Mental arithmetic

Under mental calculation refers to the solution of mathematical tasks only with the brain (without tools ). Various techniques are used that are based, among other things, on the laws of calculation .

Basics

Normally includes through the teaching of mathematics acquired skill and knowledge to perform simple addition and subtraction , the memorized little multiplication and division. The ability to do math in the head can be trained.

Magic tricks

At some magician events, rare special mental arithmetic skills are on display. Most of the time it is dealing with particularly large numbers. Often there are simple mathematical features behind this that can only be used for the specific task. They are impressive but of no use in everyday life.

Real mental arithmetic

Techniques for general mental arithmetic are rarely offered. This area usually includes all the functions that an average school calculator must master, as well as the day of the week calculation .

Well-known mental arithmetic

By Carl Friedrich Gauss , "prince of mathematicians", report some anecdotes that he could already count in my head as a child the most amazing things about the age of six with him named molecular formula or later "simple" path calculations.

The few ingenious mental calculators of today include, for example, Alexander Aitken , the Briton Robert Fountain (two-time world champion), the Dutchman Wim Klein , Jan van Koningsveld (multiple world and runner-up world champion, double Olympic champion in 2008, as well as multiple world record holders, e.g. in calendar arithmetic ), Zacharias Dase , the grand master and ten-time world champion in mental arithmetic Gert Mittring , the number artist Rüdiger Gamm and the language genius Hans Eberstark . Smith describes others in his book The great mental calculators . So-called savants can also attract attention through special mental arithmetic skills (calendar arithmetic, root tasks) or through an enormous memory (for example, they have entire telephone books in their heads).

One can achieve the title of Grand Master in mental arithmetic, such as Gert Mittring at the 9th Mind Sports Olympiad 2005 in Manchester. Since 2004 there have been official world championships in mental arithmetic that take place every two years. In 2010, eleven-year-old Priyanshi Somani from India won the World Cup in Magdeburg.

The 1st mental arithmetic world championship for children and young people under the direction of Gert Mittring took place in 2008 in Nuremberg. In 2009 there was the 1st German mental arithmetic championship for children and young people in Cologne.

Mental arithmetic methods - regardless of the type of arithmetic problem

The mental arithmetic methods make it easier to solve difficult tasks. In particular, they take into account:

• Most people cannot remember more than 7 digits straight away ( Miller's number ).
• It is difficult to keep an intermediate result in mind for a long time while doing other partial calculations.
• It is more difficult to calculate with large digits (7,8,9) than with small digits (2,3,4).

The methods are designed so that

• a complex calculation step is divided into several simpler steps,
• the order of the calculation steps puts as little stress on the memory as possible,
• a good approximate solution is achieved at an early stage.

Important calculation methods are explained below. The sorting takes place according to the type of calculation and the breadth of applicability. Generally applicable methods are explained first. At the end there are methods in which an operand is a specific number.

Direction of calculation

The preferred direction of arithmetic in mental arithmetic is from left to right , i.e. the opposite of that in written arithmetic. This thesis is quite controversial:

Gerd Mittring writes: “Some people prefer to calculate from left to right. But that is more prone to errors and you have to remember more. "

Benjamin / Shermer, on the other hand, favor from left to right: "After a little practice you will find that this is the most effective way of calculating in your head."

F. Ferrol believes that the most complex part of the math problem needs to be done first. In cross multiplication, the most complex operation is the cross product. So he suggests starting the multiplication in the middle.

In the examples, this contribution follows the thesis of Benjamin / Shermer in many cases. There are mutliple reasons for this:

• If you proceed as with written arithmetic and calculate from right to left, then the result is also produced from right to left. However, at the end the result should be said in the linguistic order: e.g. B. fifty-two-thousand-and-three-hundred-twelve. If you have calculated the calculation from right to left in your head, i.e. in the order of digits 2 1 3 2 5, this is extremely difficult. It is just as difficult as saying a phone number in reverse order.

• If you follow Benjamin / Shermer, you first calculate the 52 in the above example. Then, as a mental calculator, you can start the answer “fifty-two-thousand-and- ...” relatively early. And continue calculating for a few seconds before the 312. Maybe the estimate of 52,000 is enough - and you can just stop.

In fact, the books that favor right-to-left arithmetic often assume that you have a pen to hand and write down the calculated result numbers and then read the result at the end. The aim of these procedures (so-called fast calculation methods ) is to accelerate the written calculation and ideally to do the calculations in just one written line. However, using a pen contradicts the definition of mental arithmetic above .

