Why the rule is the rule

You can now use the order of operations confidently. This layer asks harder questions. Why does multiplication come before addition, and not the other way round? Why left to right for − and ÷? Where did brackets come from, and why do some old books use a bar instead? How do calculators and computers actually read an expression? And what happens at the edges: powers, fractions, the word "of", dividing by zero?

By the end you should be able to justify the rule, not just use it, and to simplify long expressions of the kind found in olympiad papers and Class 8 textbooks.

Multiplication is repeated addition: 3 × 4 means 4 + 4 + 4. So the expression 2 + 3 × 4 is short for

2 + 4 + 4 + 4 = 14.

Writing 3 × 4 is a compact way of writing a sum. When you see 2 + 3 × 4, the "3 × 4" is really a bundle of three 4s. It would be strange to break the bundle open and add the 2 to just the 3 (the count of the bundle), which is what (2 + 3) × 4 does: it means five 4s.

The same goes for money: 3 × ₹12 + 2 × ₹40 is ₹12 + ₹12 + ₹12 + ₹40 + ₹40. A bill is a sum of products, and sums of products are by far the most common shape of real-world calculations. The order of operations is designed so that this common shape needs no brackets at all.

There is a deeper link with the distributive property:

a × (b + c) = a × b + a × c

For example, 3 × (10 + 2) = 3 × 10 + 3 × 2 = 30 + 6 = 36. The right-hand side has no brackets, and it only means the right thing because × is done before +. The order of operations and the distributive property fit together: the distributive property tells you how to remove brackets, and the order of operations tells you how to read the result.

That is also why the syllabus pairs this topic with Properties of numbers. But note: the distributive property is a property (something always true about numbers); DMAS is a convention (an agreed way of reading). DMAS stands for Division, Multiplication, Addition, Subtraction, not "distributive".

In Class 7 you meet integers and fractions, and they give a beautiful explanation of the left-to-right rule.

Subtracting is adding the opposite. 10 − 3 is the same as 10 + (−3). So

10 − 3 + 2 = 10 + (−3) + 2.

Now everything is addition, and addition can be done in any order: 10 + (−3) + 2 = 10 + 2 + (−3) = 9. The only danger is forgetting that the − belongs to the 3. Going left to right is simply a safe way to keep each sign attached to its number.

Dividing is multiplying by the reciprocal. 8 ÷ 4 is the same as 8 × ¼. So

8 ÷ 4 × 2 = 8 × ¼ × 2 = 4,

in any order. The trap 8 ÷ (4 × 2) = 1 would be 8 × ¼ × ½: it has quietly turned the × 2 into × ½.

Just as multiplication is repeated addition, a power (also called an index or exponent) is repeated multiplication: 3² = 3 × 3 = 9 and 2³ = 2 × 2 × 2 = 8. You will study powers properly in Class 7 and 8, but they fit neatly into the order of operations.

By the same logic as before, a power is a compact bundle of multiplications, so it binds even more tightly than ×. The full order is:

1. Brackets
2. Powers (the I in BIDMAS, the E in PEMDAS; some Indian books fold this into the O of BODMAS as "orders")
3. × and ÷, left to right
4. • and −, left to right

• 2 + 3² = 2 + 9 = 11, but (2 + 3)² = 5² = 25.
• 2 × 3² = 2 × 9 = 18, but (2 × 3)² = 6² = 36.
• 10 − 2³ = 10 − 8 = 2.

Before round brackets became common, mathematicians grouped things with a horizontal line drawn over them, called a vinculum (Latin for "bond" or "chain"). A line over 5 − 2 meant exactly what (5 − 2) means today.

Some Indian textbooks still include the vinculum in "simplify" questions, and call it the bar. The full BODMAS-style order in those books is: bar first, then ( ), then { }, then [ ], then of, ÷, ×, +, −. Since the bar is always the innermost grouping, this is just "innermost first" again.

The vinculum is not extinct. You use it every day in two disguises:

• The fraction bar: in a stacked fraction with 6 + 4 on top and 2 below, the bar groups the whole top, so it means (6 + 4) ÷ 2 = 5, not 6 + 4 ÷ 2 = 8.
• The square-root sign: the line that stretches over the numbers in √(16 + 9) is a vinculum, so it means √25 = 5, not √16 + 9 = 13.

Olympiad and Class 8 questions often give long expressions with every feature at once. The method never changes:

1. Scan the whole expression. Mark every bracket pair and match them up.
2. Innermost first. Inside each bracket, use the normal order.
3. One step per line, rewriting the whole expression each time (you can copy the parts that did not change).
4. Look for shortcuts before grinding: the distributive property, cancelling, or noticing a factor of 0.
5. Check with an estimate or a second method.

A computer cannot "see" an expression the way you do. It turns it into a structure called an expression tree. The operation done last sits at the top; each operation has two branches for the two things it combines.

For 2 + 3 × 4:

• The top of the tree is + (it is done last).
• Its left branch is 2.
• Its right branch is ×, which has branches 3 and 4.

For (2 + 3) × 4 the tree is different: × at the top, with branches (+ with 2 and 3) and 4.

Working out the tree from the bottom up is exactly the order of operations. Brackets and the precedence rules are just a way of writing a tree in a single line. That is why the same expression must always give the same tree: otherwise two computers could get different answers.

