# Question about order of operations **URL:** \ **Category:** Factual Questions\ **Created:** [April 26, 2011, 4:45am UTC](https://boards.straightdope.com/t/question-about-order-of-operations/579742 "2011-04-26T04:45:56Z")\ **Posts on this page:** 8\ **Page:** 1
**Author:** ![automagic](https://avatars.discourse-cdn.com/v4/letter/a/a698b9/32.png) [@automagic](https://boards.straightdope.com/u/automagic)\ **Post date:** [April 26, 2011, 4:45am UTC](https://boards.straightdope.com/t/question-about-order-of-operations/579742/1 "2011-04-26T04:45:56Z")
Was the order of operations made the way it was due to necessity, as in it would not work any other way, or was it an agreement to avoid confusion? The reason I was thinking about this was because I was thinking why we round anything that ends in 5 up, since it was equally distance between the 2 options. We just all “agreed” that anything ending in 5 should be rounded up. Did the same thing happen with order of operations? Its obvious why anything in parenthesis needs to be calculated first but why do exponents need to be calculated first? Why does 2\*5^2=50 and not 100. Additionally, why was it decided that we need to do multiplication/division before addition/subtraction? ---
**Author:** ![dzero](https://avatars.discourse-cdn.com/v4/letter/d/5f9b8f/32.png) [@dzero](https://boards.straightdope.com/u/dzero)\ **Post date:** [April 26, 2011, 5:07am UTC](https://boards.straightdope.com/t/question-about-order-of-operations/579742/2 "2011-04-26T05:07:20Z")
> **[Order of operations](https://en.wikipedia.org/wiki/Order_of_operations)** > > In mathematics and computer programming, the order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression. > These rules are formalized with a ranking of the operations. The rank of an operation is called its precedence, and an operation with a higher precedence is performed before operations with lower precedence. Calculators generally perform operations with the same precedence from left to r... > [@](#): > > From the introduction of modern algebraic notation, where juxtaposition indicates multiplication of variables, multiplication took precedence over addition, whichever side of a number it appeared on.[[1]](http://en.wikipedia.org/wiki/Order_of_operations#cite_note-0) Thus 3 + 4 × 5 = 4 × 5 + 3 = 23. When exponents were first introduced, in the 16th and 17th centuries, exponents took precedence over both addition and multiplication, and could be placed only as a superscript to the right of their base. Thus 3 + 52 = 28 and 3 × 52 = 75. To change the order of operations, originally a [vinculum](http://en.wikipedia.org/wiki/Vinculum_%28symbol%29) (an overline or underline) was used. Today we use [brackets](http://en.wikipedia.org/wiki/Bracket_%28mathematics%29). Thus, to force addition to precede multiplication, we write (2 + 3) × 4 = 20, and to force addition to precede exponentiation, we write (3 + 5)2 = 64 ---
**Author:** ![suranyi](https://avatars.discourse-cdn.com/v4/letter/s/e36b37/32.png) [@suranyi](https://boards.straightdope.com/u/suranyi)\ **Post date:** [April 26, 2011, 5:33am UTC](https://boards.straightdope.com/t/question-about-order-of-operations/579742/3 "2011-04-26T05:33:52Z")
> [@automagic](#): > > Was the order of operations made the way it was due to necessity, as in it would not work any other way, or was it an agreement to avoid confusion? > > The reason I was thinking about this was because I was thinking why we round anything that ends in 5 up, since it was equally distance between the 2 options. We just all “agreed” that anything ending in 5 should be rounded up. Did the same thing happen with order of operations? Its obvious why anything in parenthesis needs to be calculated first but why do exponents need to be calculated first? Why does 2\*5^2=50 and not 100. > > Additionally, why was it decided that we need to do multiplication/division before addition/subtraction? Your premise is wrong. We didn’t just all “agree” that anything ending in 5 should be rounded up. We round up because if there are any more digits following the 5, then the number really is closer to the one up than the one down, hence it should be rounded up. It’s only equally distant in the case where you are rounding at the second to last digit of your measurement, and that last digit is a 5. That’s not a common situation in the real world. ---
