Brackets

The first question the student gets on this topic is “How can we define brackets”. In evaluating an expression containing a bracketed sub-expression, brackets denote a type of grouping, the operators in the sub-expression take precedence over those surrounding it. Additionally, for the different brackets, there are many uses and definitions.

Types of Brackets

The frequently used bracket types are:

• Parentheses ( )

• Square brackets [ ]

• Curly brackets { }

• Angle brackets ⟨ ⟩

Parentheses

Among the four different types of brackets used, parentheses are the most commonly used bracket type. In mathematical problems, the primary use of parentheses is to group numbers. Use the order of operations to solve the problem when we see multiple numbers and operations in parentheses.

For three key purposes, parentheses are used in mathematics:

• To divide numbers for clarification.

• To signify multiplication.

• To group numbers.

To separate numbers for clarification, parentheses may be used. For instance, if we have an additional problem with a negative number, to distinguish the two signs, parentheses will be used. To distinguish a number from its exponents, parentheses may also be used. Typically, this occurs if we lift a negative number to control.

Square Brackets

In mathematics, the square bracket symbols [ ] are employed in a variety of situations:

• Square brackets are sometimes used in place of (or in addition to) parentheses in very complex expressions, especially as a group sign outside an inner set of parentheses.

• They can signify the same thing as parenthesis, but they're meant to make things easier to read. It all depends on the situation. 

• Square brackets are used to include the number it covers while working with inclusion. 

• They can also be used to denote the least common multiple 

Curly Brackets (also known as Braces)

Left curly brackets and right curly brackets are used together in mathematical expressions. They can be replaced by square brackets or parentheses. In a nested phrase with three layers of grouping, parentheses are usually used in the innermost groupings. In the next higher level grouping, square brackets are employed, while braces are used in the outermost groups (see " Nested expressions " for an example).

Angle Brackets

The inner product of two functions is represented by an angle bracket, which is made up of a bra and a ket (bra+ket = bracket). Because angle brackets resemble the "less than" and "greater than" signs, they might seem confusing to some students. But you will get a hold of it, once you start using them in your maths practice from time to time.

What is the Use of Brackets?

• Brackets especially Parentheses () are used in elementary algebra to define the order of operations. Following the BODMAS rule, words within the bracket are evaluated first.

Ex: 5 * (2 + 4) is 30, (5 * 3) + 2 is 30.

• Brackets are often used in mathematical expressions in general to signify grouping where appropriate to prevent ambiguities and increase clarity.

• In the Cartesian system of coordinates, brackets are used to designate point coordinates.

Ex: (4,8) denotes the points on the x-y coordinate system with x-coordinate being 4 and y-coordinate being 8.

• The arguments for a function are always surrounded by parentheses.

Ex: f(x), g(x).

• For denoting an open end of an interval, a bracket may be used. 

Ex: [0,8) denotes a half-closed interval that includes all real numbers, except 8 from 0 to 8.

• Wide parentheses around two numbers denote a binomial coefficient, one above the other.

• As in (a,b,c), parentheses around a set of two or more numbers denote an n-tuple of numbers that are connected in a particular way.

• A matrix is indicated by broad brackets around an array of numbers.

• To denote the largest common divisor, parentheses are used.

BODMAS Rule

Brackets find their main application in the BODMAS or PEMDAS rule where the sequence of operations is to be performed when an expression is resolved. BODMAS or PEMDAS stands for:

B - Brackets, P- Parentheses

O - Order, E- Exponents

D - Division

M - Multiplication 

A - Addition

S - Subtraction

The BODMAS rule explains the sequence of operations to be done until an expression is resolved. According to the BODMAS law, if there are brackets ((), {},)in an expression, we first have to overcome or simplify the bracket followed by the order, then divide, multiply, add and subtract from left to right. In the wrong order, solving the issue would result in a wrong answer. 

Simply put, the four operations are crucial to arithmetic learning, and youngsters who don't know which sequence to finish them in will fail to move through the years.

Another reason BODMAS is taught in math classes is that it makes it much easier for young students to remember which operation to perform when faced with complex equations.

Basic Problems on Brackets and their Application:

1) Solve (2 + 4) - (6 - 3)

Ans: Two parentheses are involved in the given expression. We can solve both of them separately by the BODMAS rule and then combine their results.

(2 + 4) = 6……….(1)

(6 - 3) = 3………..(2)

Now subtracting (1) with (2), we get

(2 + 4) - (6 - 3) = 6 - 3 = 3

2) Solve (3 + (5 * 4)) - ((4 * 6) - 10)

Ans: Four parentheses are involved in the given expression. We will solve it by using the BODMAS rule to find the answer.

First parentheses is (5 * 4) = 20……………………………..(1)

Second parentheses is (3 + (5*4))=(3 + 20) =23………(2)

Third parentheses is (4 * 6) = 24……………………………(3)

Fourth parentheses is ((4 * 6) - 10) = (24 - 10) = 14……(4)

Now subtract (2) and (4) we get

(3 + (5 * 4)) - ((4 * 6) - 10) = 23 -14 = 9.