Understanding and correctly applying the order of operations is fundamental to solving mathematical expressions accurately. The mnemonic PEMDAS helps students and mathematicians remember the sequence in which operations should be performed. This blog post delves deep into PEMDAS, explaining each component, its importance, common pitfalls, and practical applications. By the end, you’ll have a thorough grasp of how to tackle complex mathematical expressions confidently.
What is PEMDAS?
PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. It describes the order in which mathematical operations need to be performed to ensure everyone arrives at the same, correct answer. Without a standard order, mathematical expressions could be interpreted in multiple ways, leading to confusion.
The sequence dictates:
- Parentheses (P) – Solve expressions inside parentheses or brackets first.
- Exponents (E) – Calculate powers and roots second.
- Multiplication (M) and Division (D) – Perform these operations from left to right.
- Addition (A) and Subtraction (S) – Perform these operations from left to right, last of all.
It is important to note that Multiplication and Division share the same level of priority; you do not always multiply before dividing, but instead proceed in order from left to right. The same applies to Addition and Subtraction.
Why is PEMDAS Important?
Without PEMDAS, mathematical expressions can yield ambiguous results. Consider the expression:
8+4×2
If you add first, 8+4=12, and then multiply by 2, getting 24. However, by following PEMDAS and multiplying first, 4×2=8, then adding 8, the correct answer is 16.
This example shows why understanding and applying PEMDAS is vital to avoid errors.
Breaking Down PEMDAS
Parentheses (P)
Parentheses indicate which calculation should be completed first. This also applies to brackets [ ] or braces { } in more complex expressions. Expressions inside parentheses are the highest priority and must be addressed first before moving on.
Example:
(3+5)×2
Solve inside the parentheses: 3+5=8, then multiply by 2 to get 16.
Parentheses can be nested, meaning there can be parentheses inside parentheses. The innermost set must be solved first.
Example:
((2+3)×4)+1
First, calculate 2+3=5, then multiply by 4 to get 20, then add 1 to reach 21.
Exponents (E)
Exponents or powers come next. This covers squaring numbers, cube roots, and other powers.
Example:
2^3+4=8+4=12
Multiplication (M) and Division (D)
These operations come next and are treated equally in priority. You perform them as they appear from left to right.
Example:
12÷3×2
Proceed from left to right: 12÷3=4, then, 4×2=8
Incorrectly doing multiplication first 3×2=6 would lead to: 12÷6=2, which is wrong.
Addition (A) and Subtraction (S)
Finally, addition and subtraction are computed last, with equal priority and executed from left to right.
Example:
15−5+2
Calculate:
15−5=10,
then
10+2=12
Common PEMDAS Mistakes and How to Avoid Them
Mistaking Order of Multiplication and Division or Addition and Subtraction
A common mistake is thinking multiplication always comes before division or addition always before subtraction. However, multiplication and division are performed in the order they appear, left to right.
Example:
20÷5×2
Correct: 20÷5=4, then 4×2=8.
Ignoring Parentheses
Sometimes people skip doing operations inside parentheses first, resulting in wrong answers.
Example:
4×(2+3)
Calculate inside parentheses first: 2+3=5, then multiply: 4×5=20.
Skipping parentheses by doing multiplication first gives 8+3=11, which is incorrect.
Overlooking Nested Parentheses
When parentheses are nested, solve inside-out, one layer at a time.
Example:
2×(3+(4−1))
First solve inner parentheses:
4−1=3, then 3+3=6, finally multiply: 2×6=12.
Practical Applications of PEMDAS
PEMDAS is not just academic; it applies in various real-life scenarios such as computer programming, engineering, finance, and more. Whenever multiple operations appear in one expression, following PEMDAS ensures calculations are done correctly.
In programming languages, order of operations is critical to prevent bugs. For example, in many languages such as Python or JavaScript, expressions obey PEMDAS rules.
In finance, formulas involving interest calculations and loan amortizations often rely on correct order of operations to provide accurate outputs.
Example Problems and Solutions Using PEMDAS
- 5+2×3^2
- Exponents first: 3^2 = 9
- Multiplication: 2×9=18
- Addition: 5+18=23
- (8−3)÷(2+3)×4
- Parentheses:
- 8−3=5
- 2+3=5
- Division: 5÷5=1
- Multiplication: 1×4=4
- 12−4×2+(6÷3)
- Parentheses: 6÷3=2
- Multiplication: 4×2=8
- Perform subtraction and addition from left to right:
- 12−8=4
- 4+2=6
How to Teach PEMDAS Effectively
- Use visual aids, such as color-coded steps to highlight each operation.
- Practice with real-world problems to show relevance.
- Encourage breaking down expressions step-by-step.
- Use mnemonic devices like PEMDAS or BIDMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction) for different regions.
- Reinforce the left-to-right rule for multiplication/division and addition/subtraction.
Conclusion
Understanding PEMDAS is crucial for accurately solving mathematical expressions involving multiple operations. It provides a universal standard that avoids confusion and ensures consistency in results. By mastering each component — Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction — and practicing their application, anyone can improve their math skills and avoid common errors. Whether you are a student, teacher, programmer, or just a curious learner, PEMDAS is a tool that strengthens your command over numbers. Data Science Blog
Frequently Asked Questions (FAQs)
Q1: Is multiplication always done before division?
No. Multiplication and division have the same priority and are performed from left to right as they appear in the expression.
Q2: What if there are no parentheses in a problem?
You simply follow the rest of the PEMDAS order starting with Exponents, then multiplication/division from left to right, followed by addition/subtraction from left to right.
Q3: Can PEMDAS be applied to algebraic expressions?
Yes. PEMDAS applies to all mathematical expressions, including algebraic ones.
Q4: What are some helpful tips to remember PEMDAS?
Using mnemonic phrases like “Please Excuse My Dear Aunt Sally,” practicing with examples, and breaking down problems step-by-step help reinforce PEMDAS.
Q5: Is PEMDAS used worldwide?
While PEMDAS is common in the United States, other countries use similar mnemonics such as BIDMAS or BODMAS, but the principles remain essentially the same.