Finger arithmetic

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Student doing math

As finger-reckoning to methods in which the reckoning by systematic use of the designated fingers as computing resources is performed. It is believed that such systems already existed in ancient times. The success of the decimal system can also be traced back to finger arithmetic, as humans have ten fingers and the break to the next higher digit at the end of the number of fingers was practical. Even in the sixties system, which dates back to the Sumerians and Babylonians , the sexagesimal system can be used to count and calculate with the ten fingers.

A very old form of calculation aid is the abacus , which is also decimal and easily compatible with finger calculations.

history

An early writing on finger arithmetic comes from the English Benedictine monk Beda Venerabilis (around 673–735). In his book De temporum ratione he gave a complete explanation of the finger counting method and established orderly rules for calculating with it.

Finger arithmetic was so popular with scholars, because the arithmetic means was always at hand, that an arithmetic manual in the Middle Ages was only considered complete with a description of this method. Also in Leonardo Fibonacci's famous arithmetic book Liber abaci , an encyclopedic arithmetic book that taught the western world the arithmetic calculation methods based on the Indo-Arabic place value system, at the end of the first chapter there is a sophisticated system of numbers and arithmetic rules.

The skills of finger arithmetic survived in various cultures over a long period of time and were only pushed into the background in Central Europe by the victory of written arithmetic with the Indian-Arabic numerals.

Finger arithmetic according to Anna Schnasing

One of the traditional forms is the finger arithmetic according to Anna Schnasing , a method with which the multiplication of whole numbers with the help of both hands is so simplified and split up that all the results of the multiplication table up to 10 can only be determined by simple addition. Even results from the multiplication table up to 20 can be determined with hands and feet.

The method works with all operations from the multiplication tables up to 10 without knowing the 36 results to be memorized . In addition, it can be expanded to include the multiplication tables up to 20 with the intended help of the toes, which already saves learning 145 results. The toes are moved or thought to be moved in the shoe. In any case, the user only needs to be able to add up, and with a little practice he will be able to execute many typical cash register calculations in a short time. Anna's methods are now part of finger arithmetic, which is sometimes used as a substitute in special education . Today they are no longer necessary for everyday life. But they are very suitable for dyscalculia and the disabled .

Anna Schnasing

Anna Schnasing was one of many milk sellers (so-called " Bolle girls ") at the C. Bolle dairy in Berlin owned by Carl Bolle , which was located on the Lützowufer, which was then the city limits. Schnasing probably sold fresh milk to housewives in the city center between 1879 and 1883 and became known for its special manual dexterity.

"Anna from the Spreewald" was a lively girl, but was "unable to multiply well" or "only mastered the multiplication tables up to 5", which is why she was often cheated by customers and brought less money back to the boss in the dairy . After a short vacation at home in the Spreewald she came back one day and astonished everyone with an opaque but absolutely infallible finger bill with which she could multiply and add amounts in a short time without anyone understanding how she was doing it. Only one of her customers, a reclusive private lecturer and mathematician, discovered Anna's secret through gradual observation during the sales process and wrote a report about it that was published under the humorous title Algebraic Finger Skills and caused quite a bit of amusement in the educated middle class. Where the girl got this ability from was never known. It is certain that Anna made later Bolle career, put in the accounting and manageress was promoted. Their further fate is not recorded. Her method is one of the explanations for the term milkmaid bill .

(A) Multiplication of two factors up to 5

This trivial finger calculation was used by Anna.

  1. Both closed fists are held in front of the body.
  2. The left hand extends as many fingers as the number of a factor.
  3. The right hand extends as many fingers as the number of the second factor.
  4. Now the number of the left hand is added to itself as often as the right finger can be folded in again.

(B) Multiplication of a factor up to 5 by a factor over 5

  1. Both closed fists are held in front of the body.
  2. The left hand extends as many fingers as the number of the smaller factor.
  3. The right hand extends as many fingers as the number of the larger factor, but bends the fingers again from 6 onwards. For example, the 9 has one finger outstretched and four bent fingers.
  4. Now the product of the extended fingers of the left and right hand is subtracted from ten times the number of the left hand.

