Rational Numbers - Comparison of Rational Numbers

Definition and Standard Form: A rational number is expressed in the form $\frac{p}{q}$ where $q \neq 0$. To compare two rational numbers, first ensure they are in standard form, meaning the denominator $q$ is a positive integer. If the denominator is negative, multiply both the numerator and denominator by $-1$.

Number Line Representation: On a horizontal number line, zero acts as the origin. Every positive rational number lies to the right of zero, and every negative rational number lies to the left of zero. A rational number $x$ is considered greater than $y$ if $x$ is positioned to the right of $y$ on this line.

Positive and Negative Comparisons: Every positive rational number is greater than zero and greater than any negative rational number. Conversely, zero is always greater than any negative rational number. For example, $\frac{1}{2} > 0 > -\frac{3}{4}$.

Comparing Rational Numbers with Same Denominators: If two rational numbers have the same positive denominator, the number with the larger numerator is the greater rational number. For instance, in the pair $\frac{5}{8}$ and $\frac{3}{8}$, since $5 > 3$, it follows that $\frac{5}{8} > \frac{3}{8}$.

Comparing Rational Numbers with Different Denominators (LCM Method): To compare rational numbers with different denominators, find the Least Common Multiple (LCM) of the denominators. Convert both fractions into equivalent rational numbers having this LCM as their common denominator, then compare their numerators.

Cross-Multiplication Method: For two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ where $b, d > 0$, you can compare the products $a \times d$ and $b \times c$. If $ad > bc$, then $\frac{a}{b} > \frac{c}{d}$. If $ad < bc$, then $\frac{a}{b} < \frac{c}{d}$.

Comparison of Negative Rational Numbers: When comparing two negative rational numbers, the one with the smaller absolute value is actually the greater number. On a number line, $-\frac{1}{5}$ is to the right of $-\frac{4}{5}$, therefore $-\frac{1}{5} > -\frac{4}{5}$.