Contents

# Brackets|Definition & Meaning

## Definition

**Brackets** are such symbols that are used in **pairs** to group **terms** together. They include terms such as **constants** or **varia****bles** with coefficients and separate them from the rest to carry out mathematical** operations** in order.

**Figure 1** shows different types of brackets in a numeric expression.

Note that each bracket has an **opening** end and a **closing** end.

## Importance of Brackets

**Brackets** play an essential role in solving **algebraic equations** as they provide a **systematic** way to solve the equation. Without brackets, solving the mathematical expression would be **confusing** and quite tedious.

## Types of Brackets

Brackets are **organizational tools** used to define the **order** of operation in an algebraic equation. The different **variety** of brackets makes the algebraic expression look more **comprehensible** for solving it.

The different **types** of brackets are given as follows:

### Round Brackets

Round Brackets are also known as **parentheses,** circle brackets, parens, or smooth brackets.

The term “**lunula**” came from the Latin word **“luna”** and was used for round brackets in early history as the **shape** of the round bracket is similar to the crescent **moon**.

The **singular** for parentheses is “**parenthesis**” which can be referred to the right or left round bracket. The round brackets are shown in **figure 2**.

**Parentheses** are also used in English literature to add **extra information** to the text. They are also used in expressions where the **singular** and **plural** of the word are required at the same time—for example, the **writer(s)**.

Round brackets are often **nested** in the **innermost part** of an algebraic expression. The **math operator** in the round brackets is performed first regardless of its** precedence** over the other operator with it in the **square brackets**.

For **example**, in the equation:

**[ 8 ÷ ( 5 – 3 ) ] **

The **division** should be performed first over **subtraction** according to the order of operations but the **round brackets** overshadow this precedence.

The round bracket has the **highest priority** and so the answer will be **4**.

**Parentheses** are also used to define a **function** in terms of its variable such as **f(z)** represents a function of variable **z**.

They are also used to define the **domain** and **range** of a function if there is any **discontinuity** or the graph goes up to **infinity** on the interval side.

### Square Brackets

Square brackets are also known as **closed** brackets, brackets, and **box** brackets.

They are used in algebraic equations to **enclose** more terms outside the **round brackets** and thus have the **second** precedence after the parentheses. **Figure 3** shows the square brackets.

They are also used to define an **interval** in graphs.

For example, in **[1,9)** the left** square bracket** suggests that **1** is included in the interval and the right parenthesis suggests that **9** is not included. **8.99** is included but **9** is not included.

Square brackets are also used to define **matrices** and chemical **concentrations** in solutions.

### Curly Brackets

Curly brackets are also known as **braces**, **flower** brackets, and **swirly** brackets. They have the **third** precedence after parentheses and square brackets. They are shown in **figure 4**.

Curly brackets are used in **algebraic expressions** and to define **sets** such as subsets, power sets, or supersets, etc.

### Angle Brackets

Angle brackets are also known as **broken** brackets, **pointy** brackets, **diamond** brackets, or **chevrons**. The brackets are similar to the less than and greater than signs. **Figure 5** shows the angle brackets.

Angle brackets are used to define **averages** with respect to** time** or any other variable. They are used to identify **ordered pairs**.

## Rules for Order of Operations

The order of operations is the** set of rules** that signify which mathematical **operation** should be carried out first while solving an **algebraic expression**. This order is described by the following **two acronyms**.

### BODMAS

BODMAS stands for “Brackets Orders Division Multipliplication Addition Subtraction”. **Brackets** are the first to look for while solving a mathematical **expression**.

**Orders** are the **powers** or exponents which are calculated after brackets. After these come **division** and **multiplication** which have more precedence than **addition** and **subtraction.**

Sometimes, the acronym **BIDMAS** is also used for BODMAS with “**I**” standing for **“Indices”.**

### PEMDAS

PEMDAS **stands** for “Parentheses Exponents Multiplication Division Addition Subtraction”. It is the same order of operation as** BIDMAS** or **BODMAS**. The multiplication and division and the addition and subtraction are checked from **left to right**.

## Order of Brackets In Solving Algebraic Expressions

**Brackets** are analyzed first and foremost while solving an algebraic expression. The terms in the **interior-most** bracket which is usually the **parentheses** are solved first regardless of the mathematical operation in it.

The next bracket to solve according to the **PEMDAS** rule is the **square bracket** and the last to solve is the **curly bracket**.

Although all **brackets** serve the **same purpose** of enclosing various terms, this order can be **flexible**. But most commonly, the **parenthesis** is used in the inner-most part of the expression and **first** solved.

Also, the** bar** or **vinculum** is solved even before the round brackets as they are usually enclosed in the **parentheses** if they involve **three** or more math operations.

So conclusively, **brackets** have the top **priority** in solving algebraic equations.

## Solving an Expression With the PEMDAS Rule

Solve the **algebraic expression** given below:

**5x – { 3y – [ 4x – ( 5y – 6x – 7y ) ] }**

### Solution

According to the **PEMDAS** rule, the parentheses are solved first. In the round brackets, **5y** and **7y** are **like terms** so they can be subtracted. Like terms have the **same** **variable** and **power** on the variable.

So, the **equation** becomes:

5x – { 3y – [ 4x – ( 5y – 7y – 6x ) ] }

5x – { 3y – [ 4x – ( – 2y – 6x ) ] }

The **parentheses** are not removed as the **minus sign** outside the parentheses will change the **signs** of terms inside the round brackets. So, further solving gives:

5x – { 3y – [ 4x + 2y + 6x ] }

Solving the **square bracket** and adding the like terms **4x** and **6x** gives:

5x – { 3y – [ 10x + 2y ] }

5x – { 3y – 10x – 2y }

Solving the **braces** gives:

5x – { y – 10x }

5x – y + 10x

So, the final **answer** is:

**15x – y**

*All the images are created using GeoGebra.*