# Short division with infinite decimal quotient

True or false? In short division, the quotient is always an integer.

## Short division with infinite decimal quotient

Now we are going to divide 75 by 6. If we use the short division method, we do it like this: Start with the leftmost number of the dividend, seven. Divide that seven by the six in the divisor. Six fits in seven once. Write one in the quotient.

And a small one as a memory digit. Now we use the memory digit one together with the next digit 5 - 15. Six fits into 15 two times. Write a two after the one in the quotient. The dividend is now out of numbers and integers to use.

But we still have a remainder of three. Add a three, just as usual, as a memory digit after the five… ...and a decimal point… ...and another zero in the dividend. Now we can continue to divide. Since we’re now including tenths in the division, we must also add a decimal point to the quotient. Now we use three and zero, 30, and divide by six.

Six fits in 30 five times. Write a five after the decimal point in the quotient. And it fit evenly, so we do not need to add any more zeros to the dividend. 75.0 divided by 6 is 12.5. What’s this?

A single digit number divided by a single digit number. It's easy.. One divided by six. Six fits in one zero times. Write a zero in the quotient.

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 to get more digits to divide. We also have to put a decimal point in the quotient, because we are starting to calculate with tenths. One zero, 10. Six fits in 10 one time.

Write a one in the quotient after the decimal point. And we get a remainder of four. Write a four as a memory digit after the zero in the dividend. Since it does not fit evenly, we need another digit in the dividend. We add a zero - zero hundredths.

Four zero, 40. Six fits in 40 six times. Add a six after the one in the quotient. But now we have a four left over, again. Add a new four as a memory digit.

Add a zero to the dividend. Four zero, 40. Six fits in 40 six times. Add another six after the first six in the quotient. But, there will be a four as a remainder AGAIN!

Will this ever end? No, it actually will not! One divided by six 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. One divided by six 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. Now we have learned how to calculate a division where the quotient does not become an integer. Usually, it is enough to add more numbers to the dividend until we find a finite number of digits after the decimal point.

But sometimes, decimal numbers repeat forever, and we have to round off the quotient.