How to Use Parentheses: Simple Examples and Common Mistakes
Parentheses, also known as round brackets, are symbols used in pairs to group things together. The parentheses symbol is represented as ( ). The parentheses in Math are used to group numbers, variables, or both together.
Example:
4 ( 5 - 4) = 4 1 = 4
In the above example, the parentheses group (5 - 4), tell us to calculate the bracket first.
What are parentheses in math?
Parenthese are symbols ( ) used in pairs to group things together. In general, it is important or convenient to manually choose which order of operation should occur first in Math. Generally, we evaluate exponents first, then multiplication and division in second, then finally addition and subtraction at last. We choose this order of operation, unless the given Mathematical expression is surrounded by the parenthesis.
Generally, we evaluate, 5 3 + 62 like this 5 3 + 36 = 15 + 36 = 51. However, if parentheses are added, we need to calculate the terms inside the bracket or parenthesis first.
For example,
Starting with the innermost set of parentheses and calculating, we have:
(5 (3 + 6) )2 = (5 (9))2 = (5 9)2 = (45)2 = 2025.
Parentheses Definition in Math
Parenthesis, also known as round brackets, are defined as the symbols in Math. Parenthesis are primarily used in Mathematics/Algebraic equations to modify the normal order of operation. Hence, in Mathematic expressions involving Parentheses, the terms inside the bracket or parenthesis are calculated first.
For example, In an expression like (2 + 5) 6, the part of Mathematical expression within the parenthesis is calculated first ( 2 + 5) = 7, then this result is used to calculate the rest of the expression 7 6 = 42. Therefore, (2 + 5) 6 = 42.
Parentheses Example:
What is ( (2 + 3)2 + 4 ) 7 ?
Solution:
Following the precise order of operation, we get the following:
( (2 + 3)2 + 4 ) 7 Adding terms inside the small parentheses gives
= ( (5)2 + 4 ) 7 Calculating the exponents inside the small parentheses gives
= ( 25 + 4) 7 Adding the terms inside the parentheses.
= (29) 7 Multiplying the terms gives
= 203 Result
Parentheses Rules
The four important parentheses rules are discussed below:
x + (- y) = x - y
Example: 5 + (-3) = 5 - 3 = 2
x - (- y) = x + y
Example: 5 - (- 3) = 5 + 3 = 8
x . (-y) = - xy
Example: 5 . (-3) = - 15
(-x) (-y) = xy
Example: (-5) (-3) = 5 3 = 15
Parentheses Example With Solution
1. Simplify the expression (2 + 5 7) - (3 + 4)
Solution:
Here, the expression has two parentheses. We will solve the terms inside both the parentheses separately and then combine the result to get the answer.
Let’s first solve, (2 + 5 7)
Here, according to the order of operation, we will multiply 5 and 7 first. Accordingly,
(2 + 35) = 2 + 35 = 37
Now, we will solve (3 + 4), which gives 7.
Combining, both the results, we get
(2 + 5 7) - ( 3 + 4) = 37 - 7 = 30
Therefore, ( 2 + 5 7) - ( 3 + 4) = 30
Example 2:
Solve (3 + 52)2
Solution:
The parentheses tell us we must evaluate the expression first 3 + 52 and then square it. Here, ensure to evaluate 5² first before adding. This is because according to the order of operation, we first evaluate exponents. Accordingly,
(3 + 52)2 = (3 + 25)2 = (28)2 = 784.
Therefore, (3 + 52)2 is equal to 784.
Do You Know?
In Math, the order of operation are the rules that describe the sequence in which the multiple arithmetic operations in an expression are solved. The best way to remember the order of operations is PEMDAS.
As per the PEMDAS rule, solve
P = Parentheses first
E = Exponents ( Power and Square Roots, etc)
MD = Multiplication and Division (Left to Right or whichever comes first)
AS = Addition and Subtraction (Left to Right or whichever comes first)
Conclusion:
In short, parentheses or round brackets are renown mathematical symbols used in parts to group things together or to specify the order of operation in an equation. In Math, parentheses are used in two different ways i.e. to multiply and to tell what numbers to look at first.
FAQs on What Are Parentheses in Maths? Definition & Key Uses
1. What are parentheses in Maths and what is their primary function?
In mathematics, parentheses, represented by the symbols ( ), are used to group parts of an expression together. Their primary function is to control the order of operations, ensuring that the calculations inside the parentheses are performed before any operations outside of them. This helps to avoid ambiguity and ensures that expressions are evaluated correctly and consistently.
2. What are the key uses of parentheses in mathematical expressions?
Parentheses serve several important purposes in mathematics. The main uses include:
Grouping for Order of Operations: They dictate which part of an equation to solve first, as specified by rules like BODMAS or PEMDAS. For example, in 10 - (4 + 2), you must calculate 4 + 2 first.
Indicating Multiplication: A number placed directly before a parenthesis implies multiplication. For instance, 3(4) means 3 × 4.
Separating Numbers for Clarity: They are used to separate a number's sign from an operation, such as in the expression 5 + (-2).
Function Notation: In functions like f(x), parentheses enclose the input variable.
3. How do parentheses relate to the order of operations, such as the BODMAS or PEMDAS rule?
Parentheses are a fundamental component of the order of operations, commonly remembered by the acronyms BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) or PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). In both systems, the 'B' or 'P' comes first, signifying that any expression contained within parentheses (or any type of bracket) must be evaluated before any other operation in the equation.
4. Does a number next to a parenthesis, like 5(2+3), always imply multiplication?
Yes, in standard mathematical notation, a number written directly outside a parenthesis with no operator in between implies multiplication. Therefore, the expression 5(2+3) is understood as 5 multiplied by the result of (2+3). You would first solve the expression in the parenthesis (2+3 = 5) and then multiply it by the number outside, resulting in 5 × 5 = 25.
5. What is the difference between parentheses ( ), square brackets [ ], and curly braces { } in Maths?
While all three are used for grouping, they follow a specific hierarchy when expressions are nested within each other. The standard convention is:
Parentheses ( ) are the innermost brackets.
Square Brackets [ ] are used to enclose parentheses.
Curly Braces { } are the outermost brackets, used to enclose square brackets.
For example, in the expression {10 + [5 × (3-1)]}, you would solve the parentheses (3-1) first, then the square brackets, and finally the curly braces.
6. Why is it essential to solve the calculation inside the parentheses first?
Solving the calculation inside parentheses first is essential to maintain a universal standard for evaluating mathematical expressions. This rule, a core part of the order of operations, eliminates any potential ambiguity. For example, without this rule, the expression 5 × (2 + 4) could be interpreted as (5 × 2) + 4 = 14 or 5 × 6 = 30. By prioritising the parentheses, everyone arrives at the same, correct answer of 30, ensuring consistency in mathematics.
7. How are parentheses used in other mathematical contexts, like functions and coordinates?
Beyond basic arithmetic, parentheses have specific meanings in other areas of maths. In function notation, such as f(x) = 2x + 1, the parentheses do not mean multiplication but are used to enclose the input variable 'x'. In coordinate geometry, parentheses are used to group the coordinates of a point on a plane, such as (x, y). For example, the point (3, 4) represents a location 3 units along the x-axis and 4 units along the y-axis.
8. What does it mean when a single negative number is written in parentheses, for example, -4 + (-3)?
When a single negative number is in parentheses, it is primarily for clarity. It helps to visually separate the negative sign of the number from the mathematical operation. In the example -4 + (-3), the parentheses around -3 make it clear that you are adding a negative number, rather than getting confused with two adjacent signs. This prevents misinterpreting the expression as -4 - 3, although in this specific case, the result would be the same.
