The length of any interval—whether it is open, closed, or half-open/half-closed—is precisely the difference between its upper and lower bounds. For an interval defined by the real numbers a and b, where b is the upper bound and a is the lower bound, its length is calculated as b – a.
Understanding Interval Length
In mathematics, an interval represents a set of real numbers between two specified endpoints. While the notation for intervals can vary to indicate whether these endpoints are included or excluded, the method for determining the interval's length remains consistent. It fundamentally measures the "span" or "distance" along the number line covered by the interval.
Consider any interval denoted by (a, b), [a, b], [a, b), or (a, b]. In all these cases, 'a' represents the starting point (lower bound) and 'b' represents the ending point (upper bound). The length is determined by simply subtracting the lower bound from the upper bound. This value will always be non-negative, and strictly positive if a is not equal to b.
Types of Intervals and Their Length
Different notations specify how the endpoints are treated:
- Open Interval (a, b): This includes all real numbers x such that a < x < b. The endpoints a and b are not included.
- Closed Interval [a, b]: This includes all real numbers x such that a ≤ x ≤ b. Both endpoints a and b are included.
- Half-Open/Half-Closed Intervals:
- [a, b): Includes all real numbers x such that a ≤ x < b. The lower bound a is included, but b is not.
- (a, b]: Includes all real numbers x such that a < x ≤ b. The upper bound b is included, but a is not.
Despite these distinctions in endpoint inclusion, the length of the interval is solely determined by the numerical values of a and b.
Practical Examples of Interval Length
Let's illustrate with some concrete examples:
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Example 1: Open Interval
- Interval: (3, 7)
- Here, a = 3 and b = 7.
- Length = b - a = 7 - 3 = 4
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Example 2: Closed Interval
- Interval: [-2, 5]
- Here, a = -2 and b = 5.
- Length = b - a = 5 - (-2) = 5 + 2 = 7
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Example 3: Half-Open Interval
- Interval: [0, 10)
- Here, a = 0 and b = 10.
- Length = b - a = 10 - 0 = 10
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Example 4: Half-Closed Interval
- Interval: (-1, 8]
- Here, a = -1 and b = 8.
- Length = b - a = 8 - (-1) = 8 + 1 = 9
These examples clearly demonstrate that the method of calculating length remains consistent across all interval types. The concept of interval length is fundamental in various mathematical fields, including calculus, analysis, and statistics. For more details on interval notation and properties, you can refer to resources on mathematical intervals.
Summary Table
The table below summarizes the calculation of interval length for different notations:
| Interval Type | Notation | Definition | Length Formula | Numerical Example | Calculated Length |
|---|---|---|---|---|---|
| Open | (a, b) | a < x < b | b - a | (1, 6) | 6 - 1 = 5 |
| Closed | [a, b] | a ≤ x ≤ b | b - a | [-3, 4] | 4 - (-3) = 7 |
| Half-Open | [a, b) | a ≤ x < b | b - a | [0, 9) | 9 - 0 = 9 |
| Half-Closed | (a, b] | a < x ≤ b | b - a | (-5, 2] | 2 - (-5) = 7 |
