Brackets in Math: Types & Examples

In this lesson, you will learn about the many uses of mathematical brackets – from grouping and clarity in the order of operations to specialized uses in various fields of mathematics.
Definition
Mathematical brackets are symbols, such as parentheses, that are most often used to create groups or clarify the order that operations are to be done in an algebraic expression. Some bracket symbols, however, have multiple special uses in mathematics.

Brackets and Grouping
Often, you will see mathematical brackets used for grouping. These brackets can include:

( )
[ ]
{ }
When used for grouping, brackets always come in pairs. There will be an opening bracket and a closing bracket.

Brackets are used to provide clarity in the order of operations, the order in which several operations should be done in a mathematical expression.

For example, suppose you have the following expression: 2 + 4 * 6 – 1. Despite what you might read on Facebook, there is only one correct answer to that expression. You perform multiplications and divisions, moving from left to right, before you perform additions and subtractions, also moving from left to right. Performing the multiplication first, you get 2 + 24 – 1 = 25.

What if, instead, you wanted to do the addition and subtraction first (and then multiply the results)? Use brackets. Now the problem becomes: (2 + 4) * (6 – 1) = 6 * 5 = 30. In this example, the parentheses tell you to do something different than the usual order of operations. Other times, they are simply used for visual clarity.

Multiple Levels of Grouping
You might want to do grouping within grouping. If so, then expressions like this are confusing: 2 + (1 + (3 + 2 * (4 + 5))). While there is nothing really wrong with using multiple levels of parentheses (and sometimes in computer applications you have no choice), it is a little hard to look at.

Instead, you might use different kinds of brackets for each level. In mathematics, it is most common to use round brackets for the first level (the first operation you would do), square brackets for the next level and curly brackets for the last level: 2 + {1 + [3 + 2 * (4 + 5)] } .

Either way, you do the innermost grouping first (4 + 5) and move outward from there, as follows:

2 + {1 + [3 + 2 * 9]} = 2 + {1 + [3 + 18]} = 2 + {1 + 21} = 2 + 22 = 24
It’s a bit like finding your way out of your Uncle Jerome’s mansion. First you have to find your way out of the guest suite, then you find your way off the third floor and then you navigate your way out of the house itself. You start with the innermost ‘puzzle’ first and move outward from there.

Other Uses for Brackets
Sometimes, however, brackets are used for other things than grouping. For example, if you are working with functions then f(x) means ‘the function f with x as an input.’ In this case, the parentheses are used to tell you the arguments, or inputs, of the function.

Also, parentheses can be used to indicate an ordered pair, such as (3,-1). You’ll often see this used to denote Cartesian coordinates: if you were plotting a point against x- and y- axes, (3,-1) would be the specific point on the Cartesian plane where x is 3 and y is -1.

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