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Order of operations

Numbers · level 4, lesson 7 of 7

A question to start

What is \(2 + 3 \times 4\)?

Ask a room of people and you will hear two answers. Some say 20: "2 plus 3 is 5, times 4 is 20." Others say 14: "3 times 4 is 12, plus 2 is 14." Both did correct arithmetic. They just did it in a different order.

A calculation that means different things to different people is useless: imagine a recipe, a bank, or a rocket engineer getting it wrong. So mathematicians all over the world use one agreement about the order. With it, \(2 + 3 \times 4 = 14\), always. This lesson is about that agreement, and why it was chosen this way.

The idea

A convention, not a law of nature

Some things in mathematics are true whatever anyone decides: \(3 \times 4\) is 12 no matter what. Other things are conventions: agreements, like driving on the right side of the road in Canada, or reading a page from left to right. A convention could have been chosen differently, but it is very important that everyone uses the same one.

The order of operations is a convention. But it is not a random one: there are good reasons behind each part.

The agreed order

Work out a calculation in this order:

  1. Brackets first: whatever is inside ( ) is worked out before anything else. Brackets say "do me first".
  2. Exponents (powers) next. \(5^2\) means \(5 \times 5 = 25\), and \(2^3\) means \(2 \times 2 \times 2 = 8\). The small raised number says how many copies to multiply. (Much more about powers in Level 5.)
  3. Division and Multiplication, working from left to right.
  4. Addition and Subtraction, working from left to right.

In Canada this is often remembered as BEDMAS (Brackets, Exponents, Division, Multiplication, Addition, Subtraction). Other countries say PEMDAS (Parentheses ...) or BODMAS (Brackets, Orders ...). They all mean the same order.

Careful: the word BEDMAS makes it look as if division always comes before multiplication, and addition before subtraction. It does not. D and M are equal partners, done together from left to right. A and S are also equal partners, done together from left to right. A better picture is four steps, not six:

Step What How
1 Brackets inside first (innermost brackets first)
2 Exponents powers
3 × and ÷ together, left to right
4 + and − together, left to right

Why multiplication before addition?

Think about what \(2 + 3 \times 4\) is usually describing. You buy a 2-dollar pen and 3 books at 4 dollars each. The cost is "2, plus 3 lots of 4". The "3 lots of 4" belongs together: it is one amount, 12 dollars. So the total is \(2 + 12 = 14\).

Multiplication is a shortcut for repeated addition: \(3 \times 4\) is short for \(4 + 4 + 4\). So \(2 + 3 \times 4 = 2 + 4 + 4 + 4 = 14\). Doing the multiplication first respects what it stands for. That is why multiplication "holds on tighter" than addition, and powers (repeated multiplication) hold on tighter still.

A picture helps. Every calculation has a shape, like a tree. The operation done last sits at the top:

+×234 ×+423 2 + 3 × 4 = 14(2 + 3) × 4 = 20

You always work from the bottom of the tree up. Without brackets, the agreement builds the left tree. If you want the right tree, you must write brackets: \((2 + 3) \times 4 = 20\). That is the job of brackets: they let you override the normal order.

Why left to right for × and ÷?

What is \(6 \div 2 \times 3\)?

  • Left to right: \(6 \div 2 = 3\), then \(3 \times 3 = 9\).
  • If you did the × first: \(2 \times 3 = 6\), then \(6 \div 6 = 1\).

The agreement says 9. Here is a reason it is the sensible choice. In lesson 3 of this level you saw that dividing by 2 is the same as multiplying by \(\tfrac12\). So

\[6 \div 2 \times 3 = 6 \times \tfrac12 \times 3\]

Now it is all multiplication, and multiplication can be done in any order: \(6 \times \tfrac12 \times 3 = 9\) whichever two you multiply first. The answer 9 is the one that keeps the meaning "divide 6 by 2". Doing \(2 \times 3\) first would secretly change the question to \(6 \div (2 \times 3)\).

The same goes for subtraction. \(10 - 4 + 3\): left to right gives \(6 + 3 = 9\). And subtracting 4 is adding −4 (last lesson), so it is \(10 + (-4) + 3 = 9\), in any order. Doing \(4 + 3\) first would give \(10 - 7 = 3\), which secretly changes the question to \(10 - (4 + 3)\).

Hidden brackets: the fraction bar

A fraction bar groups everything above it and everything below it, like invisible brackets:

\[\frac{8 + 4}{2 + 1} = \frac{12}{3} = 4\]

It means \((8 + 4) \div (2 + 1)\), not \(8 + 4 \div 2 + 1\) (which is \(8 + 2 + 1 = 11\)). When you type a fraction into a calculator on one line, you must type the brackets yourself.

Nested brackets

When brackets are inside brackets, work from the innermost outwards:

\[3 \times (4 + 2 \times (5 - 1)) = 3 \times (4 + 2 \times 4) = 3 \times (4 + 8) = 3 \times 12 = 36\]

Inside each bracket, the usual order applies: in \((4 + 2 \times 4)\), the multiplication comes before the addition.

