When and how to use parentheses, brackets, and braces in mathematics

Among the combination of symbols that involve arithmetic calculations or algebraic expressions, it is common to find three symbols, which are often confused in their use; parentheses ( ), square brackets [ ], and braces { }. Let’s see what is the specific application of each one together with some examples to fix ideas.

Parentheses ( ) are used to group numbers and variables, in a calculation or in an algebraic equation. When we find parentheses in the middle of various arithmetic operations, we are being told the order in which they should be done. Let’s remember that, without any other indication, multiplication and division take precedence over addition and subtraction, and exponentiation over multiplication and division. When operations with the same priority must be performed, the calculation proceeds from left to right in the mathematical expression. Let’s see the role of parentheses indicating the order of operations in the following example.

9 – 5 ÷ (8 – 3) × 2 + 6

The parentheses tell us that the operation that is proposed in its space must first be carried out, without considering the usual order of priorities in which arithmetic operations are carried out. In this example the multiplication and division operations would have to be performed before the subtraction, however since the operation 8 – 3 is enclosed in parentheses we have to perform this calculation first. Once all the calculations inside the parentheses have been carried out, in this case only 8 – 3, they are eliminated and we proceed with the other operations with the usual priorities. In this case the (8 – 3) is replaced by 5, and the resolution sequence of this calculation would be the following.

9 – 5 ÷ (8 – 3) × 2 + 6 = 9 – 5 ÷ 5 × 2 + 6

9 – 5 ÷ 5 × 2 + 6 = 9 – 1 × 2 + 6

9 – 1 × 2 + 6 = 9 – 2 + 6

9 – 2 + 6 = 7 + 6

7 + 6 = 13

The parentheses also implicitly indicate that this is a multiplication operation. For example, in the expression 3(2 + 5) the parentheses indicate that the addition must first be performed inside the space of the parentheses, 2 + 5. But there is no explicit operation between three and the space of the parentheses, so which is assumed to be a multiplication. A more general case, with two parentheses, would be the expression (6 –3)(2 + 3). Again, first we have to solve the two calculations in the space between the parentheses, that is 6 – 3 and 2 + 3, and then we assume that we have to do the product of both results. For clarity, let’s develop the calculation.

(6 – 3)(2 + 3) = (6 – 3) × (2 + 3)

(6 – 3) × (2 + 3) = (3) × (3)

(3) × (3) = 3 × 3

3 × 3 = 9

Brackets are also used when it is necessary to group numbers and variables in a calculation or in an algebraic equation, but when parentheses have already been used. That is, if it is necessary to group numbers and variables in the space that is already grouped, the inner group is indicated with parentheses and the outer one with square brackets. If another third-order grouping in the same space is necessary, braces would then be used. The sequence, which is also known as nested parentheses, would follow the following order: { [ ( ) ] }

Let’s look at an example of a mathematical expression that combines parentheses and square brackets. As in the case of the parentheses, if there is no explicit operation next to the brackets it is assumed that it is a multiplication.

4 – 3[4 – 2(6 – 3)] ÷ 3

In this expression, we first need to solve the operations inside the space of the brackets.

4 – 2(6 – 3)

This expression, in turn, has an order of priorities indicated by the parentheses; First, you have to solve the difference 6 – 3. Let’s see the complete development of the calculation sequence.

4 – 3[4 – 2(6 – 3)] ÷ 3 = 4 – 3 × [4 – 2 × (6 – 3)] ÷ 3

4 – 3 × [4 – 2 × (6 – 3)] ÷ 3 = 4 – 3 × [4 – 2 × (3)] ÷ 3

4 – 3 × [4 – 2 × (3)] ÷ 3 = 4 – 3 × [4 – 6] ÷ 3

4 – 3 × [4 – 6] ÷ 3 = 4 – 3 × [-2] ÷ 3

4 – 3 × [-2] ÷ 3 = 4 + 6 ÷ 3

4 + 6 ÷ 3 = 4 + 2

4 + 2 = 6

Now let’s look at an example that combines the three symbols.

2{1 + [4(2 + 1) + 3]}

As already mentioned, the general rule is to resolve nested parentheses from the inside out. Let’s see the calculation sequence.

2{1 + [4(2 + 1) + 3]} = 2 × {1 + [4 × (2 + 1) + 3]}

2 × {1 + [4 × (2 + 1) + 3]} = 2 × {1 + [4 × (3) + 3]}

2 × {1 + [4 × (3) + 3]} = 2 × {1 + [12 + 3]}

2 × {1 + [12 + 3]} = 2 × {1 + [15]}

2 × {1 + [15]} = 2 × {16}

2 × {16} = 32

Parentheses, brackets, and braces are also often referred to as round, square, and curly brackets respectively. In some expressions only parentheses are used even when there are multiple nested calculation spaces. This is done particularly when the nesting is greater than three levels, in which case there would no longer be symbols that differentiate the nesting levels. When only parentheses are used, special care must be taken to identify the first space between parentheses in the nesting, resolve it, and then advance to the next level.

Fountain

Samuel Selzer, Algebra and analytical geometry. Second edition. Buenos Aires, 1970.