July 23, 2015
Hiroshi Michiwaki
General long division method and division by zero
1. Introduction
Division is a calculation between two numbers a and b in which we ask how many times b
can be subtracted from a. Relatively rational calculation of such an operation generally uses
multiplication, which is considered the inverse operation of division. However, this is not thereal solution for division; it is simply a general method.
For example, when calculating a ÷ b = 15 ÷ 3,
i.
ii.
iii.

First, let us think about 3 × 4 = 12. However, this leaves a remainder. Thus,
Next, as a value slightly larger than 4, we select 6, and calculate 3 × 6 = 18. Thistime, it was a little too large, so
Finally, we find that 3 × 5 = 15. The product of the remainder and the arbitrary
selected number were consistent with the dividend, thus, we select 5 as thesolution of 15 ÷ 3.

This example is quite simple, but sufficiently shows the nature of division. In other words,
essentially, we conduct subtraction operations as follows:
15 – 3 = 12 (first time)
12 – 3 = 9 (second time)
9 – 3 = 6 (third time)
6 – 3 = 3 (fourth time)and3 – 3 = 0 (fifth time)
However, this operation is cumbersome. Thus, up to an arbitrary number of operations that is
the likely solution, we skip ahead using multiplication for rationalization. In other words, theabove operations i to iii mean:
i.
ii.
iii.

15 – 3 × 4 = 3
15 – 3 × 6 = -3
15 – 3 × 5 = 0

This means that multiplication, which has long been used as a historical solution method fordivision, is an operation in which division, defined as repeated subtraction, is abridged by theuse of the product of divisor and solution, or divisor and quasi-solution, such as 4 or 6 in theabove example, for part of the whole calculation, without using repeated subtraction.
As the example shows, defining division as an inverse operation of multiplication causes theconfusion that it is the only definition; however, in truth, it is more natural to think thatsolving division by multiplication is an extension of a shortcut involving repeatedsubtractions. Division was not originally defined as an inverse operation of multiplication,and it should be defined independently of multiplication. Its independent definition is “to finda solution that is the number of times a divisor can be subtracted from a dividend, to obtainthe smallest non–negative number as a resulting value.” This is the original definition ofdivision.
Based on this true definition of division, we show the manual method (general long division)of division below.
2. General long division 1
Definition
In the manual method of division of a dividend consisting of an arbitrary number of digits,
for a certain digit of the dividend, the divisor is subtracted once from the said dividend, andas a result, the dividend is reduced. If the result of reduction is a non-negative value, thenabove the applicable dividend digit, stack ○ upward. Continue this operation as long as
possible, adding ○s in the rows above.
If subtracting the divisor from the dividend makes no difference, or if it leads to a negative
number, add ●, then move to the next smaller digit, and repeat the above steps.
When this repeat operation is completed, count the number of ○s for each digit, which is thequotient for the applicable digit. ● is 0.
Example 1: General long division I for 234567÷543 = 431...534 is shown below.

