Written division

The long division is an algorithm to divide, which is used on the paper one number by another. To do the written division, you need the multiplication table and the written subtraction .

Since pocket calculators have become generally available, the practical significance of the method has only been minor. Nevertheless, this method of arithmetic is already taught in elementary lessons: Children usually learn it in the 4th grade of elementary school , but mostly only for single-digit divisors. The justification of this subject matter is controversial.

algorithm

Examples

Division with remainder

We divide 351 by 4.
351 is the dividend , 4 is the divisor .

351 : 4 =

We start from the left to find which section of the dividend as short as possible we can divide by the divisor.
Three cannot be divided by four. The first section that we can divide by 4 is the leftmost digits 35.
35 through 4 is 8, because 8 times 4 is 32, and the remainder 35 - 32 = 3.

351 : 4 = 8
32
--
 3

Now we pull the next digit of the dividend, the 1, down to the remainder, that is 31.
Now the 31 is divided by 4. That makes 7, because 7 times 4 is 28, and the remainder 31 - 28 = 3.

351 : 4 = 87
32
--
 31
 28
 --
  3

All digits of the dividend are now processed. We're done:

351 : 4 = 87 Rest 3
32
--
 31
 28
 --
  3

Notes on the notation

The result “87 remainder 3” is not a number , and “351: 4 = 87 remainder 3” is also not an equation in a mathematical sense . This can be seen from the fact that 438: 5 = 87 is also the remainder 3, but 351: 4 is not the same rational number as 438: 5.

The calculations only show that there is an indivisible remainder of 3 wholes left over for both tasks. Since the first case is to divide by 4, these result in 3 quarters; in the second case, however, it is divided by 5, leaving 3 fifths. The equations and show the difference. ${\ displaystyle 351: 4 = 87 + {\ tfrac {3} {4}}}$ ${\ displaystyle 438: 5 = 87 + {\ tfrac {3} {5}}}$

In order to be able to carry out the procedure of the written division with remainder mathematically correct without a fraction calculation, the division problem is sometimes written as follows:

351 = 4 · …

The calculation procedure then goes as described above, and the rest is added with a plus sign:

351 = 4 · 87 + 3

In the second example, equally correctly,

438 = 5 · 87 + 3

Written in this way, the mathematical method of division with remainder taught in elementary school is consistent with what number theory and other areas of higher mathematics understand by it.

→ Main article: Division with remainder

Multi-digit divisor

If the divisor is greater than 10, the multiplication table is not sufficient to determine the next digit of the result. We find the right numerical value by guessing and trying:

13063:32 = … -- 1:32 und 13:32 "gehen" nicht. 130:32 ist sicher mehr als 3:

13063:32 = 3… -- 3·32 rechnen wir im Kopf:
 96
 --
 34            -- der Rest ist 34 und damit größer als der Divisor; also geht 32 sogar 4-mal in 130!

So we have estimated too low, delete the last two lines and start over:

13063:32 = 4… -- 4·32 rechnen wir im Kopf oder wie bei der schriftlichen Multiplikation: 4·2=8; 4·3=12;
128
---
  26            -- der Rest ist 2; die "heruntergeholte" 6 gibt 26; 26:32 "geht" nicht; wir schreiben im Ergebnis eine Null an:

13063:32 = 40…

Now we can still calculate "0 · 32 = 0" and continue the calculation as follows:

The experienced computer sees, however, that nothing has changed on 26 and picks it up - after writing the 0! - immediately down the 3:

13063:32 = 40…
128
---
  263            -- 263:32 schätzen wir auf ungefähr 8. Wir rechnen 8·2=16, Merkzahl 1, 8·3=24, 24+1=25; also: 256
  256

There remains a remainder of 7, and the finished invoice looks like this:

13063:32 = 408 Rest 7
128
---
  263
  256
  ---
    7

- or , or , as explained above. ${\ displaystyle 13063: 32 = 408 {\ tfrac {7} {32}}}$ ${\ displaystyle 13.063 = 32 \ cdot 408 + 7}$

Division with decimal places

If we prefer a decimal fraction instead of a whole number and a remainder as the result , we write a comma after the previous result and just continue calculating as before, always adding a zero to the right of the last remainder.

