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Order of Operations (PEMDAS)

The Story Behind the Math

In the 16th century, mathematicians wrote equations like this: \(3 + 4 \times 5\). But they disagreed on what it meant. Some calculated \((3 + 4) \times 5 = 35\). Others calculated \(3 + (4 \times 5) = 23\).

The chaos: Different countries, different mathematicians, different conventions. A formula in a French text might mean something different in an English text. This wasn't just confusing—it was dangerous for engineering and navigation where precise calculations mattered.

The solution emerged gradually. By the 19th century, a consensus formed: multiplication before addition. But it wasn't arbitrary. Mathematicians realized that following this order preserved algebraic structure and made equations consistent.

The modern convention (PEMDAS/BODMAS) wasn't decreed by a king—it evolved because it works best with how we structure mathematics.

Why It Matters

Without agreed-upon order of operations:

  • Scientific formulas would be ambiguous
  • Computer programs would produce different results on different systems
  • Communication would break down
  • Calculators and programming languages couldn't exist

Every programming language, spreadsheet, and calculator follows these rules. Understanding why helps you catch errors, read formulas correctly, and write unambiguous expressions.

Prerequisites

The Core Insight

Why Multiplication Comes Before Addition

Consider the expression: \(3 + 4 \times 5\)

Interpretation 1 (left-to-right): \((3 + 4) \times 5 = 35\)

Interpretation 2 (multiplication first): \(3 + (4 \times 5) = 23\)

Which is more natural? Think about what the expression represents: - "3 apples plus 4 baskets of 5 apples each" - Total: 3 + 20 = 23 apples

Multiplication represents groups. Addition combines different quantities. We evaluate groups first, then combine them.

The Hierarchy of Operations

Operations form a natural hierarchy based on what they represent:

  1. Parentheses — Override everything (grouping)
  2. Exponents — Repeated multiplication
  3. Multiplication/Division — Groups/scaling
  4. Addition/Subtraction — Combining quantities

Each level is "stronger" than the one below it.

The Complete Rules

PEMDAS/BODMAS

Different mnemonics, same rules: - PEMDAS (US): Parentheses, Exponents, Multiplication, Division, Addition, Subtraction - BODMAS (UK): Brackets, Orders, Division, Multiplication, Addition, Subtraction

Critical detail: Multiplication and division are equal priority (left-to-right). Same for addition and subtraction.

Step-by-Step Examples

Example 1: \(3 + 4 \times 5\)

  1. Multiplication first: \(4 \times 5 = 20\)
  2. Then addition: \(3 + 20 = 23\)
\[ 3 + 4 \times 5 = 3 + 20 = 23 \]

Example 2: \((3 + 4) \times 5\)

  1. Parentheses first: \(3 + 4 = 7\)
  2. Then multiplication: \(7 \times 5 = 35\)
\[ (3 + 4) \times 5 = 7 \times 5 = 35 \]

Example 3: \(12 - 8 \div 4 + 2\)

  1. Division first: \(8 \div 4 = 2\)
  2. Left to right for addition/subtraction:
  3. \(12 - 2 = 10\)
  4. \(10 + 2 = 12\)
\[ 12 - 8 \div 4 + 2 = 12 - 2 + 2 = 10 + 2 = 12 \]

The Mathematical Foundation

Why This Order?

Exponents before multiplication:

\(2 \times 3^2\) means \(2 \times (3 \times 3) = 18\), not \((2 \times 3)^2 = 36\)

Exponents are repeated multiplication, so they bind tighter.

Multiplication before addition:

Think of \(3 + 2 \times 4\) as "3 singles plus 2 groups of 4". The groups are evaluated first.

The Distributive Property Connection

The order of operations is consistent with the distributive property:

\[ 3 \times (4 + 5) = 3 \times 4 + 3 \times 5 = 12 + 15 = 27 \]

If we did addition first inside: \(3 \times 9 = 27\) ✓

This works because multiplication distributes over addition. The order respects this algebraic structure.

Left-to-Right for Equal Precedence

Why left-to-right when operations have equal priority?

Consider: \(10 - 5 - 2\)

  • Left-to-right: \((10 - 5) - 2 = 3\)
  • Right-to-left: \(10 - (5 - 2) = 7\)

Left-to-right treats it as sequential removal: first remove 5, then remove 2.

