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The Order of Operations: PEMDAS (page 1 of 2) If you are asked to simplify something like "4 + 2×3", the question that naturally arises is "Which way do I do this? Because there are two options!": Choice 1: 4 + 2×3 = (4 + 2)×3 = 6×3 = 18 Choice 2: 4 + 2×3 = 4 + (2×3) = 4 + 6 = 10 It seems as though the answer depends on which way you look at the problem. But we can't have this kind of flexibility in mathematics; math won't work if you can't be sure of the answer, or if the exact same problem can calculate to two or more different answers. To eliminate this confusion, we have some rules of precedence called "order of operations", the "operations" being addition, subtraction, multiplication, division, exponentiation, and grouping. A common technique for remembering the order of operations is the abbreviation "PEMDAS", which is turned into the phrase "Please Excuse My Dear Aunt Sally" (or, as my husband prefers, "Purple Elephants May Destroy A School"). It stands for "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction". This tells you the ranks of the operations: Parentheses outrank exponents, which outrank multiplication and division (but multiplication and division are at the same rank), and these two outrank addition and subtraction (which are together on the bottom rank). When you have a bunch of operations of the same rank, you just operate from left to right. For instance, 15 ÷ 3 × 4 is not 15 ÷ 12, but is rather 5 × 4, because, going from left to right, you get to the division first. If you're not sure of this, test it in your calculator, which has been programmed with the Order of Operations hierarchy. For instance, typesetting this into a graphing calculator, you will get:
Using the above hierarchy, we see that, in "4 + 2×3", Choice 2 is correct, because we have to do the multiplication before the addition. (Note: Speakers of British English often instead use "BODMAS", which stands for "Brackets, Orders, Division and Multiplication, and Addition and Subtraction". Since "brackets" are the same as parentheses and "orders" are the same as exponents, the two acronyms mean the same thing.) But PEMDAS can generate its own confusion, because students tend to apply the hierarchy as though all the operations in a problem are on the same "level", but often they're not. Many times it helps to work problems from the inside out, because often some parts of the problem are "deeper down" than other parts. The best way to explain this is to do some examples:
Do the exponent before trying to add in the 4: 4 + 32 = 4 + 9 = 13
You have to simplify inside the parentheses before you can take the exponent through: 4 + (2 + 1)2 = 4 + (3)2 = 4 + 9 = 13
Don't try to do parentheses from left to right; instead, work from the inside out: 4 + [–1(–2
– 1)]2 There is no particular significance in the use of square brackets (the "[" and "]" above) instead of parentheses. Brackets and curly-braces (the "{" and "}" characters) are used when there are nested parentheses, as an aid to keeping track of which parentheses go with which. The different grouping characters are used for convenience only. (This is similar to what happens in an Excel spreadsheet when you enter a formula using parentheses: each set of parentheses is color-coded, so you can tell the pairs.) Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved Here are some more examples:
Remember to simplify inside the parentheses first:
Then 4(–2/3 + 4/3) = 8/3
Simplify from the inside out: first the parentheses, then the square brackets. Also, be careful with taking the minus through the parentheses: remember that the –3 goes on everything inside the brackets! 4 – 3[4 –2(6
– 3)] ÷ 2
Remember that, in leiu of grouping symbols telling you otherwise, the division comes before the addition, which is why this simplified in the end as "4 + 3", and not "10 ÷ 2".
Remember to simplify inside the parentheses before you square, because (8 – 3)2 is not the same as 82 – 32. 16 – 3(8
– 3)2 ÷ 5
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