Order of Operations (PEMDAS)

Remember:

P → Parentheses
E → Exponents (Powers and Roots)
MD → Multiplication and Division (left to right)
AS → Addition and Subtraction (left to right)

Key Rules

Parentheses first: Solve anything inside grouping symbols first such as (), [], or {}.
Exponents next: Handle powers and roots after parentheses.
Multiplication and Division: Work left to right; they have equal priority.
Addition and Subtraction: Also go left to right; they have equal priority.

Example

8 + 2 × (3² – 1)
→ 8 + 2 × (9 – 1)
→ 8 + 2 × 8
→ 8 + 16
→ 24

Common Mistakes

❌ Assuming “Multiplication before Division.” ➜ Correct: Process left to right. Example:
```
12 ÷ 3 × 2 = 4 × 2 = 8
```
not
```
12 ÷ (3 × 2) = 2
```
❌ Misreading fraction bars. The fraction line acts like parentheses, grouping the numerator and denominator. Example:
```
(8 + 2) / (4 – 2) = 10 / 2 = 5
```
is not the same as
```
8 + 2 / 4 – 2 = 6.5
```
➜ A written fraction a + b / c + d is not the same as a stacked fraction
```
(a + b)
-----
(c + d)
```
Always use parentheses or the fraction bar to show what is grouped.
❌ Overlooking calculator differences. Implicit multiplication such as 2(3+4) may be handled differently by devices or software. Always use explicit operators for clarity.

Ambiguity & Memes

Many “order of operations” debates stem from unclear notation.

Example:

8 ÷ 2(2 + 2)

Standard interpretation (left to right): 8 ÷ 2 × (4) → 4 × 4 → 16
Ambiguous interpretation (nonstandard): 8 ÷ [2(4)] → 8 ÷ 8 → 1

Only the first follows the formal PEMDAS rule. Confusion arises whenever notation allows more than one plausible grouping, especially in expressions that mix division, multiplication, or implicit multiplication without clear parentheses. The problem is not the math itself; it is ambiguous writing, which allows multiple valid interpretations.

Rewrite for Clarity (When in Doubt)

Ambiguous / Meme Form	Clear Intent Version	Comment
`8 ÷ 2(2 + 2)`	`(8 ÷ 2) * (2 + 2)` or `8 ÷ (2 * (2 + 2))`	Pick one; parentheses remove debate.
`a + b / c + d`	`a + (b / c) + d`	Linear form needs grouping.
`3√9x`	`3 * √(9x)` or `(3√9) * x`	Root “bar” implicitly groups.
`12 ÷ 3 × 2` gripe	(Left → Right): `(12 ÷ 3) × 2 = 8`	MD are same precedence.
`x / 2y`	`x / (2y)` or `(x / 2) * y`	Implicit multiplication not “higher.”

Guideline: If someone could argue on a forum about it, rewrite it.

Bottom Line

Left to right for MD and AS.
Use parentheses to remove doubt.
“Math memes” exploit sloppy notation, not flawed logic.

Footnote & Discussion

The order of operations is not a mathematical law. It is a socially accepted convention that emerged from centuries of written algebra. Its structure reflects the symbolic hierarchy developed by early European mathematicians for clarity and consistency, not for any fundamental reason in arithmetic itself.

In that sense, PEMDAS functions much like grammar in language:

It governs how we interpret mathematical “sentences.”
It is widely accepted because we have standardized it through shared education and notation, though the terminology that describes it varies by location.

Someone outside our cultural framework, whether an alien civilization or a non-human intelligence, could adopt an entirely different system and still perform mathematics correctly.

They would simply be speaking a different mathematical dialect.