**Author:** ![yabob](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/yabob/32/2821_2.png) [@yabob](https://boards.straightdope.com/u/yabob)\ **Post date:** [April 26, 2011, 5:40am UTC](https://boards.straightdope.com/t/question-about-order-of-operations/579742/4 "2011-04-26T05:40:40Z")
> [@](#): > > Additionally, why was it decided that we need to do multiplication/division before addition/subtraction? [Ken Iverson](http://en.wikipedia.org/wiki/Kenneth_E._Iverson) felt no such need. ---
**Author:** ![Floater](https://avatars.discourse-cdn.com/v4/letter/f/dc4da7/32.png) [@Floater](https://boards.straightdope.com/u/Floater)\ **Post date:** [April 26, 2011, 2:15pm UTC](https://boards.straightdope.com/t/question-about-order-of-operations/579742/5 "2011-04-26T14:15:56Z")
> [@automagic](#): > > Additionally, why was it decided that we need to do multiplication/division before addition/subtraction? As I understand it it’s just a convention that came about as the solution to a problem with mixed operators would be different depending on the order. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [April 26, 2011, 3:05pm UTC](https://boards.straightdope.com/t/question-about-order-of-operations/579742/6 "2011-04-26T15:05:03Z")
> [@automagic](#): > > The reason I was thinking about this was because I was thinking why we round anything that ends in 5 up, since it was equally distance between the 2 options. We just all “agreed” that anything ending in 5 should be rounded up. I’ve actually seen different conventions for how to round numbers that end in 5, but “always round up” is the simplest, and it makes sense because, as **suranyi** points out, even though 7.5 is exactly halfway between 7 and 8, if there are any more digits after the .5, the number is closer to 8 than it is to 7. It’s not much different from the reason we commonly refer to noon as “12:00 pm.” Technically, 12:00 exactly isn’t “post meridiem” (after noon), but 12:00:01 is. > [@](#): > > Was the order of operations made the way it was due to necessity, as in it would not work any other way, or was it an agreement to avoid confusion? > > Did the same thing happen with order of operations? Its obvious why anything in parenthesis needs to be calculated first but why do exponents need to be calculated first? Why does 2\*5^2=50 and not 100. > > Additionally, why was it decided that we need to do multiplication/division before addition/subtraction? I think (but am willing to be corrected) that the order of operations is a matter of convention or notation, which, like other such, is in a sense arbitrary, but some choices of notation or convention are more natural or a better fit for the way the math actually works (e.g. the distributive property). How would “combining like/similar terms” look if we did things the other way around, with addition coming before multiplication? ---
**Author:** ![KneadToKnow](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/kneadtoknow/32/3999_2.png) [@KneadToKnow](https://boards.straightdope.com/u/KneadToKnow)\ **Post date:** [April 26, 2011, 3:22pm UTC](https://boards.straightdope.com/t/question-about-order-of-operations/579742/7 "2011-04-26T15:22:19Z")
> [@dzero](#): > > [Order of operations - Wikipedia](http://en.wikipedia.org/wiki/Order_of_operations) Your exponents broke in the copy & paste. Here’s how it should have looked: > [@](#): > > From the introduction of modern algebraic notation, where juxtaposition indicates multiplication of variables, multiplication took precedence over addition, whichever side of a number it appeared on.[1] Thus 3 + 4 × 5 = 4 × 5 + 3 = 23. When exponents were first introduced, in the 16th and 17th centuries, exponents took precedence over both addition and multiplication, and could be placed only as a superscript to the right of their base. Thus 3 + 5[sup]2[/sup] = 28 and 3 × 5[sup]2[/sup] = 75. To change the order of operations, originally a vinculum (an overline or underline) was used. Today we use brackets. Thus, to force addition to precede multiplication, we write (2 + 3) × 4 = 20, and to force addition to precede exponentiation, we write (3 + 5)[sup]2[/sup] = 64 ---
**Author:** ![Mosier](https://avatars.discourse-cdn.com/v4/letter/m/7ea924/32.png) [@Mosier](https://boards.straightdope.com/u/Mosier)\ **Post date:** [April 26, 2011, 3:40pm UTC](https://boards.straightdope.com/t/question-about-order-of-operations/579742/8 "2011-04-26T15:40:22Z")
Think of it as mathematical grammar. In order for the language to make sense and mean the same thing every time it is written the same way, the rules have to be consistent.