capturing flexibility in mental calculation". Educational Studies in Mathematics. 50. Twelfth Congress of the European Society for Research in Mathematics Education (CERME12): HAL: 29–47. doi:10.1023/A:1020572803437 – via John Threlfall Flexible Mental Calculation doi:10.1023/A:1020572803437.
    5. ↑ Clawson, C.C. (1994). "The Genius Calculators". The Mathematical Traveler. Springer. pp. 233–246. doi:10.1007/978-1-4899-6014-6_14. ISBN   978-0-306-44645-0.
    6. ↑ Fehr, Thorsten; Weber, Jochen; Willmes, Klaus; Herrmann, Manfred (April 2010). "Neural correlates in exceptional mental arithmetic—About the neural architecture of prodigious skills". Neuropsychologia . 48 (5): 1407–1416. doi:10.1016/j.neuropsychologia.2010.01.007. PMID   20079753.
    7. ↑ Bousfield, W. A.; Barry, Jr., H. (April 1933). "The Visual Imagery of a Lightning Calculator" . The American Journal of Psychology . 45 (2). University of Illinois Press: JSTOR: 233. doi:10.2307/1414296. JSTOR   1414296.
    8. ↑ Chris Bills; Eddie Gray. "THE USE OF MENTAL IMAGERY IN MENTAL CALCULATION" (PDF). Education Resources Information Center . University of Warwick: Institute of Education Sciences. p. 97. What is apparent is that these children have shown a preference for concrete or abstract visual images in both calculation and non calculation contexts.
    9. ↑ O'Boyle, Michael W.; et al. (October 2005). "Mathematically gifted male adolescents activate a unique brain network during mental rotation". Cognitive Brain Research . 25 (2): 583–587. doi:10.1016/j.cogbrainres.2005.08.004. PMID   16150579.
    10. ↑ Thompson, Ian (November 1999). "Mental Calculation Strategies for Addition and Subtraction. Part 1". Mathematics in School . 28 (5): 2–4. JSTOR   30215422.
    11. ↑ Shetterly, Margot Lee (2016). Hidden Figures: The American Dream and the Untold Story of the Black Women Who Helped Win the Space Race. William Morrow and Company. pp.  115. ISBN   978-0062363596. Some of the women were capable of lightning-fast mental math, rivaling their mechanical calculating machines for speed and accuracy. Others, like Dorothy Hoover and Doris Cohen, had highly refined understandings of theoretical math, differentiating their way through nested equations ten pages deep with nary an error in sign. The best of the women made names for themselves for accuracy, speed, and insight.
    12. 1 2 "Mental Calculation World Cup - the World Championship for Mental Calculators". www.recordholders.org.
    13. ↑ Memoriad - World Mental Olympics
    14. ↑ Adke, Arti (6 May 2019). "I love to be quicker than anyone else, Hans India, May 2019". Archived from the original on 2020-12-09. Retrieved 2020-08-17.
    15. 1 2 3 4 Smith, Steven Bradley (1983). The Great Mental Calculators: The Psychology, Methods, and Lives of Calculating Prodigies, Past and Present. New York: Columbia University Press. p. 129. ISBN   0231056419.
    16. ↑ Hope, Jack A. (November 1985). "Unravelling the Mysteries of Expert Mental Calculation". Educational Studies in Mathematics. 16 (4): 355–374. doi:10.1007/BF00417192. JSTOR   3482445.
    17. 1 2 "All Hail The Nerd King". CBS News . 2004-11-24. Retrieved 2008-05-15.
    18. ↑ Macura, Wiktor K.; Weisstein, Eric W. "13th Root". MathWorld . Retrieved 14 May 2025.
    19. 1 2 Bremner, Charles (2005-04-08). "What is the 13th root of . . . 836894668823695693983732866222 5645224726780466493836677 497357558157303507570408962 528802385783156837680293 493820105634336385559593151415 04151494907094190977044493 0566026840277186962415568 8082648640933?". The Times . London. Archived from the original on 2007-02-08. Retrieved 2008-05-15.
    20. ↑ Bhattacharjee, Yudhijit (17 June 2005). "Milestones". Science . 308 (5729): 1739. doi:10.1126/science.308.5729.1739a.
    21. ↑ "How does a human calculator do it?". BBC News Magazine . 30 July 2007. Retrieved 11 May 2025.

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