Mental calculation (also known as mental computation) 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. Capacity of short-term memory is a necessary factor for the successful acquisition of a calculation, specifically perhaps, the phonological loop, in the context of addition calculations (only). 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. 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 counterexample, having experienced personal retardation. People with an unusual fastness with reliably correct performance of mental calculations of sufficient relevant complexity are prodigies or savants. 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. In a survey of children in England it was found that mental imagery was used for mental calculation. By neuro-imaging, brain activity in the parietal lobes of the right hemisphere was found to be associated with mental imaging. The teaching of mental calculation can be an element of schooling, with a focus in some teaching contexts on mental strategies.
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 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 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. 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 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. Aaryan Nitin Shukla from India currently holds 9 Guinness World Records in various disciplines of mental calculation. 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. 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 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. 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. All world records for mentally extracting a 13th root have been for numbers with an integer root:
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