Interval

Pronunciation: /ˈɪn.tər.vəl/ Explain

An interval is an unbroken range of numbers. An interval may have zero, one or two endpoints. The interval may or may not include the endpoints. An interval that includes its endpoints is called a closed interval. The interval -1 ≤ x ≤ 3 is a closed interval. An interval that does not include its endpoints, or that does not have endpoints, is called an open interval. The interval 1 < r < 3 is an open interval. If one endpoint of an interval is included, and the other endpoint is not or the interval has only one endpoint, the interval is called a half-open interval. The intervals -10 < t ≤ -5 and 2 ≤ g < 6.2 are half-open intervals.

If an interval includes an endpoint, the interval is said to be inclusive of the endpoint. If an interval does not include an endpoint, the interval is said to be exclusive of the endpoint. The interval 1 < a ≤ 9 is exclusive of 1 and inclusive of 9.

Intervals can be represented in several different ways:

• A number line can be used to represent an interval. If an endpoint is included in the interval, use a solid dot. If an endpoint is not included in the interval, use a hollow dot. A line, ray or line segment represents the rest of the interval.
  
• Inequality: a ≤ x ≤ b where '≤' represents a closed interval and '<' represents an open interval.
• Interval notation: (a,b), (a, b], [a, b) or [a, b] where '(' represents an interval open on the left, '[' represents an interval closed on the left, ')' represents an interval open on the right, and ']' represents an interval closed on the right. If an interval is unbounded in a direction, use the infinity symbol (∞). If an interval is unbounded in a direction, it is open in that direction.
• Set notation: {x ∈ ℝ | 3 < x < 7} where '≤' represents a closed interval and '<' represents an open interval.
• ISO 31-11 notation: ]3,7[ where a square bracket pointing away from the number represents an open interval and a square bracket pointing towards the number represents a closed interval.

Number Line
Inequality	Interval Notation	Set Notation	ISO 31-11 Notation	Description
3 < x < 7	(3, 7)	{x ∈ ℝ | 3<x<7 }	]3, 7[	This is an open interval between 3 and 7. The interval is exclusive of 3 and 7.
-3 ≤ x < 4	[-3, 4)	{x ∈ ℝ | -3 ≤ x < 4}	[-3, 4[	This is a half-open interval between -3 and 4. The interval includes -3 and excludes 4.
-5 < x ≤ -1	(-5, -1]	{x ∈ ℝ | -5 < x ≤ -1 }	]-5, -1]	This is a half-open interval between -5 and -1. The interval is exclusive of -5 and inclusive of -1.
2 ≤ x ≤ 8	[2, 8]	{x ∈ ℝ | 2 ≤ x ≤ 8 }	[2, 8]	This is a closed interval between 2 and 8. The interval includes 2 and 8.
-1 < x	(-1, ∞)	{x ∈ ℝ | -1 < x }	]-1, ∞[	This is an open interval from -1 to positive infinity. The interval does not include -1.
x < 3	(-∞, 3]	{x ∈ ℝ | x < 3 }	]-∞, 3]	This is a half-open interval from negative infinity to 3. The interval includes 3.
not applicable	(-∞, ∞)	{x | x ∈ ℝ }	]-∞, ∞[	This is the open interval of all real numbers.
Table 1: Representing intervals

Click on the blue points and drag them to change the figure. Click on the check boxes to change the figure. Which interval notation do you prefer? Why?

Manipulative 2 - Interval Notation Created with GeoGebra.

References

1. McAdams, David E.. All Math Words Dictionary, interval. 2nd Classroom edition 20150108-4799968. pg 101. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. Kuratowski,Kazimierz. Introduction to Calculus. U.S.A. edition. pg 21. Translated by Msielak, Julian. www.archive.org. Addison-Wesley Publishing Company Inc.. 1962. Last Accessed 8/7/2018. http://www.archive.org/stream/introductiontoca033502mbp#page/n26/mode/1up/search/open. Buy the book
3. Narayan, Shanti. Differential Calculus. 10th edition. pg 6. www.archive.org. S. Chand & Co.. 1962. Last Accessed 8/7/2018. http://www.archive.org/stream/differentialcalc031624mbp#page/n23/mode/1up/search/open. Buy the book
4. Chinn, William G. and Steenrod, N. E.. First Concepts of Topology. pg 8. Mathematical Assn of America. June 1975. Last Accessed 8/7/2018. Buy the book

Cite this article as:

McAdams, David E. Interval. 4/23/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/i/interval.html.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Revision History

4/23/2019: Updated equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
8/6/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
9/30/2010: Expanded text and examples on intervals to positive and negative infinity. (McAdams, David E.)
1/7/2009: Sprinkled examples through out. Added words half-open, inclusive, and exclusive to first paragraph (McAdams, David E.)
12/1/2008: Initial version. (McAdams, David E.)