Overview
The lecture covers the classification of real numbers and explains interval notation for expressing ranges of real values.
Types of Numbers
- Natural numbers (N) start at 1 and increase by 1 (1, 2, 3, ...).
- Whole numbers include all natural numbers and 0.
- Integers (Z) include natural numbers, their negatives, and 0 (..., -2, -1, 0, 1, 2, ...).
- Rational numbers are any numbers that can be written as fractions of integers, including repeating or terminating decimals.
- Irrational numbers cannot be written as fractions; examples are √2 and π (non-terminating, non-repeating decimals).
- Real numbers (R) comprise both rational and irrational numbers.
Interval Notation
- Interval notation is a shorthand for writing ranges of real numbers using endpoints.
- A closed interval [a, b] includes both endpoints: a ≤ x ≤ b.
- An open interval (a, b) excludes both endpoints: a < x < b.
- Mixed intervals [a, b) or (a, b] include only one endpoint: a ≤ x < b or a < x ≤ b, respectively.
- For infinite intervals, use infinity symbols: [a, ∞), (−∞, b], etc.
- Always use parentheses with ±∞ since infinity is not a real number.
- Open circles on number lines represent excluded endpoints (parentheses); closed circles represent included endpoints (brackets).
Examples of Interval Notation
- (−2, 0] means −2 < x ≤ 0 (open at −2, closed at 0).
- (3, ∞) means x > 3 (open at 3, never include ∞).
Key Terms & Definitions
- Natural numbers (N) — counting numbers starting from 1.
- Whole numbers — natural numbers plus 0.
- Integers (Z) — positive and negative whole numbers, including 0.
- Rational numbers — numbers expressible as fractions of integers.
- Irrational numbers — numbers not expressible as fractions (non-repeating, non-terminating decimals).
- Interval notation — shorthand to describe a set of numbers between two endpoints.
- Closed interval [a, b] — includes both endpoints.
- Open interval (a, b) — excludes both endpoints.
Action Items / Next Steps
- Practice writing sets of numbers in interval notation using examples from homework or textbook exercises.