Long Division with Zero, Revisited

(A new question of the week)

One of our first posts, in 2018, was about zeros in long division. But we still get many questions about this issue, and it’s time to dig in deeper. We’ll look here at two of them, answering the twin questions, “When do you put a zero in the quotient … and when do you not?”

Why add a zero?

The first is from a student who called himself CodedGamer, in late May:

Recently when I was solving some exercise problems in my book, I came across a situation where I had to divide 245/8, well at first this looked quite simple and usual to me but then when I arrived at the answer and cross checked it, it seemed like the answer was wrong. So I once again checked my answer and still got 3.625 and then being confused when I did it on my calculator I got the answer as 30.625.

The problem is I don’t understand why you add a zero before .625. Usually the cases for adding a zero before a decimal place is different from the one in this question.

The way I went about solving the problem:

8 | 245    | 3.625
(-) 24
     050
  (-) 48
       20
   (-) 16
        40
    (-) 40
         0

I also checked on this article https://www.themathdoctors.org/long-division-when-zero-gets-in-the-way/ but did not get any help from it.

All I need is a precise explanation on why we add a 0 before .625 which seems unusual to me.

The error here is the same type discussed in the old post; but it is easy for many students to miss, especially if they learned the method by rote and picked up a wrong idea of when and why you need to write a zero. The reality is that “before a decimal” makes no difference; so we can guess that Gamer is tripping over other things that happen there.

Doctor Fenton answered, starting with alternative methods for doing the same division:

Dr. Peterson’s explanation is worth reading more than once, if you don’t understand it the first time, but I will try a couple of other approaches to approach the problem.

One is that 245 divided by 8 can be written as a fraction, 245/8, and we can write

   245 = 240+5  = 240 +  5  = 30 + 5/8 = 30 + 0.625 = 30.625
    8      8       8     8

Sometimes the easiest way to do a division is to treat it as a fraction! In this case, it makes the zero hard to miss.

Never ignore a quotient

Another way to see this is to look at a similar problem:

       __31_____
     8) 255.000
    (-) 24↓
         15
     (-)  8 
          7
          :

The problem occurs at this stage.  After multiplying 3 times the 8 in the quotient to get 24, which is then subtracted from the first two digits, 25, you have a remainder of 1, and you bring down the next digit, 5, so that you have then have 15 as the new (partial) dividend.  8 goes into 15 1 time, with a remainder of 7, and so on.  The new quotient, 1, goes into the 1’s place in the overall quotient for the problem.

Note that he is using the style of long division in which the quotient is placed above the dividend, rather than to the right as Gamer did it. (Wikipedia indicates that this style is used in the English-speaking world; I have never determined where Gamer’s style is standard, but we see it often. I’ll call the former “American style”.) As we’ll see later, this can help keep track of digits better. But the key is that the new digit goes into the same place as the digit “brought down”.

Compare this with your problem:

       __3?_____
     8) 245.000
    (-) 24↓
         05

When you bring down the 5 (from the 1’s place in the dividend), you need to put the new quotient in the 1’s place in the quotient for the problem.  Here, your remainder from the previous step was 0, so when you bring down the 5, the new dividend at this step is 5, which is smaller than the divisor, 8.  You are just ignoring this problem and bringing down another digit, from the tenths place in the decimal to make a new dividend of 50.  But that is not the algorithm.  The division of 5 by 8 should be thought of as 5 = 0*8 + 5: that is, 5 divided by 8 is 0, with a remainder of 5, so the quotient at the second stage is 0, and that quotient is to be put into the 1’s place in the quotient for the problem.  You must not ignore quotients of 0 in the intermediate steps.

A zero is not nothing! It must be written into the quotient like any other digit.

A more extreme example is

       __2009
   12 ) 24108 
    (-) 24↓↓↓
         01↓↓
     (-) 00↓↓
          10↓ 
      (-) 00↓ 
          108
      (-) 108
            0

You cannot just ignore 0 quotients in the intermediate steps in the algorithm.

Does this help?

The decimal point doesn’t change things

Gamer responded:

Thank you for responding and taking my question into consideration.

I clearly understood the step where you say 5 divided by 8 is probably 5 = 0*8 + 5 with zero as the quotient and 5 as the remainder.

