Rational Numbers - Properties of Rational Numbers

Definition of Rational Numbers: A rational number is any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Visually, rational numbers can be represented as distinct points on a number line, filling the spaces between integers to represent parts of a whole.

Closure Property: Rational numbers are closed under addition, subtraction, and multiplication, meaning the result of these operations on two rational numbers is always another rational number. However, they are not closed under division because division by zero is undefined and does not result in a rational number.

Commutative Property: For any two rational numbers $a$ and $b$, addition and multiplication are commutative ($a + b = b + a$ and $a \times b = b \times a$). Subtraction and division do not follow this property, as changing the order of numbers changes the result.

Associative Property: Addition and multiplication are associative for rational numbers, meaning the grouping of three or more numbers does not change the sum or product: $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$. Like commutativity, this does not apply to subtraction or division.

Distributive Property: The distributive property of multiplication over addition and subtraction states that $a(b + c) = ab + ac$ and $a(b - c) = ab - ac$. Visually, this can be understood by calculating the area of a large rectangle by splitting it into two smaller adjacent rectangles.

Identity Elements: Zero ($0$) is the additive identity because adding it to any rational number results in the same number ($a + 0 = a$). One ($1$) is the multiplicative identity because multiplying any rational number by it results in the original number ($a \times 1 = a$).

Inverse Elements: Every rational number $\frac{a}{b}$ has an additive inverse $-\frac{a}{b}$, which is its reflection across zero on the number line ($a + (-a) = 0$). Every non-zero rational number $\frac{a}{b}$ has a multiplicative inverse (reciprocal) $\frac{b}{a}$, such that their product is $1$.

Density Property: Rational numbers are 'dense', meaning that between any two rational numbers, there exist infinitely many other rational numbers. Visually, no matter how much you zoom into a segment of the number line, you can always find more points (fractions) between any two given points.