950 : 4 = 237,5
8
-
15
12
--
 30
 28
 --
  20 -- hier bleibt ein Rest von 2; es wird aber kein Rest angeschrieben, sondern ein Komma; dann wird eine 0 "heruntergeholt".
  20                                                                                         - 20:4 geht 5-mal…
  --
   0 -- …und zwar ohne Rest, deshalb ist die Rechnung hier zu Ende.

Division of decimal numbers

If the dividend is a decimal number (and the divisor a natural number), it is first checked whether its whole number part can be divided by the divisor. If this is the case, you first divide as usual. As soon as a number after the decimal point has to be “fetched” from the dividend, a comma is added to the result.

If the whole number part of the dividend is smaller than the divisor, a zero is written in the result and a comma after it. Then the decimal places of the dividend (one after the other!) Are "brought down". Whenever the result remains smaller than the divisor, another zero is added to the result. The calculation then proceeds as described above.

Example:

1,8:5 = ?? ----- 1:5 „geht nicht“ - also: „0,...“ anschreiben und eine Nachkommastelle „herunterholen“:

1,8:5 = 0,?? ----- 18:5 „geht“, und zwar 3-mal:
1 8

1,8:5 = 0,3? ----- Der „Rest“ ist 3 — eine „unsichtbare“ 0 wird „heruntergeholt“.
1 5
---
  30 ----- 30:5 „geht“ 6-mal, und zwar ohne Rest; deshalb ist die Rechnung jetzt zu Ende:

1,8:5 = 0,36
1 5
---
  30
  30
  --
   0

If the divisor is (also) a decimal number, the comma must first be shifted, namely

so that the divisor becomes an integer,
in the same direction - that means in this case with dividends and with divisors to the right, and
by the same number of places.

If the dividend has fewer decimal places than the divisor, a corresponding number of zeros must be added to the dividend.

Then it is divided as described above.

4 : 1,6 = 
40 : 16 = 2,5
32
--
 80
 80
 --
  0

Division that results in a periodic decimal fraction

We divide 1307 by 15.

1307 : 15 = 8 -- 13:15 "geht nicht"; 130:15 schätzen wir auf 8; wir rechnen 5·8=40, 1·8=8, plus Merkzahl 4 also 12:
120           -- Die Schätzung war also richtig, und es bleibt ein Rest von 10; die 7 wird "heruntergezogen":
---
 107          -- Da 120=8·15 ist, geht die 15 in 107 offenbar 7-mal; 7·15 rechnen wir ähnlich wie eben…

... and thus have so far the following calculation:

1307 : 15 = 87
120
---
 107
 105
 ---
   2

Now there are no more digits left in the dividend - the decimal point is added to the previous solution, and we add 0. 20:15 is obviously 1, with a remainder of 5:

1307 : 15 = 87,1
120
---
 107
 105
 ---
   20
   15
   --
    5

Adding the next 0 to the remainder results in 50. 50 through 15 is 3, the remainder is again 5.

1307 : 15 = 87,13
120
---
 107
 105
 ---
   20
   15
   --
    50
    45
    --
     5

Now the remainder 5 has come out for the second time. Because there is still only 0 "to be fetched", the process is repeated; and the result is an infinite sequence of 3's.

The result is an infinite decimal fraction, because the decimal places are repeated, a so-called periodic decimal fraction.

1307 : 15 = 87,1333…

For this one usually writes and reads: "1307: 15 = 87 point 1 period 3" ${\ displaystyle 1307: 15 = 87.1 {\ overline {3}}.}$

Division for advanced

One-digit divisor

Since it is a bit cumbersome to always multiply first and then subtract the product from the number above, you can get into the habit of doing the whole thing in one go. That saves space and time:

We take the example from the beginning again:

351 : 4 =

First, let's go from left to right in dividends and see if the number is to be divided by the divisor. Then we put a tick at this point.

35'1 : 4 =       ((Der Apostroph ist eine Hilfe, damit man
                   den ersten Teilschritt besser sieht.))

The 4 goes 8 times into the 35.

35'1 : 4 = 8     Hier sagt oder denkt man jetzt:   8 mal 4 ist 32, plus  3 ist 35.
                 Nur die 3 wird hingeschrieben:

35'1 : 4 = 8
 3

nächste Stelle 1 herab:

35'1 : 4 = 8
 3 1

The 4 goes 7 times into the 31.