But let’s take another problem and compare 245/8 with it:

8 | 236 | 29
    16
 (-) 76
     72
  (-) 4

So let’s pause the problem here, here we are left with the remainder as 4 and the quotient as 29. After this I would go on and add a decimal place to the quotient. And then I can add a zero to the remainder which would become as 40 after which I can make the quotient as 29.5 and end the problem.

My point is in the case of the second example you gave me (24108/12) we add a zero to the quotient to bring another digit from the dividend to the remainder. But in the case of 245/8, when we add a decimal to the quotient; doesn’t that mean we can actually bring another digit down (in our scenario, zero) to the remainder?

I am still not sure if I understand long division properly because when I approached different sites and people and asked them about this problem, they were able to tell the right answer but not explain how they got it right. So the thing I expect is why isn’t the division of 236/8 not similar to 245/8.

Doctor Fenton replied,

Every time you “bring down” another digit, it comes down from a decimal position in the dividend: the 1’s place, the 10’s place, the hundred’s place, or the tenth’s place, the hundredth’s place, etc.

The number brought down becomes the units digit in the new dividend at that level, and you have to put the quotient of the new division in the corresponding place in the quotient on top.

When you bring down a digit from the ones place, you have to put a digit in the ones place of the quotient. To put a digit in the ones place, you have to have brought a digit down from the ones place.

In 236/8, you got the integer part of the quotient as 29, but there is a remainder of 4, which is in the same column as the 6 which was brought down to make 76.  You can stop at this point and say the answer is 29 remainder 4, or you can keep going to get a decimal answer.  If you want to keep going, the next digit comes from the 0 in the tenth’s place of 236=236.0.  Bringing down that 0 makes the new dividend 40, and the quotient of 40/8 is  5, with no remainder, so you put the 5 in the tenth’s place of the quotient on top: 29.5.  Every time you bring down a number to make a new dividend at that level, you have to put the quotient, even if it is 0, in the decimal position that the number brought down came from.

Gamer had not mentioned bringing zeros down, after the decimal point; that may be part of what he is missing. We are still following the same process.

You write

But in the case of 245/8, when we add a decimal to the quotient; doesn’t that mean we can actually bring another digit down (in our scenario zero) to the remainder?

Every time you bring down a number from the original dividend, you have to put a digit in the same position in the quotient.  You can’t skip a position and bring down two digits at one time.  In 245/8, when you bring down the 5, it came from the units position in the original dividend, and the new dividend is 05, and 5 = 0*8 + 5 (exactly, not probably!)  The new quotient at this step is 0, and the remainder is 5, so you have to put a 0 in the units place of the quotient for the problem.  Then you can bring down a zero from the tenth’s place to make a new dividend of 50, but now 50 = 6*8 + 2, so you put a 6 in the tenth’s place.  Then you bring down the 0 from the hundredth’s place to make a dividend of 20, and so on.

Comments section

Gamer had also asked his question as a comment on the post he referred to:

I came across a similar situation in long division where I had trouble with understanding zeros. In the problem 245/8 I went about solving it the usual way and got the answer as 3.625 which was wrong when I checked it with my calculator, the correct answer was 30.625. The thing I would like to know is what is the meaning of the zero before the decimal point.

I answered him there:

Hi, Gamer.

The key idea is that each time you divide by the divisor, you must put the quotient digit in the answer, even if it is a zero.

In your example, you first divide 24 by 8, and write the quotient, 3, in the answer (specifically, in the tens place, as 24 is in the tens place of the dividend – that is, it means 24 tens, not 24 ones). After multiplying and subtracting, the remainder is 0.

You then bring down the next digit, 5, and append it to the remainder to get 05. When you divide this by 8, you get 0 (because 5 < 8), and you must write this 0 in the answer. It is needed because we need a ones place in the answer, corresponding to 5 being in the ones place of the dividend.

After that, you can continue. Since the next digit you bring down is a 0 from the tenths place, the quotient you get, 6, must go in the tenths place of the answer.

So, what is the meaning of the zero? It can be considered a place holder, to ensure that the 3 you got represents 3 tens. That is because when you divided 24 by 8, you were really dividing 240 by 8, and the quotient is 30, not just 3.

We can express the work algebraically. To divide 245/8, we treat it as (240 + 5)/8 = 240/8 + 5/8 = 30 + 5/8. This corresponds exactly to what I just described.

