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Fractions have a numerator and a denominator

Fractions (and ratios) are made up of two numbers: the numerator and the denominator.

The denominator is the bottom number, and represents how many parts are possible—in other words, how many parts go into a whole.

The numerator is the top number, and represents how many parts you have. It might be less than the whole!

If the denominator is a $0$ that is bad. We are not allowed to divide by zero.

If the denominator is a $1$, our fraction is actually an integer. This suggests that we can add, subtract, multiply, or divide fractions. Adding and subtracting fractions is trickier, especially if we don't have common denominators. But it can be done.

For now though, we want to show you a visual of the different parts of a fraction.

Every fraction has a multiplicative inverse. The inverse of fraction $\frac{a}{b}$ is $\frac{b}{a}$, and is also called the reciprocal. If we multiply a fraction by its multiplicative inverse, we get $1$. The numerator and denominator cancel each other out.

(Video) Introduction to Fractions

by mathman1024

This video by mathman1024 explains fractions.

$\large{\text{Summary}}$

A fraction is a number of the form: $$\frac ab$$ where $a$ and $b$ are integers and $b\ne0$. A fraction represents the division of two numbers. The fraction symbol means the same thing as $\div$.

The number on top is called the numerator and the number on the bottom is called the denominator. This provides an alternative to writing a number with a decimal point. If you perform long division, you may get a decimal. In many cases, the fractions are easier to write.

Let's look at an example of a fraction, $\frac34$. What does this mean? Well, let's look at a picture to find out. Say we have a circle. We will break it into a number of equal parts. Just like you would slice a pizza.

The denominator tells us how many parts this circle is broken into. Since the denominator for $\frac34$ is $4$, we break the circle into $4$ equal parts.

Now the numerator tells us how many of these $4$ parts we have. Since the numerator for $\frac34$ is $3$, we shade in $3$ of the parts.

So, here's a picture representation of $\frac34$. For one circle, we have $3$ of the $4$ equal parts.

You can also use boxes to represent fractions. Look at the picture and see if you can figure out what the fraction is.

If you guessed $\frac13$ you are correct!

What about when the numerator is larger than the denominator, like $\frac52$. Well, this means that our shape is broken into $2$ equal pieces but we have $5$ of these pieces. So therefore, we will need to have more than one of our shape.

So as you see, we have our shapes broken into two equal pieces and we have $5$ of those pieces. Notice that we needed to make three diamonds to represent this. Two are filled in completely. This shows that the fraction can also be written in mixed fraction form as $2\frac12$. Both $2\frac12$ and $\frac{5}{2}$ have the same value. We can convert from mixed numbers to improper fractions and vice-versa.

A proper fraction is less than $1$. These are fractions with denominators bigger than numerators. They have smaller numerators than denominators. $$\frac12,\ \frac23,\ \frac14,\ \frac3{2017}$$

An improper fraction has a greater numerator than the denominator. $$\frac32,\ \frac73,\ \frac{11}9,\ \frac{2017}{2}$$

Any fraction, proper or improper, is a rational number. This is because every fraction is one integer divided by another. We could divide multiple pairs of numbers and get the same result each time. If this happens we have what are called equivalent fractions.

As Above so Below

By now, you've probably seen a fraction.

And now's the part where we introduce two new vocab words just to make your life harder. These two words, the numerator and the denominator, are the parts of a fraction.

• The numerator is the top number. In the picture above, the numerator is $3$.
• The denominator is the bottom number. In the picture above, the denominator is $5$.

When a numerator and a denominator love each other very much, they make a fraction.

Let's look at a real world explanation with everyone's favorite topic, pizza. Say you have a whole pizza, split into fourths, and you eat one piece.

The numerator tells us how many slices we currently have left: $3$.

The denominator tells us how many slices we had to start: $4$.

Therefore, the fraction $\frac34$ can represent the amount of pizza we have left.