In the 1920s the Polish logician Jan Łukasiewicz showed that if you write each operation before its two numbers (now called Polish notation), you need no brackets and no precedence rules at all. In the 1950s computer scientists turned it around, putting the operation after its two numbers. This is Reverse Polish Notation (RPN), or postfix:

• 2 + 3 × 4 becomes 2 3 4 × +
• (2 + 3) × 4 becomes 2 3 + 4 ×

To evaluate RPN, read left to right with a stack (a pile of numbers): push numbers onto the pile; when you meet an operation, take the top two numbers, combine them, and push the result back.

For 2 3 4 × +: pile 2; pile 2, 3; pile 2, 3, 4; × takes 3 and 4 → pile 2, 12; + takes 2 and 12 → 14.

Famous scientific calculators made by Hewlett-Packard used RPN, and computers still use stacks like this inside them to evaluate expressions.

Knowing the rule lets you prove things about numbers, not just compute. A few examples:

Claim: for any numbers, a − (b − c) = a − b + c. Reason: taking away (b − c) means taking away b, which is c too much (you only needed to take away b − c), so give c back: a − b + c.

Claim: 9 × n + n = 10 × n for every n. Reason: 9 × n + n = 9 × n + 1 × n = (9 + 1) × n = 10 × n, by the distributive property. (Try it: 9 × 7 + 7 = 63 + 7 = 70.)

Claim: the expressions (a + b) × c and a + b × c are equal only if a = 0 or c = 1. Reason: (a + b) × c = a × c + b × c. Setting it equal to a + b × c gives a × c = a, which forces a = 0 or c = 1.

In each case, the order of operations tells us exactly what the expression means, and properties of numbers do the rest.

The order of operations exists to save brackets. So here is a good test of understanding: given a fully bracketed expression, remove every bracket you can without changing the value for any numbers.

Rules of thumb that follow from the convention:

• Brackets around a × or ÷ chunk that is being added or subtracted can go: (a × b) + c → a × b + c.
• Brackets on the left of a partner chain can go, because left to right does them first anyway: (a − b) + c → a − b + c, and (a ÷ b) × c → a ÷ b × c.
• Brackets on the right of a − or ÷ must stay (unless you flip signs): a − (b + c) and a ÷ (b × c) need them.
• Brackets around a + or − that is then multiplied or divided must stay: (a + b) × c.
• Brackets around a whole sum that is added can go: a + (b + c) → a + b + c, because addition is associative.

The order of operations does not care what kind of numbers you use. Fractions, decimals and (later) negative numbers obey exactly the same levels. What changes is that the arithmetic inside each step is harder, so mistakes in the order are easier to miss.

• ½ + ⅓ × 6 = ½ + 2 = 2½. (× first: ⅓ × 6 = 2.) Reading left to right would give (½ + ⅓) × 6 = ⅚ × 6 = 5.
• 0.5 × 4 + 1.5 = 2 + 1.5 = 3.5.
• 2.4 ÷ 0.6 × 2 = 4 × 2 = 8 (left to right), not 2.4 ÷ 1.2 = 2.
• ¾ of (20 − 4) = ¾ × 16 = 12.

With fractions, the "of" and the fraction bar often do the bracketing for you: in a stacked fraction such as (3 + 5) over (6 − 2), the bar says: work out the whole top, work out the whole bottom, then divide: 8 ÷ 4 = 2.

In Class 6 and 7 you start writing rules with letters, called formulas. A formula is an expression waiting for numbers. The order of operations is what lets a formula be written without dozens of brackets.

An app-cab company charges ₹50 to start, ₹12 per km after the first 2 km, and ₹2 per minute:

fare = 50 + 12 × (d − 2) + 2 × t, where d is the distance in km and t the time in minutes.

For a 10 km, 25-minute ride: substitute d = 10 and t = 25, putting each number where its letter was:

50 + 12 × (10 − 2) + 2 × 25 = 50 + 12 × 8 + 50 = 50 + 96 + 50 = ₹196.

Notice the bracket (d − 2): without it, 12 × d − 2 would charge for every km and then take off just ₹2.

In Class 7 you meet integers, and the − sign starts doing two jobs: subtract and negative. The order of operations still works, but you must read each − carefully.

• 5 − 8 + 4: left to right, 5 − 8 = −3, then −3 + 4 = 1. (Or reorder: 5 + 4 − 8 = 1.)
• −2 × 3 + 10: the −2 is a negative number. −2 × 3 = −6, then −6 + 10 = 4.
• 10 − (−3): subtracting a negative is adding: 10 + 3 = 13.
• −3²: by the usual convention the power is done first, then the negative: −(3²) = −9. If you mean negative three, squared, write (−3)² = 9.
• 20 ÷ (−4) × 2: left to right, −5 × 2 = −10.

A good rule should say what happens in awkward corners.

• Dividing by zero. 12 ÷ (5 − 5) = 12 ÷ 0 has no value: no number times 0 gives 12. An expression that needs division by 0 is undefined, however it is bracketed.
• Zero divided by something. 0 ÷ 7 = 0 is perfectly fine.
• Non-whole results. In whole-number arithmetic, 7 + 5 ÷ 2 gives 7 + 2½ = 9½. The order of operations still works; the answer just is not whole.
• Negative results along the way. 3 − 5 + 4 = −2 + 4 = 2. With integers (Class 7) nothing breaks; if you only know whole numbers, you can reorder as 3 + 4 − 5 = 2.
• Unary minus. In −3 + 5, the − is not "subtract": it means negative three. The value is 2.
• Very long chains of the same operation. 2 − 2 − 2 − 2 = −4 by left to right; 64 ÷ 4 ÷ 4 ÷ 4 = 1.