Substituição de numeros na reta real

A proper fraction (also called a proper number) is a fraction that is between $0$ and $1$. These fractions will have a smaller number up top than on bottom.

An improper fraction is a fraction that's greater than $1$. These fractions will have a larger number up top than on the bottom.

Furthermore, zero will never be in the denominator. We're not allowed to divide by zero.

When placing a fraction on the number line we first look at the bottom number. This tells us how many spaces we need to divide up between each number.

Suppose we wanted to place $\frac58$ on the number line. We would first note that it was a proper fraction and would go between $0$ and $1$. Next, we break up the space between $0$ and $1$ into $8$ equal spaces. In other words, we are breaking the whole line segment between $0$ and $1$ into parts.

The top number is where we place our dot amongst these spaces. Starting from $0$, move $5$ ticks in. This will be the fraction $\frac58$.

For improper numbers, the process is almost identical. Say we had the fraction $\frac{21}{8}$. The first thing to look at is what two whole numbers our fraction is between. We can do this by converting it into a mixed number. $$\frac{21}{8}=2\frac58$$

Looking at the number in the front, we know our fraction is bigger than $2$. So, it will be between $2$ and $3$. Break up the space in between $2$ and $3$ into $8$ equally sized spaces.

The top number of our mixed number is where we place our dot amongst these spaces. Starting from $2$, move $5$ ticks in. This will be the mixed number $2\frac58$ or the fraction $\frac{21}{8}$.

Finally, if the number we start with has a negative sign, it is a negative number. Fractions with value between $-1$ and $0$ will still be proper fractions, while fractions with value less than $-1$ are still regarded as improper fractions. For example, $-\frac{4}{5}$ is a proper fraction because the numerator is smaller than the denominator. On the other hand, $-\frac{7}{5}$ is an improper fraction because the numerator is larger than the denominator.

(Video) Fractions on the Number Line

by mathman1024

Here's a video by mathman1024 that shows where numbers are on the number line.

$\large{\text{Summary}}$

Let's start with some vocabulary. A proper fraction is a fraction where the numerator is smaller than the denominator. $$\frac12, \frac23, \cdots$$ An improper fraction is a fraction where the numerator is larger than the denominator. $$\frac53,\frac76,\cdots$$

The first thing to note is that positive proper fractions are always between $0$ and $1$. Positive improper fractions are greater than $1$.

Negative proper fractions are always between $-1$ and $0$. Negative improper fractions are less than $-1$.

So, let's place some fractions on the number line, starting with $\frac12$. Here's our number line.

Since it's a proper fraction, it will be between $0$ and $1$. The denominator tells us how many equal parts we break the space into. The numerator tells us what line in between we go on. So, let's break it down into two equal parts.

As you can see, the space between $0$ and $1$ is broken into $2$ equal parts because our denominator is $2$. Now, the numerator tells us which line in between our fraction is located. Since the numerator is $1$ and there's only one line to begin with, we know this is our place marker for the fraction $\frac12$.

Next, let's try the fraction $\frac23$. It's proper so we know to break up the space between $0$ and $1$ into three equal parts.

Since the numerator is $2$, the second line is where our fraction $\frac23$ lies. The first line is where the fraction $\frac13$ lies.

Now, what about the improper fraction $\frac53$? Well, improper tells us it'll be bigger than $1$. So, we need to space out equal parts of $3$ into the spaces between the numbers greater than $1$. Then, our dot goes on the $5$th line. But make sure you count the line that the whole number is on because that's one of them.

So, the value $\frac53$ is a number between the whole numbers $1$ and $2$. If the fraction is even bigger (for example, $\frac83$) it will still be improper.

A fraction is between two integers on the number line.

Just like with decimals, a fraction appears between two integers on the number line. If the fraction is negative, it will have a negative sign in front. For now though let's just work with positive fractions.

The whole numbers increase as you move to the right on the number line.

With just whole numbers on it, the number line usually looks like this:

The positive fractions appear between whole numbers on the number line:

The negative fractions look just the same as the positive fractions, but with a negative sign in front. The negative fractions will be in between negative integers. Fractions and integers, positive or negative, are all examples of rational numbers.