Key Points

A number is a rational number if it can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and the denominator $q \neq 0$. Examples include $\frac{2}{3}$, $-5$ (which is $\frac{-5}{1}$), and $0$ (which is $\frac{0}{1}$).

Rational numbers are closed under addition, subtraction, and multiplication. If $a$ and $b$ are rational, then $a+b$, $a-b$, and $a \times b$ are also rational. They are not closed under division because division by zero is undefined.

Addition and multiplication are commutative for rational numbers. For any rational numbers $a$ and $b$, $a+b = b+a$ and $a \times b = b \times a$. Subtraction and division are not commutative.

Addition and multiplication are associative for rational numbers. For any rational numbers $a, b, c$, we have $(a+b)+c = a+(b+c)$ and $(a \times b) \times c = a \times (b \times c)$. Subtraction and division are not associative.

The number zero (0) is the additive identity for rational numbers. For any rational number $a$, adding zero does not change its value: $a + 0 = 0 + a = a$.

The number one (1) is the multiplicative identity for rational numbers. For any rational number $a$, multiplying by one does not change its value: $a \times 1 = 1 \times a = a$.

For every rational number $a$, there exists an additive inverse $-a$ such that their sum is zero: $a + (-a) = 0$. The additive inverse of $\frac{p}{q}$ is $-\frac{p}{q}$.

For every non-zero rational number $a = \frac{p}{q}$, there exists a multiplicative inverse (or reciprocal) $\frac{1}{a} = \frac{q}{p}$ such that their product is one: $a \times \frac{1}{a} = 1$. Zero has no reciprocal.

Multiplication distributes over addition for rational numbers. For any rational numbers $a, b, c$, the property is $a \times (b+c) = (a \times b) + (a \times c)$. This also applies to subtraction: $a \times (b-c) = (a \times b) - (a \times c)$.

Between any two distinct rational numbers, there are infinitely many other rational numbers. This is known as the density property of rational numbers.

A simple way to find a rational number between two given rational numbers $a$ and $b$ is to calculate their mean. The number $\frac{a+b}{2}$ will always lie between $a$ and $b$.