Mental calculation

Under mental arithmetic is understood solving mathematical problems in his head without the use of aids. Various techniques are used, which are based, inter alia, on the computing rules.

• 6.1 Cross Multiplication 6.1.1 Cross multiplication: older version
• 6.1.2 Cross multiplication: Ferrol'sche execution
• 6.1.3 If the one - digits are the same ...
• 6.1.4 If tens digits are the same ...
• 6.1.5 If a and b are numbers mirror, e.g. 43 × 34 ...
• 6.1.6 Squaring: multiply numbers between 30 and 70 with itself

• 6.3.1 If tens digits are the same and one digits add up to 10 ...
• 6.3.2 If the center between the factors is a round number ( least squares method ) ...
• 6.3.3 Squaring: Multiply any number by itself
• 6.3.4 Squaring of five numbers

• 6.5.1 Multiplication by 11

Basics

Many people have a basic knowledge about mental arithmetic, they have acquired in school. Normally, this knowledge includes performing simple addition and subtraction, the memorized multiplication tables and dividing. The ability to calculate in your head, can be trained.

Magic tricks

For some events of magicians rare special skills in the field of mental arithmetic are put on display. Mostly it is fiddling with particularly large numbers. Often behind it simple mathematical features that are available only for the specific task. They are impressive, but are of no use in daily life.

Genuine mental arithmetic

Rarely techniques are offered for general mental arithmetic. This area usually includes all the functions that have to master an average school calculator, as well as the weekday calculation.

Known head computer

By Carl Friedrich Gauss, " Prince of Mathematicians ", report some anecdotes that he could even as a child expect the most amazing things in mind about the age of six with the formula named after him or later "simple" path calculations.

Among the few ingenious head computers of the present example, include Alexander Aitken, Briton Robert Fountain (two -time world champion ), Dutchman Wim Klein, Jan van Koningsveld ( multiple world and vice world champion, double Olympic champion in 2008, and multiple world record holder, for example, in the calendar Computing ), Zacharias Dase, the Grand Master and eight-time world champion in mental calculation Gert Mittring, the number Rüdiger Gamm artist and linguist Hans Eberstark. In his book The great mental calculators Smith describes another. Even so-called savants can special skills in mental arithmetic (calendar arithmetic, root tasks ) or by an enormous memory ( for example, they all directories on the head) stand out.

You can win the title Grand Master in mental arithmetic, such as Gert Mittring at the 9th Mind Sports Olympiad in 2005 in Manchester. Since 2004, official World Championships in mental arithmetic, which take place every two years. On 1 July 2008 it was held in Leipzig. 2010 won the eleven-year Priyanshi Somani from India the World Cup in Magdeburg.

The first mental arithmetic World Championship for children and adolescents under the direction of Gert Mittring took place in 2008 in Nuremberg. In 2009 there were the first German Championship mental arithmetic for children and young people in Cologne.

Mental arithmetic methods - regardless of the type of arithmetic problem

The methods for mental arithmetic facilitate solving difficult tasks. They take particular account:

• Most people can not be more than 7 digits immediately notice ( Miller number).
• It is difficult to keep an intermediate result for a long time in the head, while going through the other partial invoices.
• It is more difficult with large numbers ( 7,8,9) can be expected than in small numerals (2,3,4).

The methods are designed in such a way that

• A complex computation step is divided into several simpler steps
• Memory loads the sequence of the calculation steps as small as possible,
• Early a good approximate solution is obtained.

The following are important computational methods are explained. The sorting is done according to the calculation method and the breadth of applicability. General purpose methods are declared only. At the end, there are methods in which one operand is a certain number.

Computational direction

The preferred direction for mental arithmetic computation is from left to right, that is reversed as compared to written calculations. This thesis is controversial:

This paper follows in the examples in many cases, the thesis of Benjamin / Shermer. There are several reasons:

• If you act thus as in written arithmetic and converted from right to left, the result is also the result of right to left. However, in the end the result is to be said in linguistic order: for example, fifty-two - thousand - and - three- hundred - twelve. If you have calculated the bill from right to left in the head, so in the digit sequence 2 1 3 2 5, which is extremely difficult. It is just as difficult as reciting a phone number in the reverse order.

• When Benjamin / Shermer follows, we calculate in the above example the first 52 Then you can start as a head relatively early computer the answer is " fifty-two - thousand - and - ... ". And before the 312 continue to calculate a few seconds. Perhaps the 52,000 estimate is sufficient - and you can just stop.

In fact, go the books that favor the computational direction " from right to left ," often assume that you have a pen and writes down the calculated result digits and at the end then reads the result. The aim of this method ( so-called fast rake methods) is to accelerate the written arithmetic, and ideally to do the calculations in a single line written. However, the use of a pin contradicts the above definition of ' mental arithmetic '.

But also for the consideration of F. Ferrol good reasons. Suppose you, the computing task in 2 unequal parts break down, one of the parts is more difficult to calculate. Then the following sequences are available:

In case B ) it is in great danger that during the calculation of the difficult part - forget the intermediate result from the first sub-task - which does require several seconds concentration. Case A) is therefore preferable.

Grouping of digits

For calculations with multi-digit numbers, the number of operations that you must perform in the head increases. One way is to group digits. It holds, for example, each 2 digits to a number and deals with this number as a unit. The course presupposes a very good mathematical skills.

Another advantage is that you can remember by grouping digits long series of numbers, as the human memory span of seven " chunks " at first sight suggests. That makes you are also welcome to take advantage when you want to remember a phone number. While 2 to 4 digits are often grouped into a number.

