1: Real numbers and intervals

1: Real Numbers and Intervals

The real number system forms the essential foundation for calculus. It comprises all numbers that can be represented on a continuous number line, encompassing rational numbers (expressible as fractions of integers, like $3/4$ or $-2$) and irrational numbers (which cannot be expressed as simple fractions, like $\sqrt{2}$ or $\pi$). This completeness, meaning there are no "gaps" on the line, is crucial for defining limits, continuity, and calculus concepts rigorously.

Key subsets of real numbers include:

• Natural Numbers ($\mathbb{N}$): Counting numbers ($1, 2, 3, \dots$).
• Integers ($\mathbb{Z}$): Whole numbers ($\dots, -2, -1, 0, 1, 2, \dots$).
• Rational Numbers ($\mathbb{Q}$): Numbers expressible as $p/q$ where $p$ and $q$ are integers and $q \neq 0$ (e.g., $-7$, $0.5$, $2/3$).
• Irrational Numbers: Real numbers not in $\mathbb{Q}$ (e.g., $\sqrt{2}$, $\pi$, $e$).

Intervals are fundamental subsets of the real numbers representing connected segments of the number line. They describe solution sets for inequalities and define domains/ranges of functions. Interval notation provides a concise way to write these sets using parentheses "$()$" for exclusive endpoints and brackets "$[]$" for inclusive endpoints.

• Open Interval $(a, b)$: Includes all $x$ where $a < x < b$. Endpoints $a$ and $b$ are not included. Represents a stretch of the number line without its endpoints.
• Closed Interval $[a, b]$: Includes all $x$ where $a \leq x \leq b$. Endpoints $a$ and $b$ are included.
• Half-Open (or Half-Closed) Intervals:
  • $[a, b)$: $a \leq x < b$ (includes $a$, excludes $b$).
  • $(a, b]$: $a < x \leq b$ (excludes $a$, includes $b$).
• Unbounded Intervals: Extend infinitely in one or both directions.
  • $(a, \infty)$: $x > a$
  • $[a, \infty)$: $x \geq a$
  • $(-\infty, b)$: $x < b$
  • $(-\infty, b]$: $x \leq b$
  • $(-\infty, \infty)$: All real numbers ($\mathbb{R}$).

Connections to Calculus: Understanding intervals is vital immediately. The domain of a function (the set of all valid input values) is often expressed as an interval or union of intervals. When evaluating limits (like $\lim_{x \to a} f(x)$), the behavior of the function depends critically on the values of $x$ within open intervals surrounding the point $a$, excluding $a$ itself. Continuity is defined on intervals. Solving inequalities (e.g., finding where a function is positive or differentiable) results in interval solutions. Mastery of real numbers and interval notation is indispensable for precisely describing the sets and domains central to calculus.