Rational Numbers: Definition, Types & Standard Form

• A rational number can be written as p/q, where p and q are integers, and q ≠ 0.
• Integers qualify as rational numbers because they can be expressed as a number divided by 1 (e.g., 5 = 5/1).
• A rational number is positive if its numerator (p) and denominator (q) are either both positive or both negative.
• A rational number is negative if either its numerator (p) or its denominator (q) is negative.
• Equivalent rational numbers possess the same value but have different numerators and denominators (e.g., 1/2 = 2/4 = 3/6).
• To derive equivalent rational numbers, multiply or divide the numerator and denominator of the original fraction by the same non-zero integer.
• A rational number is in its simplest form when the numerator and denominator are co-prime, meaning they share no common factors other than 1.
• To reduce a rational number to its simplest form, divide both the numerator and denominator by their highest common factor (HCF).
• The standard form of a rational number includes a positive denominator, and the numerator and denominator share no common factors other than 1.

Addition of Rational Numbers

• When adding rational numbers with the same denominator, add the numerators while keeping the denominator constant: (p/q) + (r/q) = (p+r)/q.
• For rational numbers with different denominators, find a common denominator, preferably the least common multiple (LCM). Convert the rational numbers to equivalent forms using this common denominator, and then add the numerators.

Subtraction of Rational Numbers

• To subtract rational numbers sharing a common denominator, subtract the numerators and keep the denominator the same: (p/q) - (r/q) = (p-r)/q.
• In subtracting rational numbers with different denominators, identify a common denominator (LCM), change the fractions into equivalent forms with this denominator, then perform the subtraction on the numerators.

Multiplication of Rational Numbers

• Multiplying rational numbers involves multiplying their numerators and their denominators: (p/q) × (r/s) = (p×r)/(q×s).
• Reduce the result to its simplest form by canceling common factors in the numerator and denominator.

Division of Rational Numbers

• Dividing one rational number by another requires multiplying the first rational number by the reciprocal of the second: (p/q) ÷ (r/s) = (p/q) × (s/r) = (p×s)/(q×r).
• The reciprocal of a rational number p/q is q/p.

Properties of Rational Numbers

• Closure Property: Sums, differences, and products of rational numbers remain rational. Division also results in a rational number unless dividing by zero.
• Commutative Property: The order of addition or multiplication does not change the outcome: a + b = b + a and a × b = b × a.
• Associative Property: Re-grouping rational numbers in addition or multiplication does not affect the result: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
• Distributive Property: Multiplication distributes over addition: a × (b + c) = (a × b) + (a × c).
• Identity Property:
  • Additive Identity: Adding 0 to a rational number does not alter its value: a + 0 = a.
  • Multiplicative Identity: Multiplying a rational number by 1 does not alter its value: a × 1 = a.
• Additive Inverse: For any rational number a, there exists -a such that a + (-a) = 0. -a is the additive inverse of a.
• Multiplicative Inverse (Reciprocal): For any non-zero rational number a, there exists 1/a such that a × (1/a) = 1. 1/a is the multiplicative inverse, or reciprocal, of a.

Rational Numbers on the Number Line

• Rational numbers have a position on the number line.
• To place p/q on the number line, divide the unit length into q equal segments and count p segments from 0.
• Positive rational numbers lie to the right of 0, while negative rational numbers are to the left of 0.

Rational Numbers Between Two Rational Numbers

• Infinitely many rational numbers exist between any two distinct rational numbers.
• To locate rational numbers between two given rational numbers, establish a common denominator, then identify rational numbers with numerators falling between the numerators of the two given numbers.

Comparing Rational Numbers

• When rational numbers share a positive denominator, the number with the larger numerator is greater.
• If the rational numbers have different denominators, find a common denominator before comparing the numerators.
• A positive rational number is always greater than a negative rational number.

Understand rational numbers, their positive and negative forms, and equivalent representations. Learn to simplify rational numbers to their standard form by finding the highest common factor. Explore how integers relate to rational numbers.