Written arithmetic is ambiguous until a reading order is fixed. The order everybody uses is a convention, not a theorem, and the two places it is misremembered are same-level operations and the base of an exponent.
Core ideas
Nothing in this lesson is a fact about numbers. All of it is a fact about notation, and it is agreed rather than discovered.
2 + 3 · 4 could mean 20 or 14, and both readings are coherent. The world settled on 14. Nothing forced that choice; what matters is that every book, calculator and piece of software makes the same one, so an expression names a single number.
Multiplication and division are one level, not two. So are addition and subtraction. 8 − 3 + 2 is 7, and 24 / 4 / 2 is 3. The familiar acronyms list the four operations separately and quietly suggest an order between them that does not exist.
Anything you want evaluated first goes in brackets. A fraction bar brackets its whole numerator and its whole denominator, and a radical sign brackets everything under it — which is why typing a printed fraction onto one line changes its value unless you add the brackets back.
The material
The four levels, the two operations that share a level, and the two things that look like grouping symbols because they are.
2 + 3 · 4 has to mean something definite or algebra cannot start. Read strictly left to right it is 20; read with multiplication first it is 14. The convention picks the second, and the reason is convenience rather than truth: it lets a polynomial like 3x² + 5x + 7 be written without a single bracket.
Evaluate in this order: brackets first, innermost outward; then exponents; then multiplication and division together, left to right; then addition and subtraction together, left to right.
Levels 3 and 4 each contain two operations, and within a level the rule is position, not the kind of operation.
Both wrong readings come from treating the last-named operation as the one to do first. Addition does not outrank subtraction and multiplication does not outrank division; they are read in the order they are written.
An exponent applies to the smallest thing written immediately before it. A minus sign written in front is not part of that thing unless brackets say so.
This is not a trick question and it is not a disagreement between sources; it follows from the exponent level coming before any multiplication, and −3² is (−1) · 3². The practical consequence arrives in “Evaluating Expressions”: when you substitute a negative value for a letter, write brackets round it. x² at x = −3 is (−3)² = 9.
A fraction bar groups everything above it and everything below it. When the same expression is typed on one line, those groupings have to be written as brackets or the value changes:
A radical sign groups in the same way. √(9 + 16) is √25 = 5, while √9 + √16 is 3 + 4 = 7. The bar over the radicand is a bracket drawn differently.
2 · (3 + 4)² − 10 / 5. The bracket gives 7; the exponent gives 49; at level 3, 2 · 49 = 98 and 10 / 5 = 2; at level 4, 98 − 2 = 96.
Interactive
Which operation goes first, and why
Method
Rewrite the whole expression after each step. Working in your head across two levels is where the value gets lost.
Find the bracket with no brackets inside it and evaluate it completely, then rewrite the expression without it. Repeat outward. Treat a fraction bar and a radical sign as brackets while you do this.
The base is the smallest thing the exponent is written on. −3² has base 3; (−3)² has base −3. These are −9 and 9.
One level. Sweep from the left and do whichever of the two you meet first. Do not do all the multiplications and then all the divisions.
One level again, and the same sweep. 20 − 3 + 1 is 18; grouping it as 20 − (3 + 1) gives 16 and is a misreading.
20 − 2 · (5 − 8)² / 6 + 1, rewritten in full at each stage.
The last level is where the direction mattered. Reading it as 20 − (3 + 1) gives 16, and that is the single most common way to lose this expression.
At the third level the direction happened not to matter for these numbers — 2 · (9 / 6) is also 3 — so this example does not demonstrate the rule there. 24 / 4 / 2 does: it is 3, and 24 / (4 / 2) is 12.
Had the expression read 20 − 2 · 5 − 8² / 6 + 1, the base of the exponent would be 8 and nothing else, and the answer would be different. Read the brackets before you read anything else.
Now do one with most of the guidance removed: 18 / 3 / 2 + (−2)². Write only the next operation on each line. The queue gives 18 / 3 = 6, then 6 / 2 = 3, and the answer is 7. An answer of 12 grouped the division from the right; an answer of −1 also lost the brackets around the negative base.
Evaluate −6² + 24 / 3 / 2, (−6)² + 24 / (3/2), and a fraction whose numerator and denominator each contain a sum. Rewrite the whole expression after every operation and label the invisible grouping made by the fraction bar. These shapes account for nearly every arithmetic disagreement later in the path.
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