For division: \(8 \div 4 \div 2\)

  • Left-to-right: \((8 \div 4) \div 2 = 1\)
  • Right-to-left: \(8 \div (4 \div 2) = 4\)

Left-to-right means "divide by 4, then divide that result by 2."

Common Traps and Misconceptions

Trap 1: Implicit Multiplication

\(2 \div 3x\) — does this mean \((2 \div 3) \times x\) or \(2 \div (3x)\)?

Some calculators and mathematicians treat implicit multiplication (juxtaposition) as higher priority. But PEMDAS says left-to-right: \((2 \div 3) \times x\).

Solution: Always use explicit parentheses!

Trap 2: The Division Symbol

\(6 \div 2(1 + 2)\) — is this 1 or 9?

Strict PEMDAS: 1. Parentheses: \(6 \div 2(3)\) 2. Left-to-right for multiplication/division: \(3(3) = 9\)

But many interpret \(2(3)\) as having implied higher priority, giving 1.

The controversy: Viral math problems like this get millions arguing because conventions vary!

Solution: Write clearly: - \(\frac{6}{2(1+2)} = 1\) (fraction bar groups denominator) - \(\frac{6}{2} \times (1+2) = 9\) (explicit)

Trap 3: Exponents and Negatives

\(-3^2\) — is this \(-9\) or \(9\)?

The exponent applies only to 3, not the negative sign:

\[ -3^2 = -(3^2) = -9 \]

If you want \((-3)^2 = 9\), you need parentheses!

Common Error: Following PEMDAS Literally

PEMDAS lists M before D, but they're equal:

\(8 \div 4 \times 2\)

  • Wrong (M first): \(8 \div (4 \times 2) = 1\)
  • Right (left-to-right): \((8 \div 4) \times 2 = 4\)

Similarly for addition/subtraction.

Real-World Applications

Spreadsheet Formulas

Excel, Google Sheets, etc. follow these rules exactly:

=A1+B1*C1      ← Multiplies B1*C1 first, then adds A1
=(A1+B1)*C1    ← Adds first, then multiplies
=A1^2+B1       ← Squares A1 first, then adds B1

Programming Languages

Python, JavaScript, Java—all follow order of operations:

result = 3 + 4 * 5      # 23, not 35
result = (3 + 4) * 5    # 35
result = 2 ** 3 + 1     # 9 (exponent first)
result = 10 / 2 * 3     # 15 (left-to-right)

Scientific Formulas

Einstein's mass-energy: \(E = mc^2\)

This means \(E = m \times (c^2)\), not \(E = (mc)^2\). The exponent binds to \(c\) only.

Quadratic formula: \(x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}\)

Everything in the numerator is grouped by the fraction bar. The denominator \(2a\) means \(2 \times a\).

Visualizing the Order

The "Nesting" View

Think of operations as nesting dolls:

Innermost: Parentheses
    ↓
Next: Exponents
    ↓
Next: Multiplication/Division (tie)
    ↓
Outermost: Addition/Subtraction (tie)

The "Binding Strength" View

Strongest binding (tightest grip on numbers): 1. Parentheses — "Force this to happen first" 2. Exponents — "Stick to the base" 3. Multiplication/Division — "Group things together" 4. Addition/Subtraction — "Loosest connection"

Why Not Just Always Use Parentheses?

We could! In Lisp and some computer languages, everything is explicit: (+ 3 (* 4 5))

But mathematical notation evolved to be readable. Writing \((3 + (4 \times 5))\) every time is tedious. The convention lets us write \(3 + 4 \times 5\) and know exactly what it means.

It's like grammar in language—rules let us communicate efficiently without marking everything explicitly.

Teaching Tips

For Students

  1. Think about meaning: "3 + 4 groups of 5" vs "3 + 4, then groups of 5"
  2. Use parentheses liberally when unsure
  3. Check with real values: Does your answer make sense?

Common Classroom Errors

  • Memorizing PEMDAS without understanding why
  • Thinking M always comes before D
  • Forgetting that parentheses can contain any expression
  • Not recognizing that fraction bars group

References

  • Mazur, B. (2014). "When is a thing equal to some other thing?" In Circles Disturbed: The Interplay of Mathematics and Narrative.
  • Pólya, G. (1945). How to Solve It. Princeton University Press.
  • Common Core State Standards for Mathematics, Grade 3-5, Operations and Algebraic Thinking.