35'1 : 4 = 87    7 mal 4 ist 28, plus  3 ist 31
 3 1
   3 - das ist der Rest

Multi-digit divisor

With a little practice, this abbreviated notation can also be used for multi-digit divisors. The (half-written) multiplication and subtraction are interlaced with one another.
We show this again with the example already used above:

13063:32 =

or, with the auxiliary apostrophe:

130'63:32 = 4…

We now have 4 · 32 to calculate and subtract the result from 130. To do this we calculate 4 · 2 = 8, think of these 8 as written below the zero of 13 0 , say (or think) 8 plus 2 is 10, write the 2 below the 0 and note the "borrowed" 1:

130'63:32 = 4…
  2   -   1 "geliehen"

Now the multiplication is continued with 4 * 3 = 12; the "borrowed" 1 has to be added, so 13. 13 plus 0 is 13. This 0 does not have to be written down. If the 6 is "brought down", the result is a 0 and the 3 is also "brought down", the only thing on the paper is:

130'63:32 = 40
  2 63

Since 263: 32 equals eight, we now have to calculate 8 · 32 and at the same time subtract this from 263:
8 · 2 = 16, 16 plus 7 = 23. The 7 is written to; This time 2 are "borrowed":

130'63:32 = 408
  2 63
     7

The multiplication is continued with 8 · 3 = 24, with the "borrowed" 2 thus 26; 26 plus 0 = 26. The 0 does not have to be written here either. In the case of "division by remainder", the calculation ends here. If you do without the apostrophe, it is very short;

13063:32 = 408 Rest 7
  263
    7

Alternative algorithms

In the English-speaking world, the "Big 7 Division" with the scaffolding algorithm has been used and mentioned more frequently in elementary schools in recent years, as it enables written division in simple decimal steps or any multiple of the divisor and makes the principle of repeated subtraction easier to understand. The "Big 7" only means the dividing lines like a big 7 on the paper.

In the original notation, the divisor is in front of the dividend to be divided, for example for exercise 4720: 36:

  ┌────────────┐
36│ 4720       │
   −3600       │ 100
   =1120       │
   − 360       │  10
   = 760       │
   − 360       │  10
   = 400       │
   − 360       │  10
   =  40       │
   −  36       │   1
  ----------------------
   =   4         131  R4/36  (die 36 ist der Divisor von oben, R4 das subtrahierte Ergebnis der linken Spalte, 131 der addierten rechten Spalte)

The 100's, 10's and 1's digits are added, the result is 131 remainder 4 or, correctly, as a fraction 4/36, since the remainder is always the numerator, which only makes sense with the denominator (here the divisor) for the decimal places.

For the sake of understanding, the remainder should always be written out as a fraction decimal or in the "numerator-fraction-bar-denominator" notation , optionally also calculated and added to the result. ${\ displaystyle {\ frac {Z} {N}}}$

The remainder here is 4/36, with the above method you can calculate the remainder in a second step in the same way as the places before the decimal point, but it is already easier with the first calculation to simply add zeros in the number of calculated decimal places you want afterwards to append the dividends in front of the bill. That means, we don't calculate 4720: 36, but simply z. B. 4720 000 : 36 to get three more decimal places as a result.

In the first step, we do not subtract 3600, this would take longer, but a whole 3600000, and note 100000 in the right column, because 36 · 100000 is also 3600000. The rest at the end of the calculation can then be ignored.

Optionally, depending on the task, a multiplication by 2 or other multiples of the divisor is also possible. This is the strength of the Big 7 Division as it rewards a better approximation with fewer calculation steps. However, the ability to subtract remains important, e.g. B. according to the supplementary procedure .

Web links

Individual evidence

↑ See for example Peter Bender, University of Paderborn (PDF; 89 kB) - with further references
^ Algorithms for Multiplying and Dividing Whole Numbers. (PDF) September 21, 2009, accessed on July 14, 2015 .

[1] See for example Peter Bender, University of Paderborn (PDF; 89 kB) - with further references

[2] Algorithms for Multiplying and Dividing Whole Numbers. (PDF) September 21, 2009, accessed on July 14, 2015 .