Hey it’s Professor Dave; let’s learn how to multiply large numbers.
We know what multiplication is, and hopefully we memorized our times tables.
Think fast, nine times seven!
Sixty-three, very good.
How about fourteen times eleven?
You can’t do it as quickly, can you?
Well let’s first learn how to do this rigorously on paper, but then let’s learn some easy
tricks that will help us do this in our heads much more efficiently.
The approach that we all learn in school goes like this.
Put fourteen over eleven with the places lined up, and we start by multiplying all the digits
in the top number by the rightmost digit in the bottom number.
Four times one is four, so we put a four in the units place.
Then, one times one is one, but this one represents ten, so we get a one in the tens place.
Now we are going to multiply all the digits in the top number by the next digit to the
left in the bottom number.
Because this is the tens place and not the units place, we place a zero in the next row
down for the units place, so that we can begin writing numbers in the tens place right away.
Again, four times one is four, but the one represents ten, so we are really getting forty,
which is why the four goes in the tens place.
Then, one times one is one, but both of these ones represent ten, so we are really getting
one hundred, which is why the one goes in the hundreds place.
We add these two rows together, and we get one hundred fifty-four.
This is one way to multiply these numbers together, and we should know how to use this
algorithm, but there is a much faster and easier way to do this, provided that we understand
the distributive property.
Let’s split one of these numbers up into a sum that contains numbers that are a little
rounder and easier to multiply.
Maybe we can split eleven up into ten and one.
Now instead of fourteen times eleven, we have fourteen times ten plus one.
The distributive property tells us that this is the same as fourteen times ten plus fourteen
times one, and these products are easy to get in our heads.
When multiplying by ten, we just add a zero, so one hundred forty, and multiplying by one
means the number stays the same, so we have one hundred forty plus fourteen, or one hundred
fifty-four.
Really, these two methods are doing exactly the same thing, it’s just that one follows
a rigid algorithm, while the other involves a little more creativity, as we have to look
at the numbers and develop a strategy as to how we might restate these numbers to make
things easier for ourselves.
Let’s try another one, four hundred twenty-five times twelve.
Once again, we certainly can follow the rigid algorithm.
We put the larger number on top.
Five times two is ten, so we put a zero and carry the one.
Two times two is four, plus the one gives us five, and four times two is eight.
Now we are done with the two in twelve, so let’s do the one.
Remember, we place a zero in the units place here, since we are multiplying through the
tens place.
Five times one is five, two times one is two, and four times one is four.
Now we just add these together.
A zero here, five and five is ten so put a zero and carry the one, eight and two plus
the one is eleven, so put a one and carry the one, and four plus one is five.
Five thousand one hundred is our answer.
If you like using pen and paper and are very diligent, there is no problem with this approach,
but it opens up a lot of opportunities for error, so you may prefer the simpler approach
using the distributive property.
Instead of four hundred twenty-five times twelve, let’s do four hundred twenty-five
times ten plus two.
When multiplying by ten we just add a zero, so that’s four thousand two hundred and
fifty.
Then we multiply by two.
Four hundred times two is eight hundred, and twenty-five times two is fifty, so that’s
eight hundred and fifty.
Add them together, and we get our answer.
Let’s take a moment to notice that these numbers are the same as the ones we added
together when using the other approach.
That’s because they are essentially the same approach.
When we multiply through the units place, we are multiplying by two.
When we multiply through the tens place, we are multiplying by ten.
But while the first method requires many small computations, the second method requires comparatively
fewer.
Still, through practice, you may find your own method that works for you, and as long
as you always get the right answer, it’s a good strategy.
Being able to do mental math is a great life skill to have, so give yourself a challenge,
try not to write anything down, and let’s check comprehension.