Fractions represent pieces or part of something.
In a picture, we can display this by taking a full shape like a circle and splitting it up into equal parts.
You can choose how many parts you want to split it up into. In this case, we're looking at four.
So, if we cut the circle into four equal pieces we'll have four out of four or still the whole one.
In the orange diagram in the corner, you can see you can then start looking at taking pieces
of the whole, for example, you can take one quarter away and leave three quarters.
One over four you take away and then three over four you leave behind.
When representing a fraction you always need two numbers, you need a top and a bottom.
Technically, the top is called the numerator and the bottom is called the denominator,
however, in most math classes
if you just talk about the top and the bottom people won't usually correct you.
So, here you can see an example of one over two. We've got a circle cut in half and then in
purple, we have ten over fifteen so we have ten purple parts out of fifteen total pieces.
A fraction also represents a division.
The notation for a fraction also represents a division because a
fraction and division actually works the same way.
So, anytime you see one number over top the other you can actually replace it
with any of the following kinds of notation.
You could have three divided by five using the line with two dots in the top and bottom,
you could have three divided by five using the long division notation,
you could have three over five but written with just a slash,
and you can have the notation we were using of three divided by five, three over five.
These are all different notations for division and you should try to get comfortable with all of them.
Now, whenever you have a fraction you must have equal parts.
You cannot have someone's half being bigger than the other person's half.
You also don't have to use circles as the only shape. You could use rectangles,
you could use triangles, you could use a square, anything will
work as long as you subdivide it into equal pieces.
You can use fractions to represent parts of anything.
What we'll see later is that fraction notation can also be used to describe a ratio.
Let's take a look at this example. Let's say in total we have nine pets
but those nine pets are made up of three dogs, four cats, and two pigs.
You could then have these three fractions: three dogs out of nine pets,
four over nine - four cats out of nine pets,
or two pigs out of nine pets. You could also have a test example where you wrote
a test and you answered sixteen of the twenty questions correctly
so, your overall grade of the test is sixteen out of twenty or sixteen over twenty.
Or, you could convert this to a percent which we'll see in a later unit.
Now, there's a few different kinds of fractions. Even though these different
definitions do exist, you don't have to let that scare you because
once we start doing the math all of these different fractions will behave in the same way.
So, we do have a zero fraction. A zero fraction is the first example and that's when the numerator or the top is zero.
If the numerator is zero then the fraction always represents zero
no matter what the denominator is unless you have zero as both the numerator and the
denominator then that's technically undefined
and something you would cover in a later math course.
The next one is proper fractions. So, a fraction is supposed to represent part of a whole
and that's why this is the proper fraction. When the numerator, the top, is smaller than the bottom, the denominator.
Any fraction like that will always be a proper fraction and it's smaller than one.
Next is a whole fraction. When the number in the top or the numerator is equal to the number in the bottom,
or the denominator then that is a whole fraction
because whenever you divide something by itself you get one and you have the whole part back.
Lastly, we have an improper fraction.
Sometimes the numerator or the top is bigger than the bottom and that fraction
represents something bigger than one, so, if you have seven over five for
example, you actually have one whole piece
plus two pieces more in the next hole. With those definitions in mind
we can place each of these kinds of fractions on the number line.
Maybe this illustration will help make these definitions more clear.
First, we have the zero fraction so, on the left of the number line without
getting into the negatives we see the zero fraction right at zero.
Next is a proper fraction. Any time you have a fraction that is
part of the whole just like the original definition, that is where you have a
proper fraction. Then, right at the one, you have the whole
fraction. So, if the top is equal to the bottom
then you have the whole piece. Lastly, improper fractions are anywhere
greater than one. So, any time you have a larger numerator
than denominator, you have an improper fraction. As you can see
going from left to right, a zero fraction is less than a proper fraction which
then is less than a whole fraction and those are all less than an improper fraction.