What Are Rational Numbers?

Before we dive into operations, let's make sure we're clear on what rational numbers actually are.

Rational Number

A rational number is any number that can be expressed as a fraction $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq = 0$ .

Rational numbers include:

Integers: $- 3, 0, 7$ (since $7 = \frac{7}{1}$ )
Fractions: $\frac{2}{5}, - \frac{3}{4}$
Terminating decimals: $0.75 = \frac{3}{4}$
Repeating decimals: $0. \overline{3} = \frac{1}{3}$

Analogy

Think of rational numbers as the "expressible" numbers — any number you can write as one integer divided by another. They're like LEGO bricks: you can always break them down into a ratio of two whole-number pieces.

Warning

Not every number is rational! Numbers like $π$ and $2$ cannot be written as a fraction of two integers — these are called irrational numbers.

Adding and Subtracting Fractions

Adding and subtracting rational numbers in fraction form requires a common denominator.

Common Denominator

A common denominator is a shared multiple of two or more denominators, allowing fractions to be added or subtracted directly.

Steps to add or subtract fractions:

Find the Lowest Common Denominator (LCD) of the fractions.
Rewrite each fraction as an equivalent fraction with the LCD.
Add or subtract the numerators (the denominators stay the same).
Simplify the result if possible.

Example

Calculate $\frac{2}{3} + \frac{3}{5}$

Step 1: LCD of 3 and 5 is 15.

Step 2: $\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}$ and $\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}$

Step 3: $\frac{10}{15} + \frac{9}{15} = \frac{19}{15}$

Step 4: $\frac{19}{15} = 1 \frac{4}{15}$ ✓

Example

Calculate $\frac{5}{6} - \frac{1}{4}$

Step 1: LCD of 6 and 4 is 12.

Step 2: $\frac{5}{6} = \frac{10}{12}$ and $\frac{1}{4} = \frac{3}{12}$

Step 3: $\frac{10}{12} - \frac{3}{12} = \frac{7}{12}$ ✓

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