What is the length of the interval (-2,2)
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Answer:
The length of an interval (a,b) (or [a,b], or (a,b], or [a,b)) is defined to be b−a. This is just a definition, so it requires no proof. Intuitively, it should make sense: if you think of the interval as being a "stick" cut out from the number line, then it would be b−a units long. We don't care whether the interval contains its endpoints because a single point has no length.
The following intuition may help. Suppose you have a ruler, with lengths marked in centimeters along it. If you want to measure a length of c centimeters, you would usually just measure from the start of the ruler to the point marked c. However, you could also measure from a point marked a to a point marked b, as long as b−a=c. The reason is that you could just shift the ruler forwards by a centimeters, so the point that was marked a is now at the start of the ruler and the point that was marked b is now marked b−a.
A vast generalization of this notion of "length of an interval" is Lebesgue measure on R, which is a way of defining the "length" of much more complicated sets than just an interval. In the context of Lebesgue measure, depending on your definitions, it may be a theorem that the length of (a,b) is b−a. But in calculus or basic analysis, this is usually just taken as a definition.