But there are also good reasons for F. Ferrol's consideration. Suppose you can split the math problem into 2 unequal parts, one of which is more difficult to calculate. The following sequences are then available:

A) difficult - easy

B) easy - difficult.

In case B) you run the risk of forgetting the intermediate result from the first subtask while calculating the difficult part - which requires several seconds of concentration. Case A) is therefore preferable.

Grouping of digits

When calculating with multi-digit numbers, the number of operations you have to perform in your head increases. One way out is to group digits. One summarizes z. B. combine 2 digits to form a number and treat this number as a unit. Of course, this requires very good numeracy skills.

Another advantage is that by grouping digits you can remember longer series of numbers than the human memory span of 7 "chunks" would initially suggest. People like to take advantage of this when trying to remember a telephone number. Often, 2 to 4 digits are combined into one number.

multiplication

Cross multiplication

Cross multiplication is generally applicable to multi-digit numbers and is a basic method for mental arithmetic. The method is described by different authors. The descriptions differ in the type of execution.

Two very different approaches are presented here:

Cross multiplication: older version

This is the obvious variant. It was already described in 1910 by F. Ferrol as the "older way". In order to use this embodiment of the cross multiplication for two-digit numbers, the problem a × b is represented in the following form:

${\displaystyle a=a_{1}+a_{0}}$

${\displaystyle b=b_{1}+b_{0}}$

${\displaystyle a\cdot b=a_{1}\cdot b_{1}+a_{1}\cdot b_{0}+a_{0}\cdot b_{1}+a_{0}\cdot b_{0}}$

The factors a and b are therefore broken down into 2 parts, which can be easily calculated.

Usually a ₁ , b _{1 represent} the tens and a ₀ , b _{0 represent} the ones.

The name cross multiplication is explained by the fact that in the middle part of the calculation the tens number and the ones number are multiplied by a cross.

In the following example it should be noted that the interim results of the cross multiplication are relatively easy to achieve and that you do not have to remember the interim results for long:

calculation	Explanation
18 × 32 = ...	Exercise with a 1 = 10, a 0 = 8, b 1 = 30, b 0 = 2
... = 10 × 30 + ...	300 (intermediate result)
... + 8 × 30 + ...	540 (intermediate result)
... + 10 × 2 + ...	560 (intermediate result)
... + 8 × 2 = 576	Result

In order to use this type of cross multiplication for multi-digit numbers, one has to break down the factors of the problem a × b into several parts. For example, three-digit factors are broken down into 3 parts, which are then multiplied algebraically.

Cross multiplication: Ferrol's version

Ferrol's cross multiplication is somewhat more efficient compared to the "older version". It treats the digits individually and comes up with 3 (instead of 4) calculation steps with two-digit multiplication.

The literature differs in the preferred order of the 3 calculation steps and in different notations in the didactic preparation. We'll stick to the original below. F. Ferrol explains that the determination of the number of tens z is the most complex operation and should therefore be carried out first, since it is the least stressful on the memory. Only then is the number of hundreds h and the number of ones  e determined .

The following algebraic representation of the problem a × b shows the solution:

${\displaystyle a=z_{a}\cdot 10+e_{a}}$

${\displaystyle b=z_{b}\cdot 10+e_{b}}$

${\displaystyle a\cdot b=(z_{a}\cdot e_{b}+e_{a}\cdot z_{b})\cdot 10+(z_{a}\cdot z_{b})\cdot 100+e_{a}\cdot e_{b}}$

calculation	Explanation
18 × 32 = ...	Exercise with z a = 1, e a = 8, z b = 3, e b = 2
... = (1 × 2 + 8 × 3) × 10 ...	26 tens: The cross term is done in one step.
... + (1 × 3) × 100 + ...	plus 3 hundreds = 560
... + 8 × 2 = 576	plus 16 ones = 576

All multiplications are carried out with a minimum number of digits. The respective power of ten is excluded and is only taken into account in the head before the addition.

When using Ferrol's method, certain simplifications are immediately apparent. So they don't have to be learned as separate special cases. See the following options.

If one-digits are the same ...

