Ordering and Operating with Rational Numbers

Rational numbers include integers, fractions, and decimals—both positive and negative. Understanding how to place these numbers on a number line makes comparing and ordering them much easier.

The Number Line and Opposites

On a horizontal number line, zero is the center. Positive numbers are to the right of zero, and negative numbers are to the left.

Every rational number has an opposite. Opposites are numbers that are the exact same distance from zero but on opposite sides. For any number $a$, its opposite is $-a$.

• The opposite of $5$ is $-5$.
• The opposite of $-\frac{2}{3}$ is $\frac{2}{3}$.

Comparing and Ordering Rational Numbers

When comparing negative numbers, remember that the number further to the left on the number line is always smaller. For example, $-5$ is less than $-2$ because it is further left.

To order a mix of fractions and decimals, it is usually easiest to convert them all to the same format (like decimals) first.

Example: Order $-2.5$, $-\frac{3}{4}$, $0.5$, and $-1.2$ from least to greatest.

1. Convert to decimals: $-\frac{3}{4} = -0.75$. Now we are comparing $-2.5$, $-0.75$, $0.5$, and $-1.2$.
2. Visualize on the number line:
   • $-2.5$ is the furthest to the left (the smallest).
   • $-1.2$ comes next.
   • $-0.75$ is closer to zero.
   • $0.5$ is positive, so it is the largest.
3. Write in order: $-2.5$, $-1.2$, $-\frac{3}{4}$, $0.5$.

Plotting Rational Numbers

When plotting fractions like $-\frac{3}{2}$ and $\frac{5}{4}$, converting them to mixed numbers or decimals helps you find their exact location between integers.

• Plotting $-\frac{3}{2}$: Since $-\frac{3}{2} = -1.5$, you will place this point exactly halfway between $-1$ and $-2$ on the number line.
• Plotting $\frac{5}{4}$: Since $\frac{5}{4} = 1.25$, you will place this point slightly to the right of $1$, exactly one-quarter of the way toward $2$.