1) If divisor 543 is subtracted from the target dividend 2, the result
is negative; therefore, ● is placed immediately above the target
dividend 2, and 0 is placed above (vacant position, 0), then thenext digit is considered.
2) If divisor 543 is subtracted from the target dividend, which is
now 23, the result is negative; therefore, ● is placed
immediately above the target dividend 3, and 0 above (vacant
position, 0), then the next digit is considered.
3) If divisor 543 is subtracted from the target dividend, now 234,
the result is negative; therefore, ● is placed immediately above
the target dividend 4, and 0 above (vacant position, 0), then thenext digit is considered.
4) If divisor 543 is subtracted from the target dividend, now 2345,
the result is non-negative; therefore, ○ is placed immediately
above the target dividend 5. From the target dividend 2345,
divisor 543 is subtracted once, and as in the usual manualsubtraction, a line is drawn under the applicable divisor, and thesubtracted result of 1802 is placed below the line.
5) If divisor 543 is subtracted from 1802, the result is nonnegative; therefore, ○ is placed immediately above ○ above thetarget dividend 5 in a stack, and 543 is subtracted from 1802
once. Below this, the subtracted result of 1259 is placed.
6) If divisor 543 is subtracted from 1259, the result is nonnegative; therefore, ○ is placed immediately above ○ above thetarget dividend 5 in the stack, and 543 is subtracted from 1259
once. Below this, the subtracted result of 716 is placed.
7)
If divisor 543 is subtracted from 716, the result is nonnegative; therefore, ○ is placed immediately above the top ○ above the target dividend
5 in the stack, and 543 is subtracted from 716. Below this, the subtracted result of 173
is placed.
8) If divisor 542 is subtracted from 173, the result is negative; therefore, above the top ○,● is placed in a stack. Above this, number 4 is placed, which is the number of ○s, then
the next digit is considered.
9) If divisor 543 is subtracted from the target dividend, now 1736, the result is nonnegative; therefore, ○ is placed immediately above the target dividend 6, and 543 is
subtracted from 1736 once, and as in the usual manual subtraction, a line is drawn
under the applicable divisor, and the subtracted result of 1193 is placed below the line.
10) If divisor 543 is subtracted from 1193, the result is non-negative; therefore, ○ isplaced immediately above the top ○ of the target dividend 6 in a stack, and divisor
543 is subtracted from 1193. Below this, the subtracted result of 650 is placed.
11) If divisor 543 is subtracted from 650, the result is non-negative; therefore ○ is placed
immediately above the top ○, and the divisor 543 is subtracted from 650 once. Below
this, the subtracted result of 107 is placed.
12) If divisor 543 is subtracted from 107, the result is negative; therefore, ● is placed
above the top ○ immediately above the target dividend of 6. Above this, a 3 is placed,
which is the number of ○s, then the next digit is considered.
13) If divisor 543 is subtracted from the target dividend, now 1077, the result is nonnegative; therefore, ○ is placed immediately above the target dividend of 7, and thedivisor 543 is subtracted from 1077 once, and as in the usual manual subtraction, aline is drawn under the applicable divisor, and the subtracted result of 534 is placed
below the line.
14) If divisor 543 is subtracted from 534, the result is negative; therefore, ● is placed
immediately above the top ○ above the target dividend of 7. Above this, the numberof ○s is placed, which is 1, then the next digit is considered. Because the operation
within the integer range is completed, the calculation is stopped here with 534 as theremainder. Of course, a decimal point can be added to the quotient (to the right of thetarget dividend digit 7) and calculation below the decimal point can be continued.
15) In other words, the solution of this division is 000431, with a remainder of 534;
however, according to mathematical custom, 0 above the largest digit can be omitted:
thus, the quotient is 431 with a remainder of 534.
Example 2: General long division I for 123÷0 = 0...123 is shown below.
1. If divisor 0 is subtracted from the target dividend 1, it remains at 1
without change; therefore, ● is placed immediately above thetarget dividend value of 1, and 0 is placed above (vacant space, 0).
A line is drawn below the target dividend of 1, and 1 is placed
below the line, and the next digit is considered.
2. If divisor 0 is subtracted from the target dividend, now 12, it
remains at 12 without a change; therefore, ● is placed immediately
above the target dividend of 2, and 0 is placed above (vacant space,0). A line is drawn below the target dividend of 12, and 12 isplaced below the line, and the next digit is considered.
3. If divisor 0 is subtracted from the target dividend, now 123, it remains at 123 withouta change; therefore, ● is placed immediately above the target dividend of 3, and 0 isplaced above (vacant space, 0). Then moving to the next digit, a line is drawn below
the target dividend of 123, and 123 is placed under the line. At this point, because theoperation within the integer range is completed, calculation is stopped with theremainder of 123. Of course, a decimal point can be added to the quotient (to the rightof the target dividend digit 3), and calculation can be continued to an arbitrary digit
below the decimal point.

3. General long division II
Definition○ and ● in general long division I are replaced with 1 and 0, respectively, and as quotient
value for each digit, above the first 0, the sum of the numbers in the rows below (0 to 9) is
placed.
Example 3: General long division II of 234567÷543=431...534 is shown below.
1. If divisor 543 is subtracted from the target dividend 2, theresult is negative; therefore, 0 is placed immediately above
the target dividend of 2 (vacant space, 0), and the next digit
is considered.
2. If divisor 543 is subtracted from the target dividend, now 23,
the result is negative; therefore, 0 is placed immediately
above the target dividend of 3 (vacant space, 0), and the next
digit is considered.
3. If divisor 543 is subtracted from the target dividend, now
234, the result is negative; therefore, 0 is placed above thetarget dividend of 4 (vacant space, 0), and the next digit isconsidered.
4. If divisor 543 is subtracted from the target dividend, now
2345, the result is non-negative; therefore, 1 is placed
immediately above the target dividend of 5. Divisor 543 is
subtracted from 2345 once, and as in the usual manualsubtraction, a line is drawn below the target dividend, andthe subtraction result 1802 is placed below the line.
5. If divisor 543 is subtracted from 1802, the result is nonnegative; therefore, 1 is placed immediately above the target
dividend of 5 in a stack. Divisor 543 is subtracted from 1802
once, and the subtraction result of 1259 is placed below theline.
6. If divisor 543 is subtracted from 1259, the result is nonnegative; therefore, 1 is placed immediately above the target
dividend of 5 in a stack. Divisor 543 is subtracted from 1259
once, and the subtraction result of 716 is placed below theline.
7. If divisor 543 is subtracted from 716, the result is nonnegative; therefore, 1 is placed immediately above the target
dividend of 5 in a stack. Divisor 543 is subtracted from 716
once, and the subtraction result of 173 is placed below theline.
8.
If divisor 543 is subtracted from 173, the result is negative;
therefore, 0 is placed immediately above the target dividendof 5 and above the top 1 in a stack. Above this 0, a 4 isplaced, which is the sum of the rows below 0. Then, the next
digit is considered.
9. If divisor 543 is subtracted from the target dividend, now 1736, the result is nonnegative; therefore, 1 is placed immediately above the target dividend of 6. Divisor