... the calculation of the number of tens is simplified with Ferrol's cross term:

${\displaystyle (z_{a}\cdot e+e\cdot z_{b})\cdot 10=(z_{a}+z_{b})\cdot e\cdot 10}$

And if the tens digits add up to 10, the calculation is simplified again:

${\displaystyle \ldots =e\cdot 100}$

Example with e = 2 leads to cross term = 200:

32 × 72 = 21 × 100 + 200 + 4

When tens digits are the same ...

... the calculation of the number of tens is simplified with Ferrol's cross term:

${\displaystyle (z\cdot e_{b}+e_{a}\cdot z)\cdot 10=(e_{a}+e_{b})\cdot z\cdot 10}$

And if the units digits add up to 10, the calculation is simplified again:

${\displaystyle \ldots =z\cdot 100}$

Example with z = 4 leads to cross term = 400:

43 × 47 = 16 × 100 + 400 + 21

If a and b are mirror numbers , e.g. B. 43 × 34 ...

... the cross multiplication is simplified, since it is only about the 2 digits z ₁ and z ₂ :

${\displaystyle z_{a}=e_{b}=z_{1}}$

${\displaystyle e_{a}=z_{b}=z_{2}}$

${\displaystyle a\cdot b=(z_{1}^{2}+z_{2}^{2})\cdot 10+z_{1}\cdot z_{2}\cdot 100+z_{1}\cdot z_{2}}$

This formula can be further summarized as follows:

${\displaystyle a\cdot b=z_{1}\cdot z_{2}\cdot 101+(z_{1}^{2}+z_{2}^{2})\cdot 10}$

Example:

calculation	Explanation
43 × 34 = ...	Exercise with z 1 = 3, z 2 = 4
... = 1212 + ...	3 × 4 × 101: You don't have to do the math to multiply by 101
... + 90 + 160 ...	(9 + 16) × 10: sum of squares times ten
... = 1462	Result

Squaring: Multiply numbers between 30 and 70 by yourself

Cross multiplication is the most efficient way to square numbers in your head. It is recommended to use it for two-digit numbers in the number range between 30 and 70. However, it can also be used in the same way for multi-digit numbers.

The method begins with a decomposition of the factor a . The application of the cross multiplication with b = a leads by summarizing the cross terms to the first binomial formula :

${\displaystyle a=a_{1}+a_{0}}$

${\displaystyle a^{2}=a_{1}^{2}+2\cdot a_{1}\cdot a_{0}+a_{0}^{2}}$

With a ₁ = 50, the middle term is simplified. And the result is the formula that is used to square two-digit numbers:

${\displaystyle a^{2}=(25+a_{0})\cdot 100+a_{0}^{2}}$

Examples:

calculation	Explanation
58 × 58 = ...	Exercise with a 1 = 50, a 0 = 8
... = (25 + 8) × 100 + ...	3300 (intermediate result)
... + 8 × 8 = 3364	Result

calculation	Explanation
37 × 37 = ...	Exercise with a 1 = 50, a 0 = −13
... = (25 - 13) × 100 + ...	1200 (intermediate result)
... + 13 × 13 = 1369	Result

Addition method (direct method)

The addition method is the direct method and generally applicable. In practice, however, tasks involving large numbers and large digits are often easier if you use cross multiplication instead.

To use the addition method for two-digit numbers, you have to split the number b into a sum (hence the name of the method) and then perform the calculation according to the formula:

${\displaystyle b=b_{1}+b_{0}}$

${\displaystyle a\cdot b=a\cdot b_{1}+a\cdot b_{0}}$

Usually b _{1 represents} the tens and b _{0 represents} the ones. When applied to multi-digit numbers, the number of components of b increases accordingly.

Example:

calculation	Explanation
18 × 32 = ...	Exercise with a = 18, b 1 = 30, b 0 = 2
... = 18 × 30 + ...	540 (intermediate result)
... + 18 × 2 = 576	Result

The subtraction method can be viewed as a special case of the addition method: in this case, b ₀ is a negative number. The subtraction method sometimes has advantages when a factor ends in 8 or 9.

In the following example application, a = 18, b ₁ = 40, b ₀ = −1:

18 × 39 = 18 × 40 - 18

Reference method

The reference method can be used advantageously if the two factors a and b are relatively close to one another (distance approx. <20).

In order to use the reference method, one has to represent the task a × b in the following form:

${\displaystyle a=r+a_{0}}$

${\displaystyle b=r+b_{0}}$

${\displaystyle a\cdot b=r\cdot (b+a_{0})+a_{0}\cdot b_{0}}$

The distance to a reference number r is determined for the factors a and b . The reference number is usually a round number near a and b .

