How this instrument works
Interval notation describes a run of numbers using its two endpoints and a pair of brackets that say whether those endpoints themselves belong to the set. A closed interval, written [a, b], includes both endpoints — the square brackets act like closed doors that keep the ends inside. An open interval, (a, b), uses parentheses and excludes both endpoints, leaving the doors open at each end. A half-open interval mixes the two styles, such as [a, b) or (a, b], including one end while excluding the other.
None of that bracket choice changes two plain facts about the stretch sitting between a and b: how long it is, and where its middle lies. Width is simply the right endpoint minus the left one, and midpoint is their average — figures that hold whether the interval is written closed, open, or half-open, since brackets describe which points belong to the set, not how far apart the endpoints sit. This calculator takes a and b and returns both numbers directly, without needing to know which bracket style you had in mind.
A width of zero is a useful edge case to recognize: when the two endpoints match, the 'interval' collapses to a single point rather than a genuine stretch of the line, and its midpoint is simply that point. Spotting this case matters in practice — a range reporting zero width is telling you there is no room left between its ends at all.
- Enter the smaller endpoint into Left endpoint (a) and the larger one into Right endpoint (b).
- Read Interval width for the plain distance between the two endpoints, b minus a.
- Read Interval midpoint for the single point exactly halfway between a and b.
- A width of zero signals that a and b are equal — a point rather than a genuine interval.
- Pair these two figures with whichever bracket style, closed, open, or half-open, actually fits your problem.
Worked example — the interval [2, 10]
Take the interval [2, 10], a closed interval running from 2 to 10 inclusive. Its width is b − a = 10 − 2 = 8, the plain distance covered from one end to the other. Its midpoint is (a + b) ⁄ 2 = (2 + 10) ⁄ 2 = 12 ⁄ 2 = 6, the single point sitting exactly halfway between 2 and 10, four units in from each side.
The same two formulas apply to any pair of endpoints. Over [−5, 5], width comes out to 5 − (−5) = 10 and midpoint lands at (−5 + 5) ⁄ 2 = 0, a symmetric interval centered on the origin. Push the endpoints together entirely, as in [3, 3], and width drops to 3 − 3 = 0 while midpoint stays at 3 — a single point rather than a genuine stretch of the line.
Questions
What's the difference between a closed and an open interval?
A closed interval, written with square brackets like [a, b], includes both endpoints in the set. An open interval, written (a, b), excludes both — values arbitrarily close to a and b belong, but a and b themselves don't. A half-open interval, [a, b) or (a, b], includes just one end and excludes the other.
How is the width of an interval calculated?
Subtract the left endpoint from the right one: width = b − a. For [2, 10] that comes to 10 − 2 = 8. The bracket style never enters the calculation — width measures the distance between the endpoints, not which of them technically belong to the set.
How do I find the midpoint of an interval?
Average the two endpoints: midpoint = (a + b) ⁄ 2. For [2, 10] that's (2 + 10) ⁄ 2 = 6, the point sitting exactly halfway between the ends. A symmetric interval like [−5, 5] has its midpoint at 0, right on the origin.
What does it mean if the width comes out to zero?
It means the two endpoints are equal, so the 'interval' is really a single point rather than a genuine range — [3, 3], for instance, contains only the number 3. The midpoint formula still works in that case; it simply returns that one point itself.
Can an interval stretch out to infinity?
Yes, one or both ends can be unbounded, written as (−∞, b] or [a, ∞), and infinity is always excluded with a parenthesis since it isn't an actual number the interval can include. This calculator is built for two finite endpoints, so an unbounded interval needs only its finite side entered.
Does the order I enter a and b in matter?
It should follow a ≤ b for the result to describe a genuine interval; entering the larger figure as a and the smaller as b would produce a negative width, which no longer matches how intervals are normally read. Keep the smaller endpoint in a and the larger in b for a sensible answer.