Interval

Intervals are one the most frequently used terms in mathematics. An interval is just a special case of a simple set.

We can describe an interval using plain English “all numbers between zero and ten”. Clearly, the numbers like two, seven or eight are included in the interval. It’s hard to tell if the numbers zero and ten are included in the interval. Maybe they are, maybe they aren’t. And what about the number π? Let’s define an interval properly, shall we?

  • The open interval (a, b) consists of all real numbers that are greater than a and lesser than b. I. e. the numbers a and b are not part of the interval (a, b).
  • The closed interval [a, b] consists of all real numbers that are greater than a or equal to a and lesser than b or equal to b. I. e. the numbers a and b are part of the interval [a, b].

We can also mix those two types of intervals. E. g. this interval [0, 1) consists of all real numbers greater than zero or equal to zero and lesser than one. I. e. the number zero is part of the interval and the number one is not. Such interval is called either half-closed or half-open interval.

  • The “[“ and “]” symbols means closed interval: the number corresponding to this symbol is part of the interval.
  • The “(“ and “)” symbols means open interval: the number corresponding to this symbol is not part of the interval.

We can expresses intervals in graphs. Some examples:

  • The closed interval [ − 1, 6]. The points A = − 1 and B = 6 are part of the interval.

    Closed interval from -1 to 6
  • The open interval ( − 1, 6). The points A = − 1 and B = 6 are not part of the interval.

    Open interval from -1 to 6

    You can see that the graph is the same except for the points A and B. We draw them differently.

  • The half-closed interval [ − 1, 6). The point A = − 1 is part of the interval, the point B = 6 is not.

    Half closed interval from -1 to 6
  • The half-closed interval ( − 1, 6]. The point A = − 1 is not part of the interval, the point B = 6 is.

    Half closed interval from -1 to 6

To The Infinity and Beyond

How can we express the set of all positive real numbers with an interval? The left bound is clear (0, ?), but what about the right one? We use the infinity symbol instead of a real number. Thus the interval of all positive real numbers is (0, ∞). We also use the open interval instead of closed interval. We can represent the whole set of real numbers as the interval ( − ∞, ∞).