Compare a similar problem, 236/8. Here we treat it as (160 + 76)/8 = 20 + 76/8; then we treat 76/8 as (72 + 4)/8 = 72/8 + 4/8 = 9 + 0.5. So the answer is (160 + 72 + 4)/8 = 160/8 + 72/8 + 4/8 = 20 + 9 + 0.5 = 29.5. The difference between the two problems is merely that there is a 0 in the ones place in the first example.

Does that help at all?

At this point, Gamer responded:

Thank you for responding and considering my question.

After reading Doctor Peterson’s blog post twice I pretty much understood the algorithm and the way long division works. Also I would like to thank Doctor Fenton for helping me out on this. I guess now I am quite familiar with the way long division works.

The reason all this looked different to me might be because of the way I learnt long division back in school. As in school they teach us like “follow this step and you will get the answer” which I feel is quite wrong because you don’t actually see or understand the thing that is happening under the hood. Like I never knew the reason for adding a zero in the quotient to bring down another digit from the dividend until I read Doctor Fenton’s reply.

Once again I would like to thank Doctor Fenton and Doctor Peterson for helping me out on this.

Just what we hope for!

Why not add a zero?

A similar question with different particulars had come to us from Mekhael back in March:

I want to ask about 34 739 divided by 3.

My answer is 115 790.6 but calculator says 11 579.6 .

Why is there no zero after nine? We usually put a zero before putting a decimal. I’m stuck. Can you please help.

MY WORK:

3 ) 34739 ( 115790.6
    3
    04
     3
     17
     15
      23
      21
       29
       27
        20
        18
         2

Mekhael makes the exact opposite error to Gamer, adding an extra zero, evidently thinking the decimal point requires it.

Doctor Fenton answered:

Hi Mekhael,

I think you are losing track of the decimal point in your computation.

Were you ever taught to use estimation to get an idea of the answer to a problem?  Your computation is telling you that you are dividing 34,739 into three parts, and one of those parts is over 115,790, over three times the original amount.  To check the reasonableness of the answer, you can compare your problem to an easier one, say dividing 33,000 by 3.  That is clearly 11,000, so your answer should be a bit larger than 11,000, not over 100,000.

We can check a division by estimation, or by multiplication. Both of our “patients” have checked only by using a calculator, rather than by thinking about meaning.

I think I can show you the error with a simpler example: 37 divided by 3.  If you use division with remainders, you write

           12 
       3 ) 37
           3
            7
            6
            1

which you write as 37/3 = 12 R 1  or 12 remainder 1.

But if you want a decimal answer, you keep going:

           12.33 
       3 ) 37.00
           3↓ ↓↓
           07 ↓↓
            6 ↓↓
            1 0↓
              9↓
              10 
               9  
               :

In the remainder version, the remainder must always be less than the divisor, but if you want to keep going to get the decimal answer, you don’t put 0 into the quotient.  You write the decimal point, and bring down the first digit after the decimal point and continue.

Once again, notice that showing the decimal point and following zeros in the dividend demonstrates that the same process continues.

For your problem, you could start by thinking

        ?????.?? 
    3 ) 34739.00

The dividend has five places to the left of the decimal point, so the quotient will have at most five places to the left of the decimal point.  If you line up the quotient above the dividend, you can put the decimal point in the quotient position before you even start the computation.  After you have done the division where you brought down the 1’s digit (in your computation, that is the line where you divide 29 by 3, putting a 9 in the 1’s digit of the quotient, leaving a difference of 2.  That 2 can either be the remainder, or you put a decimal point in the quotient and continue the division bringing down the first digit after the decimal point.

Does this help?

As we’ve seen, the American style makes it easier to put digits in the right place.

What’s the difference?

Mekhael wrote back,

If we compare the above question with another question, which is 21 divided by 2, the answer is supposed to be 10.5. Why do we put 0 before decimal in this question and do not put a 0 in the previous question where the answer is 1579.6. Why is it not 15790.6?

