The word "rational" contains a clue to its own meaning: ratio. At its core, a rational number is any number that can be written as a ratio, or fraction, of two integers.

This definition is simple but powerful. We can write it formally like this:

This definition covers more numbers than you might think. Integers, for example, are all rational numbers because any integer can be written as a fraction with a denominator of 1. For instance, the number 7 is rational because it's the same as $7/1$.

Fractions are obviously rational numbers, as that's their very nature. But what about decimals? Both terminating and repeating decimals are also part of the rational family.

• Terminating decimals are numbers like 0.25. This is rational because it can be written as $1/4$.
• Repeating decimals, like $0.333...$, are also rational. This infinitely repeating decimal is just another way of writing $1/3$.

We can visualize rational numbers as specific points on a number line. Integers land on the tick marks, and fractions and decimals fill the spaces in between.

One of the most interesting properties of rational numbers is their density. This means that between any two rational numbers you can pick, no matter how close they are, you can always find another rational number.

For example, what's between $1/2$ and $3/4$? We can find the average of the two numbers: $(\frac{1}{2} + \frac{3}{4}) \div 2 = (\frac{2}{4} + \frac{3}{4}) \div 2 = \frac{5}{4} \div 2 = \frac{5}{8}$. And we could repeat this process forever, always finding a new number in the middle. This is why there are infinitely many rational numbers between any two points on the line.

If a number can't be written as a simple fraction, it's called an irrational number. The decimal form of an irrational number goes on forever without ever repeating. These numbers fill in the remaining gaps on the number line, teaming up with the rational numbers to form the set of all real numbers.

Understanding this distinction is key. While we often work with neat, predictable rational numbers in daily life, the irrationals are always there, representing exact values that can't be perfectly captured by a simple ratio.