# why do we need order of operations

I ve seen this question answered before on reddit (possibly on, which would be a better place for this question) but can t find it right now. Excuse the long answer - I ve tried to summarise it in a TL;DR below. Essentially we use PEDMAS because we ve found it to be useful in arithmetic and algebra (although there are areas of mathematics where this isn t necessarily the case). There s nothing to stop us from using, say,

PSAMDE if we wanted to, but things would get very messy if we did. Let s just consider the DMAS bit. Why do multiplication and division come before addition and subtraction? Because it makes sense to do it that way. I might send you out to buy me three half-dozen boxes of eggs and two boxes containing a dozen. The total number of eggs is 3 x 6 + 2 x 12. The real-life situation this describes requires us to interpret this as (3 x 6) + (2 x 12), or 42 in total, rather than 3 x (6 + 2) x 12. Multiplication before addition occurs naturally all the time, so it makes sense to do the operations in that order. Furthermore, PEDMAS allows us to simplify algebra. We can write an expression like: and we know this means we have compute a x a x 4 and 5 x b, add these together and add 1. If the order were SAMDEP, this would have to be written as: which is less easy to read. Why do things work out this way? Well, multiplication is really repeated addition, and exponentiation is just repeated multiplication.

Suppose a = 3 in the above expression, and we expand it out: Now we have only one operation so we can do the additions in any order, but you can see that if we go backwards to the original expression, each time we collect up addends into a multiplication, we get a single product that needs to be added to another result. So we end up adding together products, meaning multiplication must come before addition. Exponentiation bundles together multiplicands ready for multiplication by other terms, hence the exponentiation needs to be done before the multiplication. If we consider integers only, division can be viewed as just repeated subtraction, and subtraction is just addition of negative terms, hence division comes at the same level as multiplication and subtraction at the same level as addition. Parentheses give us a way of overriding the existing order, so P has to come before everything else so we can more easily solve word problems like the following: How many ounces of vegetables are there in three bags of mixed vegetables each containing four ounces of carrots and six ounces of peas? (Answer: 3 x (4 + 6) oz = 3 x 10 oz = 30 oz. ) Without parentheses, we would have to write 3 x 4 + 3 x 6, essentially expanding the parentheses. Imagine if the parentheses contained some much more complicated expression - we would need to write it out in full several times over if parentheses weren t available.

TL;DR: For integers, exponentiation is repeated multiplication and collects up multiplicands ready for multiplication by or addition to other terms, while multiplication is repeated addition and collects up addends for addition to other terms. Hence it is useful to do exponentiation before multiplication (and division), and multiplication before addition (and subtraction). Parentheses give a way of overriding the order. EDIT 2: P must come first, whatever the order, or else parentheses are useless EDIT 3: Gasp! Someone s given me Reddit Gold (thank you, that person) AND this thread has hit the front page! EDIT 4: Some clarifications of disputed points 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 expression can be calculated so that you can arrive at two or more different answers. To eliminate this confusion, we have some rules of precedence, established at least as far back as the 1500s, called the "order of operations". The "operations" are addition, subtraction, multiplication, division, exponentiation, and grouping; the "order" of these operations states which operations take precedence (are taken care of) before which other operations. A common technique for remembering the order of operations is the abbreviation (or, more properly, the acronym ) "PEMDAS", which is turned into the mnemonic phrase "Please Excuse My Dear Aunt Sally".

This phrase stands for, and helps one remember the order of, "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction". This listing 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 multiplication and division outrank addition and subtraction (which are together on the bottom rank). In other words, the precedence is: When you have a bunch of operations of the same rank, you just operate from left to right. For instance, but is rather, because, going from left to right, you get to the division sign 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 the above expression into a graphing calculator, you will get: Using the above hierarchy, we see that, in the " " question at the beginning of this article, Choice 2 was the correct answer, because we have to do the multiplication before we do the addition. (Note: Speakers of British English often instead use the acronym "BODMAS", rather than PEMDAS. BODMAS 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. Also, you can see that the M and the D are reversed in the British-English version; this confirms that multiplication and division are at the same rank or level. ) The order of operations was settled upon in order to prevent miscommunication, but PEMDAS can generate its own confusion; some students sometimes tend to apply the hierarchy as though all the operations in a problem are on the same "level" (simply going from left to right), but often those operations are not "equal". Many times it helps to work problems from the inside out, rather than left-to-right, because often some parts of the problem are "deeper down" than other parts. The best way to explain this is to do some examples:. 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:. The has more worked examples examples.

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