Multiplication

Cross multiplication

The cross- multiplication is generally applicable for multi-digit numbers, and is a basic method for calculating in the head. The method is described by different authors. The descriptions differ in the method of execution.

Two very different approaches are presented here:

Cross multiplication: older version

This is the variation obvious. It was described in 1910 by F. Ferrol as the " older way." To use this version of the cross- multiplication of two-digit numbers, to set the task a × b of the form is:

The factors a and b are so divided each into two shares, which it is easy to calculate.

Normally a1 ask, b1 represents the tens number and a0, b0 the One Number dar.

The name Cross multiplication is explained by the fact that in the central part of the bill the tens number and one number multiplied crosswise with each other.

In the following example, note that the interim results of the cross multiplication are relatively easy to achieve, and that you do not have to remember the intermediate results long as:

To use this type of cross multiplication of multi-digit numbers, one has to decompose multiple proportions the factors of task a × b in accordingly. Three-digit factors are broken down for example, in 3 parts, which are then multiplied together algebraically.

Cross multiplication: Ferrol'sche execution

The Ferrol'sche Cross multiplication is compared to the " older version" slightly more efficient. It treats the digits individually and comes with a double-digit multiplication to 3 ( instead of 4 ) computational steps.

The literature differs in the preferred order of the three calculation steps and in different notations in the didactic presentation. We remain below the original. F. Ferrol states that the determination of the number of ten z is the most complex operation, and therefore should be done first, since this memory is the least loaded. First followed by the determination of the number of hundreds of hours, and the number of an E.

The following algebraic representation of the object a × b is the solution:

All multiplications are performed with a minimal number of digits. The respective power of ten is excluded and will be considered in the head just before the addition.

In applying the method Ferrol'schen certain simplifications immediately catch the eye. You do not need to be learned as a separate special cases. See the following ways.

If the one - digits are the same ...

... Simplifies the calculation of the number ten with the Ferrol'schen cross term:

And if in addition complement the tens digits to 10, the calculation even simpler:

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

If tens digits are the same ...

... Simplifies the calculation of the number ten with the Ferrol'schen cross term:

And if in addition complement the One - digits to 10, the calculation even simpler:

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

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

... Simplifies the cross multiplication, since it 's all about the 2 digits z1 and z2:

This formula can be further grouped into:

Example:

Squaring: multiply numbers between 30 and 70 with itself

The cross- multiplication provides the most efficient way to square numbers in your head. The application in two-digit numbers is recommended in the speed range between 30 and 70 but it is also applicable to multi-digit numbers in a corresponding manner.

The method starts with a separation of factor A. The use of the cross- multiplication by b = a resulting by combining the cross terms in the first binomial formula:

With a1 = 50 simplifying the medium Term And it turns out the formula that is used for squaring two-digit numbers:

Examples:

Addtionsmethode (direct method)

The addition method is the direct method and generally applicable. Practically, however, fall tasks with large numbers and large numbers often easier if you instead applies the cross multiplication.

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

Normally, the tens number b1, and b0 is the one - number dar. When applied to multi-digit numbers increases the number of components of b accordingly.

Example:

The subtraction method can be considered as a special case of the addition method: b0 is in this case a negative number. The subtraction method sometimes has advantages when a factor ends with 8 or 9.

In the following sample application is a = 18, b1 = 40, b0 = -1:

Reference method

To use the reference method, the task of a × b must be represented in the following form:

The factors a and b is the distance to a reference number r is determined. The reference number is typically a round number in the vicinity of a and b.

Note:

The above formula is identical to the following notation, which is also found in the literature. It works with numerical examples, in effect, in the same way, but requires an addition more:

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

And ... the numbers from the same decimal number derived, a simplification, as in the following example:

This calculation method can also be expressed as:

• The beginning of the result arises from the tens digit multiplied by z z 1
• And the last two digits of the result are reserved for the product of the one - digits.

If the center between the factors a round number is ( least squares method ) ...

Results for the task ... 47 × 53, the following approach when applying the reference method:

The calculation method is derived from the difference of two squares

Where m is the average of a and b, and d is the distance from the mean:

This calculation rule for a × b in English is called Quarter Squares Rule. They can be by use of m and prove d and subsequently multiplying.

Example of application:

Advantage: You must perform only single-digit multiplications for this type of task in mind.

Squaring: Multiply any number by itself

The mental arithmetic method for squaring is based on the reference method, the factors in this case are the same. The task of a × a so dissolving the fastest with:

The reference number is typically a round number r in the vicinity of a

Example:

Squaring of five numbers

It is particularly easy as the squaring of numbers ending in 5. example

One can also express this computation path for five - figures as follows:

• First digit multiply z with z 1
• And then the numerals 2 5 attach.

Factorization

The factorization method is an approach that is often possible, as you suspected. But the application is not always equal to the eye. It is applicable when both factors a and b into smaller suitable factors can be broken down so that a different arithmetic sequence takes simplification.

To use the factorization method, at the numbers a and b must therefore split and then perform the calculation according to the formula in products:

It is particularly advantageous if the product a0 × b0 a particularly easy usable result. Creativity is needed here. It would be helpful if one "simple" Candidates for the product to know the prime factors a0 × b0 is good. Here is a small selection:

Example of use:

The method is of course also applicable when only one of the two numbers a and b is a factor can be cleaved in an advantageous manner: In the following sample application ( a0 = 1, b0 = 11) one makes use of the fact that a subsequent multiplication with 11 easy is feasible in the head:

Methods for multiplication by certain numbers

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

Multiplication by 11

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