In mathematics, parentheses () are fundamental symbols that dictate the order in which operations should be performed, ensuring clarity and consistency in calculations.
Understanding Parentheses in Mathematics
Parentheses serve as grouping symbols, instructing us to prioritize the enclosed operations before tackling any operations outside of them. This rule is a cornerstone of the order of operations, often remembered by acronyms like PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) or BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction).
As stated by Study.com, "Parentheses are solved first, followed by exponents. Then, multiplication and division are solved from left to right. Finally, addition and subtraction are solved from left to right." This means that any calculation within parentheses must be completed before moving on to other parts of the expression.
The Hierarchy of Operations with Parentheses
The presence of parentheses always signals the highest priority in a mathematical expression. Here's how they fit into the standard order of operations:
- Parentheses (or Brackets):
()Operations inside parentheses are always performed first. - Exponents (or Orders):
^√Calculations involving powers or roots come next. - Multiplication and Division:
×,÷These operations are performed from left to right, after exponents. - Addition and Subtraction:
+,-These are the last operations performed, also from left to right.
Operations Within Parentheses
It's important to note that if parentheses contain multiple operations, those operations themselves will also follow the standard order of operations (PEMDAS/BODMAS) internally. For example, in (5 + 2 × 3), the multiplication 2 × 3 would be done before the addition 5 + 6.
Why Parentheses Are Crucial
Parentheses remove ambiguity from mathematical expressions. Without them, an expression like 2 + 3 × 4 could be interpreted differently depending on the chosen order (e.g., (2 + 3) × 4 = 20 or 2 + (3 × 4) = 14). With parentheses, the intent is clear:
(2 + 3) × 4clearly indicates that2 + 3is evaluated first.2 + (3 × 4)clearly indicates that3 × 4is evaluated first.
Practical Examples
Let's illustrate the rules for parentheses with some examples:
-
Example 1: Simple Parentheses
10 - (3 + 2)- First, solve the operation inside the parentheses:
3 + 2 = 5 - Then, perform the subtraction:
10 - 5 = 5 - Result:
5
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Example 2: Parentheses with Multiple Operations Inside
4 × (5 + 6 ÷ 2)- Inside the parentheses, follow the order of operations:
- Division first:
6 ÷ 2 = 3 - Then addition:
5 + 3 = 8
- Division first:
- Now, the expression is
4 × 8 - Perform the multiplication:
4 × 8 = 32 - Result:
32
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Example 3: Nested Parentheses
- Sometimes, you might encounter parentheses within other parentheses, like
8 + (6 ÷ (1 + 2)) - Start with the innermost parentheses:
(1 + 2) = 3 - The expression becomes
8 + (6 ÷ 3) - Next, solve the remaining inner parentheses:
(6 ÷ 3) = 2 - Finally, perform the addition:
8 + 2 = 10 - Result:
10
- Sometimes, you might encounter parentheses within other parentheses, like
Summary Table: Order of Operations
| Priority | Operation | Acronym Mnemonic |
|---|---|---|
| 1st | Parentheses () |
PEMDAS / BODMAS |
| 2nd | Exponents ^, Roots √ |
PEMDAS / BODMAS |
| 3rd | Multiplication × & Division ÷ (L-R) |
PEMDAS / BODMAS |
| 4th | Addition + & Subtraction - (L-R) |
PEMDAS / BODMAS |
By strictly following these rules, especially prioritizing operations within parentheses, mathematical expressions can be evaluated consistently and accurately.
