Questioning Division

Question

This may sound like a ridiculously stupid question but here it is.

Say I have $8$ divided by $4$, the answer ends up being $0.5$. Now say I have $8$ divided by $490$, the first number in the dividend is a $4$.

But instead of putting in $0.5$ as the first value in the quotient we skip $4$ and move to the next value.

Why?

Answer 1

When we do $4 \div 8$ we don't get $0.5$ right away.

We get $4$ is too small so we make it ten times bigger and add the next digit. The next digit is $0$ because $4 = 4.00000.....$. So we add the $0$ to do $40\div 8 = 5$.

It'd be the same thing if did $4.0000000007 \div 8$. We'd start with $4\div 8$ but $4$ is too small so we make it ten times bigger and add the next digit so we get $40\div 8 = 5$. and we write down $0.5$ but we aren't done because we still have the $.0000000007$ to deal with.

Let's do $490 \div 8$ first.

First note $490 = 490.00000000000.....$ we may need to go into the zeros on the other side of the decimal point.

We start with $4 \div 8$ and we can't do that so we make it ten times bigger and add the next digit to do $49\div 8$. (I guess this is what you mean by "skipping").

And we get $49\div 8 = 6$ with one remainder. So we write down the six and consider the remainder.

$1$ is too small so we make it ten times bigger and add the next digit which is $0$. So we do $10\div 8$. That is $1$ and a two as a remainder. So we write down the $1$ and go on to deal with the $2$.

$2$ is too small so we make to ten times bigger and add the next digit. We've reach the end of the place values we need to use the zeros from the other side of the decimal point. So we note a decimal point and take a $0$ to get $20$.

$20\div 8$ is $2$ with $4$ remainder. So we write down the $2$.

The remainder $4$ is too small so we make it ten times bigger and add the next digit which is a $0$.

We get $40\div 8 = 5$ exactly so we are done we write down $5$ and have $490 \div 8 = 61.25$.

....

But $4\div 8$ is actually exactly the same!

$4 = 4.00000000....$

We start we $4\div 8$ but $8 > 4$ so we "skip" and make it ten times bigger and add the next digit. We write a decimal point because we've reached the ends of the place value.

Then we do $40 \div 8= 5$ so we write down $5$ and we hae $.5$.

Notice we did not ever actually do $4\div 8 = 0.5$. We did $40\div 8=5$.

Answer 2

"But instead of putting in 0.5 as the first value in the quotient we skip 4 and move to the next value. "

That thing to realize is that we are doing the exact same thing when we say $4\div 8 = .5$. We aren't doing $4\div 8 = 0.5$ off the bat. We are really doing:

$4\div 8 = $

$4*1 \div 8 = $

$(0*1 + 40*0.1)\div 8=$

$0*1 + 5*0.1 = 0.5$.

And when we do $490\div 8$ we are doing the exact same thing:

$490 = 4*100 + 9*10 = 40*10 + 9*10 = 49*10 = 48*10 + 1*10$ so

$490\div 8 = $

$(4*100 + 9*10)\div 8 =$

$ (0*100 + 49* 10)\div 8=$

$(0*100 + 48*10 + 1*10)\div 8 = $

$(0*100 + 48*10 + 10*1)\div 8=$

$(0*100 + 48*10 + 8*1+ 2*1)\div 8=$

$(0*100 + 48*10 + 8*1 + 20*0.1)\div 8=$

$(0*100 + 48*10 + 8*1 + 20*0.1)\div 8=$

$(0*100 + 48*10 + 8*1 + 16*0.1+ 4*0.1)\div 8=$

$(0*100 + 48*10 + 8*1 + 16*0.1+ 40*0.01)\div 8=$

$(0*100 + 6*10 + 1*1 + 2*0.1 + 5*0.01 = 61.25$.

.....

Now we could do $4\div 8 = 0.5$ and $1\div 8 = 0.125$ right off the bat if we wanted to:

$490 = 4*100 + 9*10= 4*100 + 8*10 + 1*10$

so $490\div 8=$

$(4\div 8)*100 + (8\div 8)*10 + (1\div 8)*10 =$

$0.5*100 + 1*10+0.125*10 = $

$50 + 10 + 1.25 = 61.25$.

But it's easier and simply to realize $4 < 8$ so we don't want to deal with decimals but to deal with remainders.