Start Learning Numbers 11 | Rational Numbers (Ordering) [dark version]

Rational numbers need a rigorous “less than or equal to” rule before they can be placed in order on the number line. The core task is defining when one fraction should be considered smaller than another—like why 1/4 sits to the left of 1/3—without relying on intuition that can fail once signs and denominators get involved.

The ordering for rational numbers is built directly from the ordering already known for integers. For integers, a ≤ b holds when there exists a nonnegative integer K such that a + K = b, capturing the idea that moving from a to b requires only steps to the right. For fractions, the comparison is reduced to the integer case by clearing denominators: for two fractions A/B and C/D, the inequality is determined by multiplying across so the decision depends on the corresponding integer inequality. This approach matches the “compare ratios” intuition—if 3 ≤ 4, then 1/4 ≤ 1/3—while avoiding contradictions that would arise if denominators could be negative.

Because fractions represent equivalence classes, the definition excludes problematic cases like negative denominators, ensuring the ordering is well-defined across all equivalent ways to write the same rational number. With that in place, the rational numbers inherit an ordering that fits the number line: it is reflexive (every number is ≤ itself), antisymmetric (if x ≤ y and y ≤ x, then x = y), and transitive (if x ≤ y and y ≤ z, then x ≤ z). These properties are essential for consistent comparison.

The ordering also behaves predictably with arithmetic. If x ≤ y, adding the same rational number z to both sides keeps the inequality unchanged—so the order is compatible with addition. Multiplication is similar but with a crucial condition: multiplying both sides by a rational number preserves the inequality only when the multiplier is positive; if the multiplier is negative, the inequality flips direction. The transcript also notes how to read the “flipped” inequality correctly using the standard ≥ / ≤ conventions.

Beyond compatibility, the rational numbers form a total order: any two rationals can always be compared, meaning either x ≤ y or y ≤ x holds. Finally, the Archimedean property rules out both “infinitely large” and “infinitely small” gaps. For any positive rational X and any positive step size ε, repeatedly adding ε to 0 eventually surpasses X, and there is no step size so tiny that it can never accumulate enough. This property matches the number-line picture and prevents pathological order behavior.

With these ordering properties in place—alongside the field structure—rational numbers are ready for the next step: constructing the real numbers by adding one additional property, reserved for later videos. The ordering is the bridge that turns fractions from algebraic objects into a properly ranked continuum foundation for analysis and beyond.