Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the last lecture, we saw how ancient people came up with the idea of fractions
as a way to represent numbers between zero and one.
At first, the only type of fractions used were "unit fractions",
one-half, one-third, one-fourth, and so on.
The denominator of a unit fraction, which we will represent by the letter q,
can be any natural number, but the numerator must be one.
Eventually, unit fractions were replaced with "common fractions".
The numerator of a common fraction can be any integer,
which we will represent by the letter p.
When representing the value of a fraction on the number line,
the denominator specifies how many equal parts each unit length is subdivided into,
and the numerator specifies the number of those subdivisions in the fraction.
Using the number line, we can visualize arithmetic operations with fractions
in much the same way we visualize arithmetic operations with integers.
Integers are represented as multiples of the unit length.
However, to represent fractions, we use multiples of the subdivisions determined by their denominators.
For example, if we add the fractions three-fifths and four-fifths,
we use five subdivisions per unit length,
since these fractions both have denominators of five.
Each subdivision is one-fifth of a unit.
The fraction three-fifths represents three of these subdivisions
and the fraction four-fifths represents four subdivisions.
Therefore, the total length adds up to seven subdivisions, or seven-fifths.
If we subtract four-fifths from three-fifths,
we start out at a position three subdivisions to the right of the origin
and then move four subdivisions to the left.
We end up one subdivision to the left of the origin, at negative one-fifth.
If we want to multiply an integer times a fraction,
for example, four times three-fifths,
this is the same as adding three-fifths four times.
On the number line, we can see that this adds up to twelve-fifths.
Multiplying an integer times a fraction doesn’t change the size of the subdivisions,
so the denominator is not affected.
We simply multiply four times the numerator, three, to get the numerator of the result.
So when multiplying an integer and a fraction,
we multiply the integer times the fraction’s numerator.
For example, to multiply three times one-fifth
we multiply three times the numerator, one,
which give us three fifths.
Likewise, five times two-thirds is five times the numerator, two,
or ten thirds.
What happens when we multiply a fraction times a fraction?
We can visualize this by starting with a square with a height and width of one unit.
The area of a square is equal to its height times its width.
So this square has an area of one.
Now if we divide the square into four equal pieces, each piece will have an area of one-fourth.
Since the height and the width of each piece is one-half,
we can see that one-half times one-half
is one-fourth.
If we divide the square into thirds horizontally,
we will have six equal pieces, each with an area of one-sixth.
Since each piece has a height of one-half and a width of one-third,
we see that one-half times one-third is one-sixth.
Likewise, if we divide the square into thirds horizontally and vertically,
we will have nine equal pieces, each with an area of one-ninth.
Since each piece has a height and width of one-third,
we see that one-third times one-third is one-ninth.
Notice that in each case, when we multiplied two fractions
we could have obtained the denominator of the result
by multiplying the denominators of the two fractions.
So to multiply two unit fractions, we multiply their denominators.
As an example, to multiply the fractions one-third times one-seventh,
we multiply the denominators three and seven.
So the product of these two fractions is one twenty-first.
But what if instead of multiplying two unit fractions,
we want to multiply two common fractions?
When multiplying common fractions,
the numerators are multiplied, creating the product’s numerator
and the denominators are multiplied, creating the product’s denominator.
But why does this work?
We can see the reason if we use a series of small steps that we already know work.
Let's start over and do this one step at a time.
We’ll start by splitting each common fraction into an integer times a unit fraction.
Using the commutative property of multiplication, we can rearrange the factors
and using the associative property, we can group the two integers, and the two unit fractions.
As we saw, the two unit fractions are multiplied by multiplying their denominators.
So now we have the product n times p, which is an integer,
times the unit fraction one over m times q.
To multiply an integer times a unit fraction, we multiply the integer times the numerator.
Of course, n times p times one is just n times p.
So when multiplying two common fractions,
the numerators are multiplied, creating the product’s numerator
and the denominators are multiplied, creating the product’s denominator.
Let’s do this one more time,
but instead of using letters which can represent any integers,
we’ll use specific numbers.
Once again, we start by splitting each common fraction into an integer times a unit fraction.
Then using the commutative and associative properties,
we rearrange the factors so that the two numbers which were the numerators are grouped together
and the two unit fractions are grouped together.
We then multiply the unit fractions by multiplying their denominators
and we multiply the two and five which were the numerators.
Of course, ten times one twenty-first is ten twenty-firsts.
So to multiply the fractions two-thirds times five-sevenths
we can just take the short cut of multiplying the numerators two and five,
and the denominators three and seven.
And the result is ten twenty-firsts.
This rule works for any type of fraction.
If the fractions are unit fractions,
multiplying the numerators and the denominators still gives us the correct result.
So multiplying fractions is simple.
We multiply the numerators and the denominators.
In the next lecture, we will see how to divide fractions.