PEMDAS and BODMAS are the same rule wearing different words. Parentheses or Brackets, Exponents or Orders, then Division and Multiplication, then Addition and Subtraction. Four tiers, not six — and the acronym causes as much trouble as it prevents, because it lists six letters and people read them a...
The M does not outrank the D. PEMDAS lists multiplication before division because the word fits, not because it goes first — the two share one tier and run left to right, which is why 8 ÷ 2 × 4 is 16 and not 1. The same applies to the A and the S: 10 − 4 + 3 is 9, not 3. That misreading is not a rare edge case. Across randomly generated expressions containing both a × and a ÷, taking the acronym literally changed the answer 36.8% of the time. Every alternative reading on this page is a real parser with its own precedence table, run against your expression — not a canned wrong answer.
8 ÷ 2 × (2 + 2)
16
3 operations, carried out in the order shown on the right. Worked in exact fractions, so nothing was rounded on the way.
OTHER READINGS OF THE SAME LINE
1 — the acronym taken literally — M before D, A before S
Each of these is a real parse with its own precedence table, not a guess at a wrong answer. All of them are misreadings of the standard rule.
OPERATIONS
3
things to actually do
FIRST TO HAPPEN
2 + 2
innermost brackets before anything outside them
EXACT VALUE
16
a whole number
READINGS THAT DIFFER
1
shown above
WHAT IS HAPPENING
2 + 2 = 4 — innermost brackets before anything outside them
8 ÷ 2 = 4 — same tier, so left to right
4 × 4 = 16 — the last operation left
The tiers are not a queue. The rule is not that every multiplication happens before every addition anywhere in the line — it is that each operation waits for the branches feeding it. Two multiplications on opposite sides of a plus are independent, and the addition between them waits for both. Brackets do not add a fifth tier; they cut the expression into pieces that each get the same four.
Worked in exact fractions · every alternative reading is a real parse, not a canned answer
Type the expression, or tap one of the examples. Numbers, brackets and + − × ÷ ^ are all understood, and * and / work too if that is what your keyboard offers.
Read the steps on the right. Each one shows the whole expression with the piece being worked on highlighted, what it became, and why that operation went before everything else still waiting.
Look at the tree underneath. The numbers on the operations are the order they happened, and the shape shows why: nothing can be carried out until both branches below it are numbers.
Check the amber panel if it appears. It lists every other reading that would give a different answer — each one produced by a real parser, so you can see exactly what a misreading costs.
Take 6 ÷ 2 + 3 × (5 − 1)². Brackets first, wherever they are and whatever they contain: 5 − 1 is 4. The expression is now 6 ÷ 2 + 3 × 4². Note that the subtraction inside the brackets went first even though subtraction is the lowest tier — brackets outrank tiers entirely. Powers next: 4² is 16, giving 6 ÷ 2 + 3 × 16. Now the multiply-and-divide tier, left to right. The division is further left, so 6 ÷ 2 is 3 before the multiplication happens at all. Then 3 × 16 is 48. The expression is 3 + 48. Addition last: 51. Five operations, and the order was decided three different ways — brackets for the first, tier for the second and fourth, position for the third. That is the entire rule. Now the famous one: 8 ÷ 2(2+2). Brackets first: 2 + 2 is 4, leaving 8 ÷ 2(4). The multiplication here is implied — nobody wrote a × — and that is where the argument starts. Reading the line strictly left to right, ÷ and the implied × share a tier, so 8 ÷ 2 is 4 and then 4 × 4 is 16. Reading juxtaposition as binding tighter than the division, 2(4) is 8 and then 8 ÷ 8 is 1. Both are defensible. The strict left-to-right reading is what calculators and programming languages implement. The tighter reading is what many physics and mathematics journals use for expressions like 1/2π, where writing 1/(2π) every time would be tiresome. What is not defensible is insisting the other side is stupid: the expression is ambiguous, and the fix is to write (8 ÷ 2)(2+2) or 8 ÷ (2(2+2)) and remove the argument. One more worth doing by hand: −3². The power binds tighter than the leading minus, so this is −(3²) = −9. If you want the minus included in the base you have to write (−3)², which is 9. Spreadsheets take the opposite view and return 9 for −3^2, which means a formula copied out of a textbook can quietly change value on its way into a cell.
| Rule | What it says | Why |
|---|---|---|
| The tiers | brackets · powers · × ÷ · + − | Four levels, not six. PEMDAS and BODMAS name the same four. |
| PEMDAS = BODMAS | P/B and E/O are the same | Parentheses or Brackets, Exponents or Orders. Different words, identical rule. |
| × and ÷ rank equally | 8 ÷ 2 × 4 = 16 | The M before the D in the acronym is alphabetical accident, not priority. |
| + and − rank equally | 10 − 4 + 3 = 9 | Same trap, lower tier. Not 3. |
| Within a tier | left to right | The only tie-break there is. It decides both cases above. |
| Brackets first | innermost outward | A bracket outranks everything outside it, whatever tier it holds. |
| Powers group right | 2^3^2 = 2^9 = 512 | Right to left, unlike every other operator. Not 64. |
| A leading minus | −3² = −9 | The power binds tighter, so it applies to 3 alone. (−3)² is 9. |
| Spreadsheets differ | −3^2 = 9 in Excel | Excel and Google Sheets give the minus priority. Write (−3)^2 or −(3^2). |
| Implied × | 2(3+4) means 2 × 7 | Whether it binds tighter than a preceding ÷ is genuinely disputed. |
| The viral one | 8 ÷ 2(2+2) | 16 read strictly left to right; 1 if juxtaposition binds tighter. Write brackets. |
| A fraction bar | groups on its own | Everything above the bar is one bracket, everything below another. |
| Same tier, no reorder | a − b + c ≠ a − (b + c) | Rearranging a subtraction chain changes the value. |
| When in doubt | add brackets | An expression needing a convention to settle it is badly written, not clever. |
PEMDAS is Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. BODMAS is Brackets, Orders, Division and Multiplication, Addition and Subtraction. They are the same rule — parentheses and brackets are the same thing, exponents and orders are the same thing, and the two middle pairs each share a single tier. BIDMAS and BEDMAS are the same again with Indices or Exponents in the middle. Four tiers, whichever acronym you were taught.
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Last updated: August 5, 2026 · Evaluated in exact fractions, never floating point · × and ÷ share a tier, as do + and −.