Checking with a calculator

A scientific calculator (and the calculator on most phones, turned sideways) follows the order of operations: type 2 + 3 × 4 = and it shows 14. But some very simple calculators work out each step as soon as you press the next key, and show 20. Test yours with \(2 + 3 \times 4\) before you trust it.

The safest habit: when in doubt, add brackets. Extra brackets never change a correct answer, and they make your meaning clear to everyone. You may have seen puzzles online like \(6 \div 2(1 + 2)\) that start big arguments; they are written badly on purpose. With a clear sign, \(6 \div 2 \times (1 + 2) = 9\) by the agreement, but a careful writer would avoid the doubt with brackets: \((6 \div 2) \times (1 + 2)\) or \(6 \div (2 \times (1 + 2))\).

Worked examples

1. \(20 - (3 + 2) \times 3\). Brackets: \(3 + 2 = 5\). Then ×: \(5 \times 3 = 15\). Then −: \(20 - 15 = 5\).

2. \(5 + 2^3 \times 2\). Exponent: \(2^3 = 8\). Then ×: \(8 \times 2 = 16\). Then +: \(5 + 16 = 21\).

3. \(18 \div 3 \times 2 - 4\). × and ÷ left to right: \(18 \div 3 = 6\), \(6 \times 2 = 12\). Then \(12 - 4 = 8\).

4. \(48 \div (2 \times (9 - 3))\). Innermost: \(9 - 3 = 6\). Then \(2 \times 6 = 12\). Then \(48 \div 12 = 4\).

5. \(100 - 5^2 \times 3\). Exponent: \(5^2 = 25\). Then \(25 \times 3 = 75\). Then \(100 - 75 = 25\).

6. Where do the brackets go? Put brackets into \(12 - 2 \times 3 + 1\) to make the answer 40.

Without brackets: \(12 - 6 + 1 = 7\). Try \((12 - 2) \times (3 + 1) = 10 \times 4 = 40\). ✓

Try it

  1. \(7 + 6 \times 5\).
  2. \((7 + 6) \times 5\).
  3. \(36 \div 6 \times 3\) and \(36 \div (6 \times 3)\).
  4. \(15 - 6 + 4\) and \(15 - (6 + 4)\).
  5. \(2 \times (3 + 4)^2 - 10\).
  6. \(\dfrac{30 - 6}{2 \times 4}\). Then write it on one line, with brackets, as you would type it into a calculator.
  7. Put brackets into \(8 + 4 \times 3 - 1\) to make (a) 35 (b) 16.
  8. Harder: Using each of 2, 3, 4 and 5 once, with any operations and brackets, make 1, then make 50.
Answers
  1. × first: \(6 \times 5 = 30\), then \(7 + 30 = 37\).
  2. Brackets first: \(13 \times 5 = 65\).
  3. Left to right: \(36 \div 6 = 6\), \(6 \times 3 = 18\). With brackets: \(36 \div 18 = 2\).
  4. Left to right: \(15 - 6 = 9\), \(9 + 4 = 13\). With brackets: \(15 - 10 = 5\).
  5. Bracket: \(3 + 4 = 7\). Exponent: \(7^2 = 49\). ×: \(2 \times 49 = 98\). −: \(98 - 10 = 88\).
  6. Top: \(30 - 6 = 24\). Bottom: \(2 \times 4 = 8\). \(24 \div 8 = 3\). On one line: (30 - 6) / (2 * 4), or \((30 - 6) \div (2 \times 4)\). Without the brackets a calculator would do \(30 - 6 \div 2 \times 4 = 30 - 12 = 18\).
  7. (a) \((8 + 4) \times 3 - 1 = 12 \times 3 - 1 = 36 - 1 = 35\). ✓ (b) \(8 + 4 \times (3 - 1) = 8 + 4 \times 2 = 8 + 8 = 16\). ✓
  8. Many answers work. For 1: \((5 - 4) \times (3 - 2) = 1 \times 1 = 1\). For 50: \((4 + 3 \times 2) \times 5 = (4 + 6) \times 5 = 50\). Each number is used exactly once in each.

Common mistake

Reading BEDMAS as six separate steps. In \(12 - 4 + 2\), people who think "A before S" do \(4 + 2 = 6\) first and get 6. The right answer is \(12 - 4 = 8\), then \(8 + 2 = 10\). Addition and subtraction are partners, done left to right. The same goes for multiplication and division: \(6 \div 2 \times 3 = 9\), not 1.

Typing a fraction without brackets. \(\tfrac{30 - 6}{2 \times 4}\) is \((30 - 6) \div (2 \times 4)\). Typed as 30 - 6 / 2 * 4, a calculator gives 18 instead of 3.

Where it leads