Rational Numbers - Operations and Word Problems

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 include all fractions, terminating decimals, and repeating decimals, appearing as distinct points between integers on a number line.

Addition and Subtraction of rational numbers require a common denominator. If denominators are different, find the Least Common Multiple (LCM). Visually, this process is like dividing different-sized segments on a number line into smaller, equal-sized units so they can be combined or compared accurately.

The product of two rational numbers is found by multiplying their numerators and their denominators separately: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$. Simplification can be done by 'cross-canceling' common factors between any numerator and any denominator before multiplying.

Division by a rational number is the same as multiplying by its reciprocal. The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$ (where $a, b \neq 0$). On a number line, dividing a length into $n$ parts is the same as multiplying that length by the unit fraction $\frac{1}{n}$.

Additive and Multiplicative Identities: Zero ($0$) is the additive identity because adding it to any rational number does not change its value ($x + 0 = x$). One ($1$) is the multiplicative identity because multiplying any rational number by it leaves the value unchanged ($x \times 1 = x$).

Additive Inverse and Multiplicative Inverse: For every rational number $\frac{a}{b}$, there exists an additive inverse $-\frac{a}{b}$ such that their sum is $0$. The multiplicative inverse (or reciprocal) of $\frac{a}{b}$ is $\frac{b}{a}$, such that their product is $1$.

The Density Property states that between any two rational numbers, there are infinitely many other rational numbers. To find a rational number exactly halfway between two numbers $x$ and $y$, use the formula $\frac{x+y}{2}$. This shows that the number line is 'dense' with rational points.

Distributive Property: For any three rational numbers $a, b,$ and $c$, the relationship $a(b + c) = ab + ac$ holds true. This property is vital for simplifying complex algebraic expressions and solving word problems involving shares or parts.