3.5: Interval Notation and Infinity

Last updated
Save as PDF

Page ID: 45161

Victoria Dominguez, Cristian Martinez, & Sanaa Saykali
Citrus College via ASCCC Open Educational Resources Initiative

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

What is infinity? Infinity is not a real number. Infinity is larger than any number that can be imagined. It is an idea of having no bounds. A line is an example of having no bounds. For instance the number line has arrows at the end to represent this idea of having no bounds. The symbol used to represent infinity is $\infty$. On the left side of the number line is $−\infty$ and on the right side of the number line is $\infty$ to describe the boundless behavior of the number line.

Interval Notation	Number Line
a) $(12,\infty)$	$clipboard_e80433e1e62fd9286767d7fccd338cf9b.png$ Figure 3.5.1
b) $[-5,\infty)$	$clipboard_eb315c734eff17646d4b6150afe365542.png$ Figure 3.5.2
c) $(−\infty,4)$	$clipboard_e347ccb8b609ce4a6a16613afe5ef2c4b.png$ Figure 3.5.3
d) $(−\infty,0]$	$clipboard_e960bf6751735e85b398571075c5f0bf9.png$ Figure 3.5.4
e) $(−\infty,\infty)$	$clipboard_e65f0217f121cc70ff0562beb8ad1573f.png$ Figure 3.5.5

Note: Since $\infty$ is not a real number, it is required to use parentheses (,). $\infty$ can’t be included in the interval.

Exercise 3.5.1

Draw a number line that corresponds to the following intervals.

$(−\infty,5)$
$[-5,\infty)$
$(−\infty,\infty)$
$(-3,\infty)$
$(−\infty,-2]$

Interval Notation	Number Line
a) \((12,\infty)\)	$clipboard_e80433e1e62fd9286767d7fccd338cf9b.png$ Figure 3.5.1
b) \([-5,\infty)\)	$clipboard_eb315c734eff17646d4b6150afe365542.png$ Figure 3.5.2
c) \((−\infty,4)\)	$clipboard_e347ccb8b609ce4a6a16613afe5ef2c4b.png$ Figure 3.5.3
d) \((−\infty,0]\)	$clipboard_e960bf6751735e85b398571075c5f0bf9.png$ Figure 3.5.4
e) \((−\infty,\infty)\)	$clipboard_e65f0217f121cc70ff0562beb8ad1573f.png$ Figure 3.5.5