Evaluating expressions and the order of operations

Work out 4 + 3 · 7 from left to right and you get 49; multiply first and you get 25. Only one of them can be the value, so the order is agreed in advance.

The left-to-right habit gives 49

OpenStax opens its order of operations section with exactly this expression, 4 + 3 · 7, and two groups of students. The first group reads the way we read a sentence: 4 + 3 is 7, and 7 · 7 is 49. The second group multiplies first: 3 · 7 is 21, and 4 + 21 is 25. Both groups did correct arithmetic, so the disagreement is not about adding or multiplying. It is about which operation comes first. If the same expression could mean two different numbers, nobody could write a formula for anyone else to use. So mathematicians fixed one order, and the answer is 25.

The left-to-right habit is not silly. It is right for a row of additions and subtractions, and right for a row of multiplications and divisions. It goes wrong when the two kinds are mixed, because multiplication binds more tightly than addition. Think of 4 + 3 · 7 as 4 plus three sevens: the 3 · 7 is one chunk, a single amount, and the 4 is added to that chunk. Read that way, 25 stops looking like a rule to memorise and starts looking like the meaning of the expression.

The agreed order, step by step

1. Parentheses and other grouping symbols, innermost first.
2. Exponents.
3. Multiplication and division, from left to right.
4. Addition and subtraction, from left to right.

English-speaking classrooms remember the list as PEMDAS, from the silly sentence OpenStax quotes, “Please Excuse My Dear Aunt Sally”. The pairing matters more than the order of letters. “My Dear” sits together because multiplication and division have equal priority, and you do whichever comes first from the left. So 18 ÷ 9 · 2 is 2 · 2 = 4, not 18 ÷ 18 = 1. The same goes for “Aunt Sally”: 10 − 3 + 2 is 9, not 5. A longer example from the same book is 18 ÷ 6 + 4(5 − 2). The bracket gives 3, then 18 ÷ 6 is 3 and 4 · 3 is 12, and 3 + 12 = 15.

Brackets inside brackets are worked from the inside out. In 5 + 2³ + 3[6 − 3(4 − 2)], start with 4 − 2 = 2. The square bracket then holds 6 − 3 · 2 = 0, and three times zero is 0. The power 2³ is 8, so the whole thing is 5 + 8 + 0 = 13. Write each step on its own line under the one before. OpenStax recommends this habit of working down the page, and it is the easiest way to find where a mistake crept in, because each line is a complete, shorter expression with the same value as the line above.

Putting a number in place of the letter

To evaluate an expression means to find its value when the variable is replaced by a given number. You substitute, then simplify with the order of operations. For 9x − 2 with x = 5, the expression becomes 9 · 5 − 2 = 45 − 2 = 43. With x = 1 it becomes 9 − 2 = 7. Same expression, different values, because the value depends on the number you put in. Write the multiplication sign back in when you substitute. With x = 4, 3x is 3 · 4 = 12, and anyone who writes the 4 straight after the 3 sees 34, which is a different number.

Negative numbers go in with brackets. If x = −3, then x² is (−3)² = (−3) · (−3) = 9. The expression −x² is different: the power is done first and the minus sign after, so at x = −3 it is −9, and at x = 3 it is −9 as well. Brackets also protect subtraction: 5 − x at x = −2 is 5 − (−2) = 7. A spreadsheet is a handy way to evaluate one expression for many values at once, and the lab below asks you to do exactly that.