Then the formula is used. It is interesting that with this method you end up very close to the result in the first calculation step.

calculation	Explanation
39 × 33 = ...	Exercise with r = 40, a 0 = −1, b 0 = −7
... = 40 × 32 + ...	1280 (intermediate result) For the left factor +1. To compensate for the right factor −1.
... + 1 × 7 = 1287	Result

Annotation:

The calculation formula given above is identical to the following notation, which can also be found in the literature. In the end, it works in the same way with numerical examples, but requires one more addition:

${\displaystyle a\cdot b=r\cdot (r+a_{0}+b_{0})+a_{0}\cdot b_{0}}$

If tens digits are the same and units digits add up to 10 ...

... and the numbers come from the same series of ten, the result is a simplification, as in the following example:

43 × 47 = 40 × 50 + 3 × 7

This calculation can also be expressed as follows:

• The beginning of the result results from the tens digit z multiplied by z +1
• and the last two digits of the result are reserved for the product of the ones digits.

If there is a round number in the middle between the factors (square method) ...

... if you use the reference method, you get the following solution for problem 47 × 53:

47 × 53 = 50 × 50 - 3 × 3

The calculation method results from the difference between two squares

${\displaystyle a\cdot b=m^{2}-d^{2}}$

where m is the mean of a and b and d is their distance from the mean:

${\displaystyle m=\left({\frac {a+b}{2}}\right)^{2}}$

${\displaystyle d=\left({\frac {a-b}{2}}\right)^{2}}$

This calculation rule for a × b is called the Quarter Squares Rule . You can prove it by inserting m and d and then multiplying them out.

Application example:

calculation	Explanation
18 × 22 = ...	Exercise with a = 18, b = 22
m = 20 d = 2	Mean of a and b Distance from mean
20 × 20 - 2 × 2 = 396	Calculation and result

Advantage: You only have to carry out single-digit multiplications in your head for this type of task.

Squaring: Multiplication of any number by itself

The mental arithmetic method of squaring is based on the reference method , in which case the factors are the same. The fastest way to solve the problem a × a is:

${\displaystyle a=r+a_{0}}$

${\displaystyle a\cdot b=r\cdot (a+a_{0})+a_{0}^{2}}$

The reference number r is usually a round number near a .

Example:

calculation	Explanation
33 × 33 = ...	Exercise with r = 30, a 0 = 3
... = 30 × 36 + ...	1080 (intermediate result) For the left factor −3. To compensate for the right factor +3.
... + 3 × 3 = 1089	Result

Squaring five numbers

This makes it particularly easy to square numbers that end in 5 . example

35 × 35 = 30 × 40 + 25

You can also express this calculation method for five-digit numbers as follows:

• Multiply first digit z by z +1
• and then append the digits 2 5.

If that is a number that ends in 5, it can be represented as ${\displaystyle a}$

${\displaystyle a=10\cdot z+5}$ With ${\displaystyle z\in \mathbb {N} }$

and it follows

${\displaystyle a^{2}=(10\cdot z+5)^{2}=100\cdot z^{2}+100\cdot z+25=100\cdot z\cdot (z+1)+25}$.

Factoring method

The factoring method is an approach that can be used more often than expected. But the application doesn't always catch the eye. It can be used if the two factors a and b can be broken down into smaller factors in a suitable manner, so that a different calculation sequence brings the simplification.

To use the factorization method, you have to split the numbers a and b into products and then perform the calculation according to the formula:

${\displaystyle a=a_{1}\cdot a_{0}}$

${\displaystyle b=b_{1}\cdot b_{0}}$

${\displaystyle a\cdot b=(a_{1}\cdot b_{1})\cdot a_{0}\cdot b_{0}}$

It is particularly advantageous if the product a ₀ × b ₀ provides a result that is particularly easy to use. Creativity is required here. It is helpful if you know the prime factorization of “simple” candidates for the product a ₀ × b ₀ well. Here is a small selection:

10 = 2 × 5

20 = 2 × 2 × 5

100 = 2 × 2 × 5 × 5

102 = 2 × 3 × 17

201 = 3 × 67

301 = 7 × 43

1001 = 7 × 11 × 13

Application example:

calculation	Explanation
86 × 14 = ...	Exercise with a 1 = 2, a 0 = 43, b 1 = 2, b 0 = 7 and 43 × 7 = 301
... = (2 × 2) × 301 ...	This is of course very easy ...
... = 1204	Result

The method can of course also be used if a factor can advantageously be split off from just one of the two numbers a and b : In the following example application ( a ₀ = 1, b ₀ = 11) one makes use of the fact that a subsequent multiplication by 11 is very easy to do in your head:

17 × 66 = 17 × 6 × 11

Methods of multiplying by certain numbers

Jakow Trachtenberg systematized methods for multiplication with special numbers between 2 and 12. However, the same methods are often described by other authors with little variation.

