It is merely a matter of convention (not a matter of right or wrong), giving us a standardized way to decide whether 6/2*3 is (6/2)*3 or 6/(2*3). Next, multiplication and division get carried out from left to right. Note that division can be thought of as still carrying out a multiplication since a/b = a*(1/b), so it makes a certain sense that neither division nor multiplication takes precedence over one another. Finally, additions and subtractions get carried out from left to right.
The convention has been accepted just so that when different people see an equation they can agree on its meaning, even if they hate each other’s guts. Subtraction can be thought of as being like addition since a – b = a (-b) so it makes sense that subtraction is treated on an equal footing with addition.
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Mathematician: Now that the Physicist and I have answered questions like these ones at least 9 times via email, I figure we should get this horrible topic out of the way once and for all with a short post.
This is a contradiction, and hence the order of operations form a paradox. Without a convention, it has many possible interpretations, such as a^(b*(c/(d e)))-f or (a^b)*((c/(d e))-f) which could be horribly confusing for people trying to communicate with each other.
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It is easy to get confused by these mnemonics though because they don’t capture the fact that multiplication is treated the same way as division (carried out left to right), and addition is treated the same way as subtraction (carried out left to right).
That’s why I prefer: “Puke Exhaust Mud And Dolls Autumn And Sasquatch” One interesting thing to note about the order of operations is that there is nothing interesting about them whatsoever.
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