Understanding Rational Numbers

A rational number is any number that can be written as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. This 'family' is much larger than just whole numbers! It includes positive and negative integers, fractions, and decimals. The word 'rational' actually comes from the word ratio. For example, the integer $7$ is rational because it can be written as $\frac{7}{1}$. Even a decimal like $0.25$ is rational because it is equivalent to the ratio $\frac{1}{4}$. If you can write it as a simple fraction, it belongs in this group!

We use number lines to visualize where these numbers live. On a horizontal line, positive numbers are to the right of zero, and negative numbers are to the left. On a vertical line, like a thermometer, positive is 'up' and negative is 'down.' Rational numbers allow us to describe the points between the integers. To find $-\frac{1}{2}$, you look exactly halfway between $0$ and $-1$. The further a negative number is from zero, the 'smaller' its value is, even if the digit looks big!

Every rational number has a decimal 'secret identity.' To find it, we treat the fraction bar as a division symbol. To convert $\frac{p}{q}$, you simply calculate $p \div q$. This results in two types of decimals: terminating decimals, which end (like $0.5$), or repeating decimals, which go on forever in a pattern (like $0.333...$). For repeating decimals, we use bar notation—a small line drawn over the digits that repeat—to save time and space.