You've seen it in your feed. A simple line of numbers and symbols. Which means 2 + 4 + 6 × 8. The comments section explodes. In real terms, people are certain* the answer is 96. Others scream 54. Someone brings up "new math" versus "old math." Another person insists calculators are lying That alone is useful..
It's just arithmetic. Third-grade stuff, technically. So why does it start arguments?
What Is This Expression Actually Asking
The expression "the sum of two and four and six times eight" translates to 2 + 4 + 6 × 8. On the flip side, that's it. That's why three numbers, two operators. But the phrasing — "sum of two and four and six times eight" — is where the trouble starts. Think about it: human language is ambiguous. Mathematical notation is not.
In plain English, "sum" suggests addition. "Times" suggests multiplication. On top of that, the second "and" connects that result to "six times eight. But the word "and" appears twice, doing different jobs. " Or does it connect four to six? Also, the first "and" connects two and four. Language doesn't have parentheses.
Mathematical notation solves this with a convention called order of operations. In practice, you learned it as PEMDAS or BODMAS or BEDMAS depending on where you went to school. Parentheses (or brackets), exponents (or orders), multiplication and division (left to right), addition and subtraction (left to right).
The official docs gloss over this. That's a mistake.
Multiplication binds tighter than addition. Practically speaking, always. But not sometimes. Here's the thing — not "unless you feel like adding first. " Always.
So 6 × 8 happens before anything gets added. Forty-eight. Then 2 + 4 + 48. Fifty-four Easy to understand, harder to ignore..
The Ambiguity of Natural Language
Here's the thing most people miss: the phrase "the sum of two and four and six times eight" is genuinely ambiguous in English. A linguist would tell you it has multiple valid parse trees Simple, but easy to overlook..
Parse one: (the sum of (two and four) and (six times eight)) → (2 + 4) + (6 × 8) = 54 Parse two: (the sum of two and four and six) times eight → (2 + 4 + 6) × 8 = 96 Parse three: the sum of two and (four and six times eight) → 2 + (4 + 6 × 8) = 54
Natural language doesn't enforce precedence. Plus, mathematical notation does. That's the entire point of the notation — to remove the ambiguity that language introduces That's the whole idea..
Why It Matters / Why People Care
You might think: who cares? In practice, it's a Facebook engagement bait post. But the underlying confusion reveals something real about how people understand — or don't understand — mathematical structure.
The Viral Math Problem Phenomenon
These problems go viral every few months. 8 ÷ 2(2 + 2). 6 ÷ 2(1 + 2). In real terms, 2 + 4 + 6 × 8. They rack up millions of comments. News sites write explainers. Mathematicians sigh Easy to understand, harder to ignore..
Why? Because they hit a sweet spot: simple enough that everyone feels qualified to answer, ambiguous enough (in natural language) that smart people disagree, and consequential enough that being "right" feels like it matters.
But the real stakes aren't about internet points. The confusion mirrors a genuine gap in math education. On top of that, many adults were taught order of operations as a memorized acronym, not as a logical consequence of how notation works. They remember "Please Excuse My Dear Aunt Sally" but not why multiplication comes before addition.
Real-World Consequences
This isn't just academic. Financial calculations drift. A developer who writes total = 2 + 4 + 6 * 8 expecting 96 gets 54. Bugs like this ship to production. Now, their code runs — silently, correctly, differently than intended. Programming languages follow the same precedence rules. Inventory counts go wrong.
Spreadsheets too. Think about it: excel and Google Sheets follow standard precedence. =2+4+6*8 returns 54. Here's the thing — if you meant 96, you need parentheses: =(2+4+6)*8. People build entire budgets on formulas they don't fully understand.
I've seen a small business owner calculate pricing as cost + markup * quantity when they meant (cost + markup) * quantity. The difference was thousands of dollars over a year. They blamed the spreadsheet. The spreadsheet was following rules they never learned.
How It Works: Order of Operations Deep Dive
Let's break down why the convention exists. It's not arbitrary. Practically speaking, it's not a conspiracy by textbook publishers. It emerges from how we write polynomials and how multiplication relates to addition It's one of those things that adds up..
Multiplication as Repeated Addition
Multiplication is shorthand. 6 × 8 means 8 + 8 + 8 + 8 + 8 + 8. Six groups of eight.
2 + 4 + (8 + 8 + 8 + 8 + 8 + 8)
The parentheses are implicit in the notation. The multiplication is a grouped sum. Which means it would be bizarre to add the 2 and 4 into the middle of that repeated addition. You'd be breaking the group.
Polynomials Need This Convention
Consider a polynomial: 3x² + 5x + 7
When x = 2, this becomes 3(2)² + 5(2) + 7
If addition happened before multiplication, you'd need parentheses everywhere: (3 × (2²)) + (5 × 2) + 7. Every term would need wrapping. The convention lets us write polynomials cleanly — the most common algebraic expressions — without clutter Easy to understand, harder to ignore. Still holds up..
The convention serves the notation's primary use case. That's not accidental.
Left-to-Right for Same Precedence
Multiplication and division share precedence. Think about it: addition and subtraction share precedence. When operators share precedence, you evaluate left to right.
8 ÷ 4 × 2 = (8 ÷ 4) × 2 = 2 × 2 = 4
Not 8 ÷ (4 × 2) = 8 ÷ 8 = 1.
This matters for 2 + 4 + 6 × 8 only in the final addition step: 2 + 4 + 48. Consider this: right to left would give 2 + (4 + 48) = 54. Day to day, left to right gives (2 + 4) + 48 = 54. Still, left to right: (8 - 4) + 2 = 6. But 8 - 4 + 2? Right to left: 8 - (4 + 2) = 2. Addition is associative, so it doesn't change the result here. Different answers. The convention picks one.
Implicit Multiplication Controversy
This is where the convention gets messy. This is implicit multiplication* — no symbol between the number and the variable. In algebra, we often write 2x, meaning 2 × x. Many people, especially those trained in mathematics, treat this as having higher precedence than explicit division.
Consider 1/2x. Now, is this (1/2)x or 1/(2x)? Now, the algebraic interpretation often leans toward the latter. The 2x is seen as a single, tightly-bound term. This is why calculators and some programming languages give different answers for 6÷2(1+2) — the 2(1+2) is interpreted by some as a grouped unit Worth knowing..
Still, in most programming languages and strict computational contexts, implicit multiplication does not exist. You must write the * explicitly. And when you do, it follows standard precedence. The ambiguity is a clash between human algebraic notation and machine-parsable rules.
The Real Lesson
The order of operations isn't a sacred law. Also, it's a set of social agreements that make mathematical communication efficient. It solves a real problem: how to interpret a string of symbols without a forest of parentheses.
The goal isn't memorization. It's understanding why the rules exist so you can apply them with judgment.
Conclusion: Write for Humans, Not Just Machines
The convention exists to serve clarity. Plus, when you're writing code or a spreadsheet formula, you are communicating with both a computer and your future self. The computer will follow the rules exactly. Your future self, or a colleague, will try to understand your intent.
It sounds simple, but the gap is usually here.
The most solid approach is to use parentheses liberally. Write (2 + 4 + 6) * 8 even if you're certain of the precedence. Also, it removes any ambiguity and makes your intent unmistakable. It's the difference between relying on a convention everyone should know and ensuring your meaning is clear beyond all doubt Most people skip this — try not to..
In the end, the order of operations is a tool. Used with understanding, it's the foundation of clear technical communication. But used without thought, it leads to silent errors. The parentheses are not a sign of weakness; they are the hallmark of a careful writer.
This changes depending on context. Keep that in mind.
