It’s Professor Dave; let’s learn how to do long division.
After we learned about the concept of multiplication, we had to learn how to multiply not just small
numbers together, but also large ones, and we saw some different strategies for how to
do this.
The same goes for division.
Sometimes it’s easy to divide two numbers.
Twelve divided by three is clearly four, since four goes into twelve three times.
But what about six hundred twenty-four divided by three?
Just as there is an algorithm for the multiplication of large numbers, there is one for division
as well, and it is called long division.
This technique is usually remembered with great disdain from childhood, but if we look
carefully, we can see the elegance of this technique, and appreciate the patterns that
it reveals.
It works just like our multiplication method, simply in reverse.
Instead of multiplying one number through the units place of another number, then the
tens place, and so forth, we will divide one number by another starting with the largest
place value of the larger number, and then the next largest, and so forth, moving down
until we get to the units place.
Let’s try this with the example we mentioned.
We place the first number, the dividend, inside this symbol here, and the other number, the
divisor, goes to the left.
Now we just go through the dividend one number at a time.
Three goes into six two times, or in other words, six divided by three is two, so we
put a two here in the hundreds place of the quotient.
Now we do two times three, which is six, place that directly under the first six, and then
subtract.
Six minus six is zero, so we draw a line and write a zero here.
Then we drop the next digit down and repeat the process.
The next digit is a two.
Three does not fit into two, or rather, two divided by three is less than one.
So let’s put a zero here.
Zero times three is zero, two minus zero is two, and then we drop down the five.
Three fits into twenty-five a little more than eight times, because three times eight
is twenty-four, so let’s put an eight here, then eight times three is twenty-four as we
said, and subtracting from twenty-five we get one.
Now there are no more digits to drop down, and three doesn’t fit into one, so this
is called the remainder.
Three fits into six hundred twenty-five, two hundred and eight times, with one left over,
or we can say that six hundred twenty-five divided by three is two hundred eight remainder
one.
So that’s all there is to long division, we go one digit at a time just like we did
with multiplication, except we go left to right instead of right to left, since division
is meant to be the reverse of multiplication.
Let’s try one more for practice, three hundred ninety seven divided by eleven.
This time we have to divide by a two-digit number.
Eleven doesn’t fit into three, so let’s put a zero, get a zero down here, which leaves
us with three.
Then we bring down the nine.
Eleven does fit into thirty-nine three times, so we put a three, multiply to get thirty-three,
subtract to get six, and then pull down the seven.
Eleven fits into sixty-seven six times, so we put a six, multiply to get sixty-six, and
subtract to get one.
Eleven doesn’t fit into one and we’ve run out of digits in the dividend, so that
will be thirty-six with a remainder of one.
If we don’t like these remainders, we could keep going to get decimal places.
Just put a decimal here and add a zero, drop it down to get ten.
Eleven doesn’t fit in ten, so we put a zero, multiply to get a zero, and subtract to get
ten.
Then we pull down another zero, and we get one hundred.
Eleven fits into a hundred nine times, so we place the nine, multiply to get ninety-nine,
and subtract to get one.
We would soon find out that this process could go on forever; giving alternating zeroes and
nines forever, and this is our first introduction to repeating decimals, which will come back
later when we look at the difference between rational and irrational numbers.
For now, let’s check comprehension.