Lesson 2: Adding Rational Numbers

Property

To add the integers $p$ and $q$: begin at zero and draw the line segment (arrow) to $p$.
Starting at the endpoint $p$, draw the line segment representing $q$. Where it ends is the sum $p + q$.
An arrow pointing right is positive, and a negative arrow points left.
Each arrow is a quantity with both length (magnitude) and direction (sign).

Examples

• To calculate $3 + 4$, start at 0, move 3 units right, and then move 4 more units right. You land at 7. So, $3 + 4 = 7$.
• To find $-6 + 4$, start at 0, move 6 units left to $-6$, then move 4 units right. You land at $-2$. So, $-6 + 4 = -2$.
• To compute $-3 + (-5)$, start at 0, move 3 units left to $-3$, then move 5 more units left. You land at $-8$. So, $-3 + (-5) = -8$.

Explanation

Think of adding on a number line as taking a journey. Positive numbers are steps to the right, and negative numbers are steps to the left. Your final position is the sum of the integers.

Property

We can add fractions of the form $a/c$ and $b/c$ by adding the numerators:

Examples

• To add $\frac{2}{5} + \frac{1}{10}$, find a common denominator, which is 10. Rewrite $\frac{2}{5}$ as $\frac{4}{10}$. Now add: $\frac{4}{10} + \frac{1}{10} = \frac{5}{10}$.
• To add $\frac{3}{4} + \frac{2}{3}$, the LCM of 4 and 3 is 12. Rewrite the fractions: $\frac{9}{12} + \frac{8}{12} = \frac{17}{12}$.
• To add $\frac{5}{6} + \frac{3}{8}$, the LCM of 6 and 8 is 24. Rewrite the fractions: $\frac{20}{24} + \frac{9}{24} = \frac{29}{24}$.

Explanation

To add fractions, their denominators must be the same. This means the 'pieces' you're adding are the same size. Find a common multiple for the denominators, convert the fractions, and then just add the numerators.