Rational Numbers - Equivalent Rational Numbers

A rational number is defined as a number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Visually, if you look at a number line, rational numbers fill the gaps between integers, representing parts of a whole.

Equivalent rational numbers are different representations of the same value. For example, on a number line, the fractions $\frac{1}{2}$, $\frac{2}{4}$, and $\frac{4}{8}$ all point to the exact same spot halfway between $0$ and $1$.

To find an equivalent rational number, you can multiply both the numerator and the denominator by the same non-zero integer. Visually, this is like taking a shaded region of a rectangle and dividing every part into smaller, equal pieces; the total shaded area remains the same.

You can also find equivalent rational numbers by dividing both the numerator and the denominator by their common divisor. This process is often called 'simplifying' or 'reducing' the fraction to its lower terms.

A rational number $\frac{p}{q}$ is in its standard form if the denominator $q$ is a positive integer and the numerator $p$ and denominator $q$ have no common factor other than $1$. In a diagram, this represents the most basic way to express that specific ratio.

Two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ are equal if and only if $a \times d = b \times c$. This is known as the cross-multiplication test for equivalence. If you visualize two proportional rectangles, the product of the extremes equals the product of the means.

Every rational number has an infinite number of equivalent rational numbers. You can keep scaling the numerator and denominator by any integer $m \neq 0$ to generate a new version of the same value.