543 is subtracted from 1736 once. As in the usual manual subtraction, a line is drawn
below the target dividend, and the subtraction result of 1193 is placed below the line.
10. If divisor 543 is subtracted from 1193, the result is non-negative; therefore, 1 isplaced immediately above the target dividend of 6 in a stack. Divisor 543 is subtracted
from 1193 once, and the subtraction result of 650 is placed below. If divisor 543 is
subtracted from 650, the result is non-negative; therefore, 1 is placed immediately
above the target dividend 6 above the top 1 in a stack. Divisor 543 is subtracted from
650 once, and the subtraction result of 107 is placed below.
11. If divisor 543 is subtracted from the target dividend, now 107, the result is negative;
therefore, 0 is placed immediately above the target dividend of 6 above the top 1, andabove the 0, a 3 is placed, which is the sum of the rows below 0. Then the next digit isconsidered.
12. If divisor 543 is subtracted from the target dividend, now 1077, the result is nonnegative; therefore 1 is placed immediately above the target dividend of 7. Divisor
543 is subtracted from 1077 once, and as in the usual manual subtraction, a line isdrawn below the target dividend, and the subtraction result of 534 is placed below.
13. If divisor 543 is subtracted from 534, the result is negative; therefore, 0 is placed
immediately above the target dividend of 7 above the top 1. Above 0, place 1, whichis the sum of the rows below 0. Then we move to the next digit. Because operation in
the integer range is complete at this point, calculation is stopped with the remainder of534. Of course, a decimal point can be added to the quotient (to the right of the target
dividend digit 7) and calculation can be continued to an arbitrary digit below thedecimal point.
14. The quotient is 000431 and the remainder is 534, but the 0 above the largest digit isomitted: the quotient is 431 and the remainder is 534.
Example 4: General long division I of division 123÷0 = 0...123 is shown below.
1)If divisor 0 is subtracted from the target dividend of 1, the result
remains 1 without a change; therefore, 0 is placed immediately above
the target dividend of 1 (vacant space, 0). A line is drawn under thetarget dividend of 1, and the subtraction result of 1 is placed below
the line. Then the next digit is considered.
2)If divisor 0 is subtracted from the target dividend, now 12, the result
remains 12 without a change; therefore, 0 is placed immediately
above the target dividend of 2 (vacant space, 0). A line is drawn under
the target dividend of 12, and the subtraction result of 12 is placed
below the line. Then the next digit is considered.
3) If divisor 0 is subtracted from the target dividend, now 123, the result remains 123
without a change; therefore, 0 is placed immediately above the target dividend of 3. Aline is drawn below the target dividend of 123, and the subtraction result of 123 isplaced below the line. At this point, the operation within the integer range iscomplete; thus, calculation is stopped with the remainder of 123. Of course, a decimal
point can be added to the quotient (to the right of the target dividend digit 3) andcalculation can be continued to an arbitrary digit below the decimal point.
4) The solution of this division is a quotient of 000 with a remainder of 123. However,according to the mathematical custom, 0s before the largest digit are omitted: aquotient of 0 with a remainder of 123.
4. General long division III

