Learning Objectives
- Understand the properties of rational numbers
- Find rational numbers between two given rational numbers
- Represent rational numbers on the number line
- Apply closure, commutative, associative, and distributive properties
Key Concepts
What are Rational Numbers?
A number that can be expressed in the form p/q, where p and q are integers and q ≠ 0, is called a rational number. Examples: 1/2, -3/4, 5, 0, -7.
Properties of Rational Numbers
Closure Property: Rational numbers are closed under addition, subtraction, and multiplication. This means if a and b are rational numbers, then a + b, a - b, and a × b are also rational numbers. Rational numbers are NOT closed under division (division by zero is not defined).
Commutative Property: For rational numbers a and b: a + b = b + a (addition) and a × b = b × a (multiplication). Subtraction and division are NOT commutative.
Associative Property: For rational numbers a, b, and c: (a + b) + c = a + (b + c) for addition, and (a × b) × c = a × (b × c) for multiplication. Subtraction and division are NOT associative.
Distributive Property: a × (b + c) = a × b + a × c. Multiplication distributes over addition and subtraction.
Additive and Multiplicative Identities and Inverses
Additive Identity: 0 is the additive identity. a + 0 = 0 + a = a for every rational number a.
Multiplicative Identity: 1 is the multiplicative identity. a × 1 = 1 × a = a for every rational number a.
Additive Inverse: For a rational number a/b, the additive inverse is -a/b, such that a/b + (-a/b) = 0.
Multiplicative Inverse (Reciprocal): For a non-zero rational number a/b, the reciprocal is b/a, such that (a/b) × (b/a) = 1.
Rational Numbers Between Two Rational Numbers
There are infinitely many rational numbers between any two given rational numbers. To find rational numbers between two rationals, you can find their average (mean) or convert them to equivalent fractions with a larger denominator.
Summary
Rational numbers form the set of all numbers expressible as p/q where q ≠ 0. They satisfy closure, commutativity, and associativity for addition and multiplication. Zero is the additive identity, and one is the multiplicative identity. Between any two rational numbers, there exist infinitely many rational numbers.
Important Terms
- Rational Number: A number of the form p/q where p, q are integers and q ≠ 0
- Additive Identity: Zero (0), adding it to any number gives the same number
- Multiplicative Identity: One (1), multiplying any number by it gives the same number
- Additive Inverse: The number which when added gives zero
- Reciprocal: The multiplicative inverse; for a/b it is b/a
- Closure Property: The result of an operation on two numbers in a set also belongs to that set
Quick Revision
- Rational numbers are closed under +, -, × but not ÷ (division by zero undefined)
- Commutative for + and ×; NOT for - and ÷
- Associative for + and ×; NOT for - and ÷
- 0 is additive identity; 1 is multiplicative identity
- Infinite rational numbers exist between any two rational numbers
- Distributive: a × (b + c) = ab + ac