Rational Numbers: Properties and Operations

• Rational numbers are numbers that can be expressed in the form p/q, where p and q are integers and q is not equal to zero.

Properties of Rational Numbers

• Closure Property:

  • Rational numbers are closed under addition, subtraction, and multiplication; the sum, difference, and product of two rational numbers is always a rational number.
  • Rational numbers are not closed under division because division by zero is not defined.

• Commutative Property:

  • Addition and multiplication are commutative for rational numbers, the order in which rational numbers are added or multiplied does not affect the result (a + b = b + a and a * b = b * a).
  • Subtraction and division are not commutative for rational numbers.

• Associative Property:

  • Addition and multiplication are associative for rational numbers, the grouping of rational numbers when adding or multiplying does not affect the result ((a + b) + c = a + (b + c) and (a * b) * c = a * (b * c)).
  • Subtraction and division are not associative for rational numbers.

• Identity Property:

  • 0 is the additive identity for rational numbers (a + 0 = a).
  • 1 is the multiplicative identity for rational numbers (a * 1 = a).

• Inverse Property:

  • Every rational number a has an additive inverse -a, such that a + (-a) = 0.
  • Every non-zero rational number a has a multiplicative inverse 1/a, such that a * (1/a) = 1.

• Distributive Property:

  • Multiplication is distributive over addition for rational numbers: a * (b + c) = a * b + a * c.

Representation of Rational Numbers on a Number Line

• Rational numbers can be represented on a number line.
• Representing a rational number p/q involves dividing the unit length into q equal parts and marking the pth part from zero.

Rational Numbers Between Two Rational Numbers

• An infinite number of rational numbers exist between any two distinct rational numbers.
• The average (mean) of two numbers can be used to discover the rational numbers between them.
  • If a and b are two rational numbers, then (a + b) / 2 is a rational number between them.

Standard Form of a Rational Number

• A rational number p/q is in standard form if q is positive and p and q are co-prime (i.e., their greatest common divisor is 1).

Operations on Rational Numbers

• Addition:

  • When adding rational numbers with the same denominator, add the numerators and keep the denominator the same ((a/c) + (b/c) = (a + b) / c).
  • When adding rational numbers with different denominators, find a common denominator, convert each fraction to an equivalent fraction with the common denominator, and then add the numerators.

• Subtraction:

  • When subtracting rational numbers with the same denominator, subtract the numerators and keep the denominator the same ((a/c) - (b/c) = (a - b) / c).
  • When subtracting rational numbers with different denominators, find a common denominator, convert each fraction to an equivalent fraction with the common denominator, and then subtract the numerators.

• Multiplication:

  • To multiply rational numbers, multiply the numerators and multiply the denominators ((a/b) * (c/d) = (a * c) / (b * d)).

• Division:

  • To divide rational numbers, multiply by the reciprocal of the divisor ((a/b) ÷ (c/d) = (a/b) * (d/c) = (a * d) / (b * c), where c ≠ 0).