Multiplication by 11

Multiplications by 11 are a classic. The method is explained using examples:

calculation	Explanation
11 × 13 = ...	task
1 3	the first and last digits are (almost) fixed
1 4 3	the middle digit is the sum of the other two. In the event that the total is> 9, a carryover is made to the left number.
... = 143	Result

calculation	Explanation
11 × 123 = ...	Task with a three-digit factor
1 3 5 3	The 2nd digit is the sum of the first two. The 3rd digit is the sum of the last two.
... = 1353	Result

literature

• F. Ferrol: Ferrol's new method of calculation - 8 letters . 5th edition. FJ Huthmacher, Bonn 1913.
• Walter Lietzmann: Oddities in the realm of numbers . Dümmlers Verlag, Bonn 1947.
• Arthur Benjamin, Michael Shermer: Math Magic - Amazing tricks for lightning fast mental arithmetic and a phenomenal number memory. 6th edition. Heyne, 2007, ISBN 978-3-453-61502-1 . (American first edition 2006)
• Gert Mittring : Arithmetic with the world champion - math and memory training for everyday life. 4th edition. Fischer, 2012, ISBN 978-3-596-18989-2 . (First edition 2011)
• Ronald W. Doerfler: Dead reckoning - Calculating without instruments . Gulf Publishing, London a. a. 1993, ISBN 0-88415-087-9 .
• Ann Cutler, Rudolph McShane: The Trachtenberg Speed ​​System of Basic Mathematics . Souvenir Press, London 2011, ISBN 978-0-285-62916-5 . (First edition 1962)
• Jagaduru Swami Sri Baharati Krsna Tirthaji Maharaja: Vedic Mathematics . Motilal Banarsidass Publishers, Delhi 2010, ISBN 978-81-208-0164-6 . (First edition 1965)
• Armin Schonard, Cordula Kokot: The math hit. Rediscovered faster and easier calculation methods. Ingeniously simple - simply ingenious. 2nd Edition. Self-published, 2011, ISBN 978-3-00-017801-6 . (First edition 2006)
• Robert Fountain, Jan van Koningsveld: The Mental Calculator's Handbook. 1st edition. Self-published, 2013, ISBN 978-1-300-84665-9 .

Web links

Wiktionary: Mental arithmetic  - explanations of meanings, word origins, synonyms, translations

Individual evidence

1. ↑ Eleven year old is faster than a calculator. In: The time. June 8, 2010.
2. ↑ Gerd Mittring: Calculating with the world champion , p. 53.
3. ↑ Arthur Benjamin, Michael Shermer: Mathe-Magie , p. 53.
4. ↑ F. Ferrol: The Ferrol'sche new accounting practices , letter I
5. ↑ F. Ferrol: The Ferrol'sche new accounting practices , letter I
6. ↑ Ronald W. Doerfler: Dead reckoning , p. 11.
7. ↑ Gerd Mittring: Calculating with the world champion. P. 94.
8. ↑ F. Ferrol: The Ferrol'sche new calculation method , Letter II, p. 37.
9. ↑ ^{a b c} Walter Lietzmann: Weirdos in the Realm of Numbers , p. 25.
10. ↑ Ronald W. Doerfler: Dead reckoning , p. 16.
11. ↑ Arthur Benjamin, Michael Shermer: Mathe-Magie , p. 78.
12. ↑ Arthur Benjamin, Michael Shermer: Mathe-Magie , p. 83.
13. ↑ Ronald W. Doerfler: Dead reckoning , p. 12.
14. ↑ Ronald W. Doerfler: Dead reckoning , p. 15.
15. ↑ Arthur Benjamin, Michael Shermer: Mathe Magie , p. 67.
16. ↑ Arthur Benjamin, Michael Shermer: Mathe-Magie , p. 86.