Doctor Fenton replied,

Writing

         1?.? 
     2 ) 21.0
         2↓
         01

the 1 we just brought down is the last digit before the decimal point, and the difference of the first step was 0, so the next step divides 2 into 1.  2 goes into 1   0 times, so we have

         10.? 
     2 ) 21.0
         2↓
         01
         00
          1

That 1 is now the remainder, so you can write 21/2 = 10 R 1 (10 remainder 1), or you can keep going to get a decimal answer:

         10.? 
     2 ) 21.0
         2↓ ↓ 
         01 ↓
         00 ↓ 
          1 0

and the quotient now continues after the decimal point.  2 goes into 10  5 times, so

         10.5 
     2 ) 21.0
         2↓ ↓ 
         01 ↓
         00 ↓ 
          1 0 
          1 0
            0

and the division terminates.

Last time there was no reason for a zero; this time there is. That is the difference!

You have been writing successive digits in the quotient as quotients of smaller problems, and the 0 before the decimal is one of these quotients.  You haven’t arbitrarily inserted a 0.  If the problem had been 23/2, we would write

         11.5 
     2 ) 23.0
         2↓ ↓ 
         03 ↓
         02 ↓ 
          1 0
          1 0
            0

Do you see the difference?  Now there is a 1 in the units place of the quotient, because the 3 in the units place in the dividend gives a digit of 1 in the quotient because 3 is larger than 2.  When there was a 1 in the units place of the dividend, the divisor 2 is larger than 1, so the digit it contributes to the quotient is 0, which we write in the units digit of the quotient.   There is nothing arbitrary about this step.

Recap

Mekhael tried to summarize:

I just came to know that if we take down a digit from the original number and it’s small then only we add a zero but if we have the remainder that is small then we don’t put a zero, is that correct? Am I getting it right?

Doctor Rick joined in to clarify:

Hi, Mekhael. I am going to add some thoughts.

Here is how I would say what I think you are saying. When you find the remainder, the remainder must be small, in the sense that if it’s greater than the divisor, you picked too small a digit for the quotient. But if, after you’ve brought down the next digit from the dividend, what you’ve got is still less than the divisor, then the divisor goes into that number 0 times, and you write this 0 as the next digit of the quotient.

The issue isn’t a small digit brought down, or a small remainder, but a zero quotient.

Here’s your work on the original problem up to the point just before you added the 0 incorrectly:

3 ) 34739 ( 11579
    3
    04
     3
     17
     15
      23
      21
       29
       27
        20

Here you had a remainder of 2, which is less than 3 as it should be. Then you brought down a zero (the unwritten tenths digit of the dividend), making 20 as your new partial dividend. Dividing 20 by 3, you get 6, which you then write as the next digit of the quotient. (You need to remember to put the decimal point in the quotient first. This is easier to remember when you put the quotient above the dividend, as Doctor Fenton has been doing, rather than at the right.)

Again, using the American style and writing the decimal point in the dividend helps you remember.

You would only write a 0 in the quotient if you say, “3 doesn’t go into my partial dividend.” You don’t ask that question when all you have is the remainder from the previous step; you wait until after you brought down a new digit.

This really has nothing to do with where the decimal point is; I think the real reason you got confused is that you brought down a 0. Here’s another problem where you need to bring down a 0:

3 ) 34039 ( 11346 r 1
    3
    04
     3
     10
      9
      13
      12
       19
       18
        1

When I brought down the 0, I didn’t write a 0 in the quotient. I saw that 3 goes into 10 3 times, so I put 3 into the quotient.

Again, it is not bringing down a zero, but having a small partial dividend, that results in a zero.

Here is one more example, where you do write a 0 in the quotient:

3 ) 317 ( 105 r 2
    3
    01
     0
     17
     15
      2

When I brought down the 1, I had a partial dividend of 01, or 1, which is less than 3. Since 3 does not go into 1 (that is, it goes in zero times), I wrote 0 in the quotient. That’s the only time you should write 0 in the quotient!

Gamer made a different mistake:

Students probably have more trouble remembering to write the 0 than writing 0 when they shouldn’t. That’s because we often shorten the work to this:

3 ) 317 ( 105 r 2
    3
    017
     15
      2

Since subtracting 0 from a number doesn’t change the number, we can save work by not writing the subtraction. But we’re still doing it, and we still need to write our quotient digit 0.

I hope this helps!

The more you write, the safer you are; the less you write, the more you have to think!

Mehkael concluded:

Thanks a lot Dr Rick and Dr Fenton. I found your prompt replies very helpful. And they did clear my concept. Thanks again.

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