Unit 1 · Lesson 1.3

Order of operations

Read 2 + 3 × 4 two different ways and you get two different answers. Add first: 2 + 3 is 5, then 5 × 4 is 20. Multiply first: 3 × 4 is 12, then 2 + 12 is 14. Both used real arithmetic, but they can't both be right, because a written expression has to mean exactly one thing. So everyone agrees on one order to read it in, and that agreement is the order of operations.

Here's the order, top to bottom:

1. Parentheses and other grouping symbols
2. Exponents
3. Multiply and divide, equally important, done left to right
4. Add and subtract, equally important, done left to right

Within the bottom two lines you work in reading order, left to right, the way you read a sentence. Multiply and divide aren't ranked against each other; they're equal, so you just take them in the order they come. Same for add and subtract.

There's a short word that holds this whole order in your head: PEMDAS. It's a memory hook, one letter for each operation in the order you handle them. P is for Parentheses, E for Exponents, M for Multiplication, D for Division, A for Addition, and S for Subtraction. Say it once or twice and it sticks, and then the letters remind you what comes before what.

The one thing to know about PEMDAS is that the six letters aren't six separate steps. M and D are a single step, and so are A and S, exactly like the four-line list above. The clearest way to hold that is to stack the letters in four rows, one row per line of the list:

• P
• E
• MD
• AS

You read the rows top to bottom: parentheses first, then exponents, then the MD row, then the AS row. Inside the MD row, multiply and divide are equal, so you take them left to right; inside the AS row, add and subtract are equal, so you take those left to right too. Stacking it this way is what keeps PEMDAS from looking like six ranked steps, when it's really four.

Now watch the order work on the spots that trip people, reading each one left to right within its line:

• 12 ÷ 2 × 3. Divide and multiply are on the same line, so read left to right: 12 ÷ 2 is 6, then 6 × 3 is 18.
• 8 − 3 + 2. Subtract and add are on the same line too, so left to right again: 8 − 3 is 5, then 5 + 2 is 7.
• 2 × 3². Exponents come before multiplying, so square first: 3² is 9, then 2 × 9 is 18.

Grouping, the top line, can change the answer outright. Compare 2 + 3 × 4 = 14 with (2 + 3) × 4 = 20: same digits, but the parentheses force the addition first. Square brackets [ ] group exactly the way parentheses do.

And the fraction bar is a grouping symbol in disguise: in (12 − 2)/(3 + 2) the bar means "finish the whole top and the whole bottom first, then divide," giving 10/5, then 2.