Example: 3 times 9

  1. The left hand extends three fingers.
  2. The right hand counts up to 9. Then one finger of this hand is extended.
  3. The number of outstretched fingers on the left hand is 3, so 30 is the summand of the intermediate result.
  4. Then 3 × 1 is calculated, i.e. 3, and subtracted from that.
  5. 30 - 3 = 3 × 9 = 27

Proof:

If

  • a = number of outstretched fingers on the left hand
  • b = number of outstretched fingers on the right hand

then applies

Multiplication of two factors, both between 5 and 10

Multiplication with fingers
Example 7 × 8

This method was described among others by the Persian writer Beha Ad-Din Al'Amuli, who dealt with algebra at the end of the 16th and early 17th centuries and is still used today. B. in the Hebrew language area. The French mathematician Nicolas Chuquet also dealt with this type of multiplication in Triparty en la science des nombres in the 15th century .

The method is used to calculate a multiplication of numbers between 5 and 10, ie for the ABCs from the number 5. The fingers and the thumb the significance of the numbers have (open all fingers) 5 to 10 (closed all fingers) as shown on the picture.

  • One hand stretches out as many fingers as the number of a factor, but bends the fingers again from 6 onwards. For example, the 9 has one finger outstretched and four bent fingers. In other words: As shown in the illustration, all fingers with a label smaller than and equal to the number are bent.
  • The second hand does the same with the second factor.
  • Now all buckled fingers of both hands are counted and noted as 10's, i.e. the first summand in the intermediate result.
  • The outstretched fingers of each hand are counted and multiplied together, this gives the second summand in the intermediate result.
  • The summands of both intermediate results are added up to obtain the result

The procedure is explained using the example of the multiplication 7 × 8:

  1. On the first hand, fingers 6 and 7 are closed: this hand represents the 7th.
  2. On the second hand, fingers 6, 7 and 8 are closed: this hand represents the 8.
  3. You now count the number of closed fingers: that is 5 fingers (2 thumbs, 2 forefingers and 1 middle finger). This number results in the first digit (tens place) of the solution: 5
  4. Now count the outstretched fingers per hand: We have two fingers (ring finger and little finger) on one hand and 3 fingers (middle finger, ring finger and little finger) on the other. These two numbers of fingers are multiplied 2 × 3 = 6. This gives the 2nd digit (ones place) of the result: 6
  5. Solution: 56

This example is shown in the 2nd illustration, whereby the fingers were not bent here.

Proof: if

  • a = number of retracted fingers of the left hand
  • b = number of retracted fingers of the right hand

then applies

Corresponding extensions apply to the use of the toes. The process is much faster to learn than the multiplication tables or the multiplication table.

Chisanbop

Chisanbop is a Korean finger calculation method.

literature

  • Pay please! Milkmaid bill. In: NZZ Folio . No. 5, May 1999.
  • M. Wedell: Actio - loquela digitorum - computatio. On the question of the number between offers of order, forms of use and modalities of experience. In: M. Wedell (ed.): What counts. Order offers, forms of use and modalities of experience of the numerus in the Middle Ages. Pictura et Poesis 31, Cologne a. a. 2012, pp. 15–63, color tables (on number gestures from late antiquity to the 17th century).
  • Volker Wieprecht, Robert Skuppin: Berlin popular errors (on Anna Schnasing).
  • Karl-August Wirth: Finger numbers . In: Real Lexicon on German Art History . Vol. 8, 1986, Col. 1225-1309.

Web links

Individual evidence

  1. ^ V. Wieprecht, R. Skuppin: Berlin popular errors: A lexicon. Berlin 2005, p. 143 ( Google Books ).
  2. ↑ medienwerkstatt-online.de - Bolle: a Berlin institution
  3. ^ Account , Wikibook in Hebrew
  4. ↑ Georges Ifrah: Universal history of numbers , 2nd edition, Campus Verlag, Frankfurt / Main, New York 1991, special edition 1998 Parkland Verlag, Cologne, ISBN 3-88059-956-4 , p. 97 middle