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Math Lesson 2.2.2 - Intervals and Segments
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Welcome to our Math lesson on Intervals and Segments, this is the second lesson of our suite of math lessons covering the topic of Upper and Lower Bounds. Intervals and Segments, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Intervals and Segments
An Interval is a set of numbers in which all values are included except the bounds. It is expressed through the curved brackets, (). For example, if we see the following written somewhere (3, 8), we read "the interval that includes all values between 3 and 8 without these two bounds."
A Segment on the other hand, represents a set of numbers that besides the in-between values includes the two bounds as well. A segment is expressed through the square brackets [ ]. Thus, if we see written [4, 11], we read "the segment that includes all values between 4 and 11 including the bounds."
Intervals are shown in the number line by white (empty) dots, while segments by black (filled) dots. Look at the figure below.
We say the ends of an interval are open, while those of a segment are closed.
On the other hand, when the lower bound (the left end) is open and the upper bound (the right end) if closed, we have a half-interval. Likewise, when the lower bond (the left end) is closed and the upper bound (the right end) is open, we have a half-segment. As you see the first (left) bound gives the name to the structure. Look at the figure below.
Example 6
Express the number sets (-2, 7], [1, 4], (0, 5) and [1, 8) in the number line.
Solution 6
(-2, 7] represents a half-interval. It is open at the left and closed at the right end. It is shown in the number line as
[1, 4] is a segment. It is closed at both sides and is shown in the number line as
(0, 5) is an interval. It is open at both ends and is represented in the number line as follows
[1, 8) is a half-segment. It is closed at the left end and open at the right one.
When one end of a number set extends to infinity, we represent that part using the symbol of interval, as it is impossible to find the exact value of infinity. For example, the set that extends from 4 to infinity (including 4), is a half segment that is symbolically written as [4, +∞), where ∞ is the symbol of infinity. When shown in the number line, the part that goes towards infinity is open, as shown in the figure.
The same thing occurs when the interval points toward negative infinity. In this case, the arrow points toward left. For example, if we see written (-∞, 6], we read "the set of numbers extending from minus infinity to 6, including the number 6" or simply "the half-interval from minus infinity to 6".
Example 7
Show the following number sets in the number line.
- [-3, +∞)
- (-∞, 2)
- (-∞, +∞)
Solution 7
- We have a half-segment that starts from -3 and extends on the right towards positive infinity. Hence, when this set is shown in the number line, we get
- We have an interval that starts from negative infinity and extends to the right up to the number 2, without including this number. Hence, when this set is shown on the number line, we get
- This set extends to infinity on both sides, so it is indicated by a double-sided arrow in the number line.
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