Mathematicians frequently want to talk about intervals of real numbers such as “all real numbers between \(1\) and \(2\)”, without mentioning a variable. As an example, “The range of the function \(f:x\mapsto \sin x\) is all real numbers between \(-1\) and \(1\)”.

A compact notation often used for these intervals of real numbers is as follows:

  • \((1,2)\) means all real numbers between \(1\) and \(2\), excluding the endpoints

  • \([1,2]\) means all real numbers between \(1\) and \(2\), including the endpoints

We can also write these intervals using set notation as \(\{x:1<x<2\}\) and \(\{x:1\le x\le 2\}\) respectively.

If needed, we can also mix the two types of bracket, so \((1,2]\) means the interval \(\{x:1<x\le 2\}\) and \([1,2)\) means \(\{x:1\le x<2\}\).

The interval “all real numbers greater than \(-5\)” is written as \((-5,\infty)\), and “all real numbers less than or equal to \(7\)” is written as \((-\infty,7]\). This does not mean that \(\infty\) is a number; it is just a convenient shorthand.

Although the notation \((1,2)\) is exactly the same as the notation for coordinates, the two are rarely confused because the context will make it clear which is meant.