I know some places use BIDMAS as well. Overall though, to someone who knows little about mathematics, why is this the correct order? What’s the exact logic behind it
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Everything in the way we represent mathematics is convention. There is no true natural and correct way to write the concept of adding two numbers together. We have agreed to use “+” to convey that, but we could just as easily have decided upon “×” instead. The symbols only have the meaning we assign to them – but consider how confusing it would be if half the world used “+” for addition and “×” for multiplication, and half used the reverse. So – we agree on a consensus for what the symbols mean and the order to apply them so that we can share and communicate mathematical concepts with less confusion.
It has to do with how we decided to write equations, as other people have stated. What I haven’t seen stated is why that order.
Basically, you can build operations out of “smaller” or “weaker” operations. Multiplication is just repeated addition, exponentiation is just repeated multiplication, tetration is just repeated exponentiation, pentration is just repeated tetration, and so on. Technically, addition is just repeated counting.
If you want to take a LONG time, you can reduce each operation down to addition or even counting, but that is a lot of effort. Instead, you start with the “highest order” operation and work your way down. If you don’t have any tetration or higher, you start with exponents, then multiplication, then addition.
Subtraction, division, exponential roots, and so on are inversions of operations, so they are done at the same time as their “parent” operation.
Everything in the way we represent mathematics is convention. There is no true natural and correct way to write the concept of adding two numbers together. We have agreed to use “+” to convey that, but we could just as easily have decided upon “×” instead. The symbols only have the meaning we assign to them – but consider how confusing it would be if half the world used “+” for addition and “×” for multiplication, and half used the reverse. So – we agree on a consensus for what the symbols mean and the order to apply them so that we can share and communicate mathematical concepts with less confusion.
It has to do with how we decided to write equations, as other people have stated. What I haven’t seen stated is why that order.
Basically, you can build operations out of “smaller” or “weaker” operations. Multiplication is just repeated addition, exponentiation is just repeated multiplication, tetration is just repeated exponentiation, pentration is just repeated tetration, and so on. Technically, addition is just repeated counting.
If you want to take a LONG time, you can reduce each operation down to addition or even counting, but that is a lot of effort. Instead, you start with the “highest order” operation and work your way down. If you don’t have any tetration or higher, you start with exponents, then multiplication, then addition.
Subtraction, division, exponential roots, and so on are inversions of operations, so they are done at the same time as their “parent” operation.
Parenthesis or Brackets
The purpose is to group and create a higher order operation, so they need to go first since that’s their whole job.
If this were lower than it makes the problem harder because you have to add, multiply, etc over every term inside it.
Exponents
This is basically shorthand for self-multiplication, so it seems reasonable that we need to work it out earlier.
This could happen before parenthesis, but that get notibly harder to do. Could also happen after multiplication, but then if you divide first you’ll have to apply it to fractions which isn’t always fun.
Multiplication and Division
These happen together because they are similar but inverse operations; and because of certain properties of the operations they need to happen before addition and subtraction.
Addition and Subtraction
These can basically be done whenever, but because everything else was more important, they get shoved to the lowest priority.
In the end this priority is agreed upon because if you change the order, then you don’t always get the same result from the same equation everytime.
The same logic behind grammar and spelling rules. To be consistent and prevent ambiguity and misunderstanding when communicating with others.
Math is just like any other language we use to describe the world around us, with it’s own set of “grammar” rules.
You even have an equivalent of PEMDAS in the English language, the order of adjectives. It goes:
Opinion->Size->Age->Shape->Colour->origin->material->purpose + noun.
Here are 2 examples:
“My neighbour drives a beautiful little old red Italian sports car”
Vs.
“My neighbour drives a little beautiful red sports Italian old car”
Or:
“The big old brown cat shat on the sofa”
Vs.
“The old brown big cat shat on the sofa”
Although both sentences can be used to convey information, you intuitively know one is correct and the other is not. Why is one more correct than the other? Just because we agreed it is, nothing more.
Similarly, deviating from PEMDAS doesn’t result in mathematical impossibilities, or causes math to implode in on itself, but it yields results that just don’t feel quite right…
Take the equation 4*5+3
If we add before we multiply we just get:
4×5+3 = 4×8 = 32 instead of 4×5+3 = 20+3 = 23.
If both yield valid, real answers, why is one correct over the other? Just because we all agreed it is.
The same logic behind grammar and spelling rules. To be consistent and prevent ambiguity and misunderstanding when communicating with others.
Math is just like any other language we use to describe the world around us, with it’s own set of “grammar” rules.
You even have an equivalent of PEMDAS in the English language, the order of adjectives. It goes:
Opinion->Size->Age->Shape->Colour->origin->material->purpose + noun.
Here are 2 examples:
“My neighbour drives a beautiful little old red Italian sports car”
Vs.
“My neighbour drives a little beautiful red sports Italian old car”
Or:
“The big old brown cat shat on the sofa”
Vs.
“The old brown big cat shat on the sofa”
Although both sentences can be used to convey information, you intuitively know one is correct and the other is not. Why is one more correct than the other? Just because we agreed it is, nothing more.
Similarly, deviating from PEMDAS doesn’t result in mathematical impossibilities, or causes math to implode in on itself, but it yields results that just don’t feel quite right…
Take the equation 4*5+3
If we add before we multiply we just get:
4×5+3 = 4×8 = 32 instead of 4×5+3 = 20+3 = 23.
If both yield valid, real answers, why is one correct over the other? Just because we all agreed it is.
The same logic behind grammar and spelling rules. To be consistent and prevent ambiguity and misunderstanding when communicating with others.
Math is just like any other language we use to describe the world around us, with it’s own set of “grammar” rules.
You even have an equivalent of PEMDAS in the English language, the order of adjectives. It goes:
Opinion->Size->Age->Shape->Colour->origin->material->purpose + noun.
Here are 2 examples:
“My neighbour drives a beautiful little old red Italian sports car”
Vs.
“My neighbour drives a little beautiful red sports Italian old car”
Or:
“The big old brown cat shat on the sofa”
Vs.
“The old brown big cat shat on the sofa”
Although both sentences can be used to convey information, you intuitively know one is correct and the other is not. Why is one more correct than the other? Just because we agreed it is, nothing more.
Similarly, deviating from PEMDAS doesn’t result in mathematical impossibilities, or causes math to implode in on itself, but it yields results that just don’t feel quite right…
Take the equation 4*5+3
If we add before we multiply we just get:
4×5+3 = 4×8 = 32 instead of 4×5+3 = 20+3 = 23.
If both yield valid, real answers, why is one correct over the other? Just because we all agreed it is.
Parentheses have to go first, because otherwise there’d be no reliable way to indicate when you want readers to do the operations in a particular order. For example, if it were MPEDAS, you wouldn’t be able to tell someone to add two to three and then multiply by five, because in (2+3)×5, the multiplication would have to be done before the addition.
Other than that, it doesn’t matter the order so long as everyone agrees what it is. We need to use the same order so you can know in advance how people will interpret the mathematical expression you’re writing. But if we’d all agreed to use PADMES instead, that’d work equally well.
Parentheses have to go first, because otherwise there’d be no reliable way to indicate when you want readers to do the operations in a particular order. For example, if it were MPEDAS, you wouldn’t be able to tell someone to add two to three and then multiply by five, because in (2+3)×5, the multiplication would have to be done before the addition.
Other than that, it doesn’t matter the order so long as everyone agrees what it is. We need to use the same order so you can know in advance how people will interpret the mathematical expression you’re writing. But if we’d all agreed to use PADMES instead, that’d work equally well.
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