Rational Numbers

A rational number is any number that can be expressed in the form p/q, where p and q are integers and q is not equal to zero. Rational numbers include integers, fractions, and terminating or repeating decimals.

Examples of Rational Numbers:
3/4, -5/7, 2 (= 2/1), 0 (= 0/1), -3 (= -3/1), 1.5 (= 3/2)

Positive and Negative Rational Numbers:
A rational number p/q is positive if p and q have the same sign (both positive or both negative). It is negative if p and q have opposite signs. Zero is neither positive nor negative.

Standard Form:
A rational number is in standard form when the denominator is positive and the numerator and denominator have no common factor other than 1 (i.e., it is in lowest terms with a positive denominator).

Equivalent Rational Numbers:
a/b = (a x k)/(b x k) for any non-zero integer k. Multiplying or dividing both numerator and denominator by the same non-zero number gives an equivalent rational number.

Comparing Rational Numbers:
To compare, convert to equivalent fractions with the same denominator (LCM of denominators), then compare numerators.

• Operations:
• Addition/Subtraction: Get a common denominator, then add/subtract numerators.
• Multiplication: Multiply numerators together and denominators together: (p/q) x (r/s) = pr/qs.
• Division: Multiply by the reciprocal: (p/q) / (r/s) = (p x s)/(q x r).

• Properties:
• Rational numbers are closed under all four operations (except division by zero).
• Addition and multiplication are commutative and associative.
• 0 is the additive identity, 1 is the multiplicative identity.
• The additive inverse of p/q is -p/q.
• The multiplicative inverse (reciprocal) of p/q is q/p (p not equal to 0).

Rational Numbers on a Number Line:
To represent p/q on the number line, divide the unit segment into q equal parts and count p parts from 0.