For example,
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\([3,7]\) means all numbers between 3 and 7, including both 3 and 7.
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\((3,7)\) means all numbers between 3 and 7, but not including 3 nor 7.
Section 10.1 Interval Notation
In mathematics, we often work with ranges of numbers. A interval is a range of numbers between two points.
Mathematicians developed a notation to write intervals of numbers concisely and clearly. An interval is made up of:
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Two numbers, that represent the start and end of the interval.
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Round brackets or square brackets, that indicate if the numbers at the start or end are included or not.
Example 10.1.1.
The type of bracket tells you if the endpoint is included or not:
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Square brackets indicate the endpoint is included.
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Round brackets (or parentheses) indicate the endpoint is not included (i.e. is excluded).
Example 10.1.2.
For example,
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\([-2,4)\) means all the numbers from \(-2\) to 4, including \(-2\) but not including 4.
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\((-2,4]\) means all the numbers from \(-2\) to 4, not including \(-2\) but including 4.
Some intervals have no maximum value, no minimum value, or both. For an interval which has no endpoint, we say that it extends to infinity, and use \(\infty\) as the endpoint. We use round brackets for \(\infty\) or \(-\infty\text{,}\) since \(\infty\) is not a real number, so they are not numbers we can include or “reach” in our interval.
Subsection 10.1.1 Summary of Interval Notation
In summary,
Definition 10.1.4.
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\(\displaystyle [a,b] = \set{x \in \mathbb{R} : a \leq x \leq b}\)
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\(\displaystyle (a,b) = \set{x \in \mathbb{R} : a \lt x \lt b}\)
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\(\displaystyle (a,b] = \set{x \in \mathbb{R} : a \lt x \leq b}\)
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\(\displaystyle [a,b) = \set{x \in \mathbb{R} : a \leq x \lt b}\)
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\(\displaystyle (a,\infty) = \set{x \in \mathbb{R} : x \gt a}\)
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\(\displaystyle [a,\infty) = \set{x \in \mathbb{R} : x \geq a}\)
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\(\displaystyle (-\infty,b) = \set{x \in \mathbb{R} : x \lt b}\)
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\(\displaystyle (-\infty,b] = \set{x \in \mathbb{R} : x \leq b}\)
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\((-\infty,\infty) = \mathbb{R}\) (the entire real line)
In short, an interval is a shorthand notation for describing all numbers between two endpoints.
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Put the smaller number first, and the bigger number second.
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Use square brackets when the endpoint is included.
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Use round brackets when the endpoint is not included, including for \(\infty\) or \(-\infty\text{.}\)
Note that the notation for an interval \((a,b)\) is the same as a point in the \(xy\)-plane \((a,b)\text{.}\) This is an overloading of notation, however in most cases, the context should make clear which one it’s referring to.