Here, as general long division, we present a manual method that partially expands generallong division II compared to the traditional manual method of division. As we discussed in
the introduction, in usual division, subtractions are repeated as an essential operation ofdivision; however, to omit this operation somewhat rationally, multiplication is used to skip
ahead to the likely solution to compress the subtraction operation. This is the same in the
manual method of division.
As an example, let us follow the division 234567÷543 discussed in general long division I
and II in steps.
Example 5:1) Considering the target dividend 2 and divisor 543, 2 < 543; thus
we move to the next digit.
2) Considering the target dividend 23 and divisor 543, 23 < 543;
thus we move to the next digit.
3) Considering the target dividend 234 and divisor 543, 234 < 543;
thus we move to the next digit.
4) Considering the target dividend 2345 and divisor 543, 2345 ≥
543; thus, we select a quasi-solution of 4 where the product ofmultiplying the value of a suitable digit (quasi-solution number)and divisor 543 would likely be equal to or slightly less than thetarget dividend of 2345, and place this quasi-solution
immediately above the target multiplicand of 5. After placing
the product 2172 immediately below the target dividend 2345, aline is drawn under the product 2172, and the subtraction resultof 173 is placed below the line. Then we move to the next digit.
5) Considering the target dividend of 1736 and divisor 543, 1736 ≥ 543; thus, we select aquasi-solution of 3 where the product of the arbitrary quasi-solution and divisor 543
would likely be equal to or slightly less than the target dividend of 1736, and place
this quasi-solution immediately above the target multiplicand of 6. After placing the
product 1629 immediately below the target dividend 1736, a line is drawn under the
product 1629, and the subtraction result of 107 is placed below the line. Then wemove to the next digit.
6) Considering the target dividend 1077 and divisor 543, 1077 ≥ 543; thus, we select aquasi-solution of 1 where the product of the arbitrary quasi-solution and divisor 543
would likely be equal to or slightly less than the target dividend of 1077, and place
this quasi-solution immediately above the target multiplicand of 7. After placing the
product 543 immediately below the target dividend 1077, a line is drawn under the
product 543, and the subtraction result of 534 is placed below the line. At this point,
operation within the integer range is completed; thus, calculation is stopped with thesubtraction result of 534 as the remainder. Of course, a decimal point can be added tothe quotient (to the right of the target dividend digit 7) and calculation can becontinued to an arbitrary digit below the decimal point. The quotient is 431 and theremainder is 534. The quotient of 431 should show 000431 to indicate vacant spaces
as we moved down the digits, but these 0s are omitted.
Now, as we can see from the operation procedure of the usual manual method of division(especially 4), 5), and 6)), selecting the quasi-solution is simply selecting the optimum

number by multiplying the divisor with several arbitrary one-digit numbers as a trial
calculation. This is because we wouldn’t know how many times a divisor can be subtracted in
the repeated operation until the operation is performed. In other words, deciding the solution
for each digit is just selecting an appropriate number that works in multiplication as aninverse operation.
Thus, let us show a calculation method that is able to move the calculation operation in longdivision in a rational manner by introducing procedures of general long division II into theusual manual method of division.
Example 6:
Step 1.
First, instead of forcefully selecting a quasi-solution, choose a quasi-solution with asufficiently large remainder, and place a quasi-solution of 3 immediately above the target
dividend of 5. Place the product of the divisor 543 and the quasi-solution of 3, 1629,
immediately below the target dividend of 2345. A line is drawn under this product, and thesubtraction result of 716 is placed under the line.

Step 2.
Next, because the subtraction result above is 716, and 716 ≥ 543, subtracting the divisor 543
from 716 results in non-negative number. Therefore, 1 is placed immediately above thequasi-solution of 3 in a stack. The divisor 543 is subtracted from 716 once, and thesubtraction result of 173 is placed below. Then we move to the next digit.

Step 3.
Next, though the target dividend of 1736 means 1736 ≥ 543, we would not force the selectionof a quasi-solution. Instead, we choose a quasi-selection with a sufficiently large remainder.
We place a quasi-solution of 2 immediately above the target dividend of 6. The product of thedivisor 543 and the quasi-solution 2, 1086, is placed immediately below the target dividend
1736. A line is drawn below, and the subtraction result of 650 is placed under the line.

Step 4.
Next, because the subtraction result above is 650 and 650 ≥ 543, if we subtract the divisor of
543 from the subtraction result of 650, the result is non-negative. Therefore, we place 1
immediately above the above quasi-solution of 2 in a stack. We subtract the divisor of 543
from the subtraction result of 650 once, and place the subtraction result of 107 below. Then
we move to the next digit.

Step 5.
Next, though the target dividend is 1077, and 1077 ≥ 543, we do not forcefully select a quasisolution, but find a quasi-solution with sufficiently large remainder. We place the quasisolution of 1 immediately above the target dividend of 7. The product of the divisor 543 andthe quasi-solution of 1, 543, is placed immediately below the target dividend of 1077. A line
is drawn below, and the subtraction result of 534 is placed below the line.

Step 6.
Next, because the target dividend is 534, and 543 < 543, we move to the next digit. Here,
operation within the integer range is completed, thus the calculation is stopped with thesubtraction result of 534 as the remainder. Of course, a decimal point can be added to thequotient (to the right of the target dividend digit 7) and calculation can be continued to anarbitrary digit below the decimal point. We draw a line above the top row of the quasisolution number and present the sum of the quasi-solution numbers above this line.

As shown above, the solution of this division is a quotient of 431 with a remainder of 534.
If we follow the above procedure, the manual method of division can rationally progress inaccordance with the traditional method. Traditionally, it was common to erase the solution

when one selected an incorrect (not optimal) quasi-solution, but with this method, such a
process is unnecessary. If a quasi-solution is not too small but too large, by handling thequasi-solution number added above other quasi-solutions as a negative number instead of apositive number, these numbers can be cancelled. However, in such a case, instead ofsubtracting the number below from the number above, these numbers are added, whichmakes the calculation confusing. Therefore, the initial quasi-solution should be relatively
small.