Order of Operations – PEMDAS – Mathematical Mysteries

Definition

The order of operations in mathematics is the sequence in which a problem is solved. Explore the definition and examples of the order of operations in math, discover the steps involved, and learn the shortcut for remembering the steps defined by the acronym PEMDAS.¹

Who

For those who are confused on how to apply PEMDAS. This topic appears to be the most misunderstood topic I have been asked to explain.

What

In mathematics, we perform operations like addition, subtraction, multiplication and division, which are governed by an order of operation. The PEMDAS rule is one of the rules which is equivalent to the BODMAS rule.

Why

See Theoretical Knowledge Vs Practical Application.

How

Order of Operations Steps

The steps we use to solve any mathematical expression are:¹

Simplify all of the parentheses. This includes all forms of grouping symbols, such as brackets and braces, in addition to parentheses.

“Parentheses can be nested, and should be evaluated from the inside outward. For legibility, outer parentheses can be made larger than inner parentheses. Alternately, other grouping symbols, such as curly braces { } or square brackets [ ], are sometimes used along with parentheses ( ).” [3]

For example: [(1 + 2) ÷ (3 + 4)] + 5 = (3 ÷ 7 ) + 5
Simplify all exponents.
Simplify all multiplication and division from left to right. When simplifying the multiplication and division, work from left to right.
Simplify all addition and subtraction from left to right. Again, when simplifying the addition and subtraction, work from left to right.

By following this order, we can all solve the problem and get the same solution.

PEMDAS

Remember that the steps for multiplication and division is one step. We work all of the multiplication and division in one step from left to right. Multiplication does not always come before division; they are worked in the order that they appear. This is also true for addition and subtraction. They are worked in the same step from left to right. An easy way for me to remember these steps is to remember the phrase Please Excuse My Dear Aunt Sally, where the:²

P – Parentheses – Please
E – Exponents – Excuse
M – Multiplication – My
D – Division – Dear
A – Addition – Aunt
S – Subtraction – Sally

$\left( PLEASE-ExCUSE \right) ^2\,\,+\,\,MY\left( \frac{DEAR}{AUNT}\,\,+\,\,SALLY \right)$

Examples

42 – (24 / (4 * 2))
Ans: 42 – (24 / (4 * 2)) = 42 – (24 / (8)) = 42 – (3) = 39

9 + (12 + 1)^2
Ans: 9 + (12 + 1) ^ 2 = 9 + (13) ^ 2 = 9 + 169 = 178

7 + [–5 × (–10 – 1)] ^ 3
Ans: 7 + [–5 × (–10 – 1)] ^ 3 = 7 + [–5 × (–11)] ^ 3 = 7 + [55] ^ 3+ = 7 + 166375 = 166382

12 / 6 × 3 / 2
Ans: 12 / 6 × 3 / 2 = (12 / 6) × 3 / 2 = 2 × 3 / 2 = 6 / 2 = 3

8 + (16 × 52 – 10)
Ans: 8 + (16 × 52 – 10) = 8 + ((16 × 25) – 10) = 8 + (400 – 10) = 8 + (390) = 398

7 x 3 + 10 x (25 ÷ 5)
Ans: 7 × 3 + 10 × (25 ÷ 5) = 7 × 3 + 10 × (5) = 21 + 50 = 71

6 ÷ 2 × 3
Ans: (6 ÷ 2) × 3 = 3 × 3 = 9

Many of the References and Additional Reading websites, and Videos will assist you with using PEMDAS.

As some professors say: “It is intuitively obvious to even the most casual observer.”

References

[1] “What Is The Order of Operations in Math? – Definition & Examples”. 2021. study.com. https://study.com/academy/lesson/order-of-operations.html.

[2] “What Is Order Of Operations? – Definition, Facts & Example”. 2021. splashlearn.com. https://www.splashlearn.com/math-vocabulary/algebra/order-of-operations.

[3] ⭐ “Order of Operations.” 2024. Wikipedia. Wikimedia Foundation. April 19. https://en.m.wikipedia.org/wiki/Order_of_operations#:

Additional Reading

“Order Of Operations – PEMDAS”. 2021. mathsisfun.com. https://www.mathsisfun.com/operation-order-pemdas.html.

“PEMDAS Explained – How Does PEMDAS Work?”. 2021. thecalculatorsite.com. https://www.thecalculatorsite.com/articles/math/how-does-pemdas-work.php.

PEMDAS is a mnemonic acronym for the order of operations in math: parentheses; exponents; multiply or divide; add or subtract. When there are several operations in a single expression, it’s important to calculate them in the proper order (parenthesis first, exponents second…) to get the correct outcome.

⭐ “PEMDAS Rule: Definition, Order, PEMDAS vs BODMAS.” 2024. GeeksforGeeks. GeeksforGeeks. March 31. https://www.geeksforgeeks.org/pemdas-rule/.

In this article, we will learn about the PEMDAS Rule with its order of operations, its applications, examples, and others in detail.

⭐ Stapel, Elizabeth. 2023. “Order of Operations: What It Is & Why We Need It.” Purplemath. Accessed October 19. https://www.purplemath.com/modules/orderops.htm.

To eliminate confusion, we have some rules of precedence, established at least as far back as the 1500s, called the “order of operations”. The “operations” are addition, subtraction, multiplication, division, exponentiation, and grouping; the “order” of these operations states which operations take precedence over (that is, which operations are taken care of before) which other operations.

Videos

What is the value of 3 x 3 – 3 ÷ 3 + 3?

Intro to order of operations

This example shows the steps and clarifies the purpose of order of operations: to have ONE way to interpret a mathematical statement.

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.