Long division with infinite decimal quotient

Now we are going to divide 75 by 6. We’ll use the long division method. Draw a line. Write the denominator, 6, below the line, to the left. Write the dividend, 75, to the right below the line.

Now let’s calculate. We start with the leftmost number of the dividend, 7. How many times does 6 fit in 7? 6 fits in 7 once. Write 1 above the line.

Now, let’s multiply 6 by 1. This is 6. And then the seven here minus 6. This is 1. And then move down the five.

Now we use 1 together with 5 - 15. Six fits in 15 two times. Write a two after the one above the line - this will be the quotient of the calculation. Then we carry on with the long division. 6 times 2 is 12, and 15 minus 12 is 3.

The dividend is now out of digits to use. But we still have a remainder of three. Add a decimal point… ...and a zero in the dividend. The zero makes it possible to keep on dividing. Now we are including decimals in the division, so we can expect decimals in the quotient.

So we add a decimal point above the line. And then we move down the decimal zero so it ends up next to the three. The three and zero together forms the number 30, and now we divide by 6. 6 fits in 30 five times. Write 5 after the decimal point in the quotient.

6 times 5 is 30, and 30 minus 30 is zero. Now it fits evenly, so we do not need to add any more zeros to the dividend. 75.0 divided by 6 is 12.5. Here we have a single digit number divided by another single digit number. 1 divided by 6.

6 is larger than 1. So it does not fit even once in 1. Zero times. Write 0 in the quotient. 6 times 0 is 0, and 1 minus 0 is 1.

We get a remainder of one. Now there are no more digits in the dividend, so we do as usual and put a decimal point and a zero in the dividend. We also have to put a decimal point in the quotient, because we are starting to calculate with tenths. Move down the zero. 1 and 0, this makes 10.

Six fits in 10 one time. Write a one in the quotient after the decimal point. 6 times 1 is 6, and 10 minus 6 is 4. We get a remainder of 4. Since it does not fit evenly, we need another digit in the dividend.

We add a zero - zero hundredths. Move down the zero. 4 and 0, 40. 6 fits in 40 6 times. Add a six after the one in the quotient.

6 times 6 is 36. 40 minus 36 is 4. But now we have a four left over, again. Yes, it can turn out this way. Just carry on.

We add another zero to the dividend. Move down the new zero. 4 and 0, 40. 6 fits in 40, 6 times, like we saw a minute ago. We add another six after the first six in the quotient.

But, there will be a four as a remainder AGAIN! Because 6 times 6 is 36, and there are four left to make 40. Will this ever end? No, it actually will not! 1 divided by 6 has an answer with an infinite number of decimals.

It does not matter how many zeros we add to the dividend. When we try to calculate one divided by six, we will never get a decimal number that ends. We simply have to round to a certain number of decimals, for example two. 1 divided by 6 is about 0.17 because six is rounded up to a seven. If we want to state the exact value of this number, it is better to write ‘one-sixth’, a fraction, instead of a decimal.

A number written as a fraction is always exact.