Advanced Analysis of Interval Notation
In real analysis and calculus, an interval is a subset of the real numbers $\mathbb{R}$ containing all numbers between any two numbers of the set. Beyond mere notation, intervals possess geometric properties including width, midpoint, and radius.
Geometric Characteristics of Bounded Intervals
For any bounded interval with lower endpoint $a$ and upper endpoint $b$:
- Length (Width):
L = b - a
- Midpoint (Center $m$):
m = \frac{a + b}{2}
- Radius ($r$):
r = \frac{b - a}{2} = \frac{L}{2}
The Absolute Value Connection
A symmetric interval $[a, b]$ can be expressed as a neighborhood around its center:
|x - m| \le r
This equivalence is crucial for epsilon-delta proofs in calculus and tolerance engineering, where a nominal dimension $m \pm r$ defines accepted production boundaries.
Comparison Table: Types of Intervals on $\mathbb{R}$
| Interval Type | Notation | Midpoint | Radius | Compact in $\mathbb{R}$? |
|---|---|---|---|---|
| Closed | $[a, b]$ | $(a + b)/2$ | $(b - a)/2$ | Yes (Heine-Borel) |
| Open | $(a, b)$ | $(a + b)/2$ | $(b - a)/2$ | No |
| Left-Closed | $[a, b)$ | $(a + b)/2$ | $(b - a)/2$ | No |
| Right-Closed | $(a, b]$ | $(a + b)/2$ | $(b - a)/2$ | No |
| Rays | $(a, \infty)$ or $(-\infty, b]$ | Undefined | Infinite | No |