How do decimal remainders work in terms of long division?

Question

I was taught long division a long time ago, and I had this question. Simple enough so far. But then, instead of writing the remainder, I was taught to turn the answer into a decimal by adding zeroes, like so. And further more. I understand it has something to do with the numbers being tenths and hundredths, but I can't fully grasp it. How does this work?

Answer 1

The first three significant figures of $22/7$ are the same as the first three significant figures of $2200/7$ $-$ just the position of the decimal point has changed. So you just have to 'pretend' that you are dividing $2200$ by $7$. And then $22000$, and so on.

Answer 2

$22 = 3\times 7 + 1$ so $22 \div 7 = (21 + 1)\div 7 = 21\div 7 + 1 \div 7 = 3 + 1\div 7$.

That is $3$ and $1$ remainder. That $1$ remainder must in turn be divided by $7$. One way of doing that is to simply to use fractions and say $22 \div 7 = 3 \frac 17$. This is sensible. If we divide a $1$ into seven parts each part is $\frac 17$.

But how to do it in decimals? The issue is the same we still have $22 \div 7 = 3 + (1 \div 7)$ and we have to figure out what $1\div 7$ in decimals is.

Well. $1$ can be written as $\frac {10}{10}$. That is we divide $1$ into $10$ pieces and then divide these $10$ pieces by $7$.

We can think of it this way. $1 \div 7 = \frac {10}{10} = 10\cdot \color{red}{\frac 1{10}}\div 7 = (10\div 7)\cdot\color{red}{\frac 1{10}}$

In a way... if you have $10 \color{red}{\ THINGS}$ and you divide them by $7$ you will have $10 \div 7 \color{red}{\ THINGS}$ and it doesn't matter what the $ \color{red}{\ THINGS}$ are. The could be apples, they could be bricks. Or they could by $ \color{red}{\frac 1{10}}$s.

(In fact this is exactly how long division of big numbers work. To do $822 \div 6$ you start by saying $822$ is $8$ hundreds plus some stuff. $8\div 6$ is $1$ with $2$ remainder so I have $1$ hundred and then $2$ hundreds extra. So $822\div 6 = (8\times 100 + 22)\div 6 = (8\div 6)\times 100 + (22\div 6) = 1\times 100 + (2\times 100 + 22)\div 6= 100 + (222\div 6)$. Then you do the same for $222\div 6$. That's $22$ tens plus some stuff; so you divide $22$ by $6$ to get $3$ tens and $4$ remaining tens and so on.... the only difference here is was are using $\color{red}{\frac 1{10}}$s and not $\color{red}{100}$s.)

And so $(10\div 7) = 1 R 3$ so we get $1$ of these $\color{red}{\frac 1{10}}$s with $3$ of these $\color{red}{\frac 1{10}}$s remaining.

So.... what have we got so far?

$22 \div 7 = (21 + 1)\div 7 = 21\div 7 + 1 \div 7 = 3 + 1\div 7=$

$3 + (10\cdot\color{red}{\frac 1{10}}\div 7 = 3 + (10\div 7)\cdot\color{red}{\frac 1{10}}=$

$3 + (1 \text{with 3 remaining})\cdot\color{red}{\frac 1{10}}= 3 +1\cdot\color{red}{\frac 1{10}} + (3\div 7)\cdot\color{red}{\frac 1{10}}=$

$3 + \frac 1{10} + (\frac 3{10}\div 7)$.

Now we we can rewrite $\frac 3{10}$ as $\frac {30}{100}$ and we now have $30$ of small $\color{orange}{\frac 1{100}}$ pieces. We do the same thing over again.

We have $30$ of these pieces and we divide the $30$ pieces by $7$ and we will have $4$ groups of $7$ of these pieces and $2$ remaining.

$22\div 7 = (21 + 1)\div 7 =$

$3 + 1\div 7 = 3 + \frac{10}{10}\div 7 =$

$3 + 1\cdot \frac 1{10} + \frac {3}{10}\div 7 =$

$3 + 1\cdot \frac 1{10} + \frac {30}{100}\div 7 = $

$3 + 1\cdot \frac 1{10} + 4\cdot \frac 1{100} + \frac {2}{100} \div 7 =$

$3 + 1\cdot \frac 1{10} + 4\cdot \frac 1{100} + \frac {20}{1000}\div 7 =$

$3 + 1\cdot \frac 1{10} + 4\cdot \frac 1{100} + 2\cdot \frac 1{1000} + \frac 6{1000}\div 7 = $

$....$

As we can write $\frac 1{10}, \frac 1{100}, etc$ as $0.1$ or $0.01$ or etc we get

$22\div 7 =3 + 1\cdot \frac 1{10} + 4\cdot \frac 1{100} + 2\cdot \frac 1{1000} +....= 3 + 0.1 + 0.04 + 0.0002 + ..... = 3.142......$