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For example $2+2\times2$ is $6$ not $8$.

Actually $+$ and $\times$ are binary operations on $\mathbb Z$. but here there is an triple $(2,2,2)$ which we sent to $2+2\times 2$. So we have to put and order for applying operations as binary such that $$a+b\times c \: : \: (a,b,c) \to a+(b\times c)$$ with assuming that $(\cdots)$ has priority more.

My question is why multiplication has priority more than addition? We could also define $$a+b\times c \: : \: (a,b,c) \to (a+b) \times c.$$

student forever
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  • You're right, we could also define it the other way, as long as we're consistent and clear about what we mean by the expressions we write. It's just a convention - choosing one convention or the other is just a notational choice, with no mathematical significance. An interesting question is why this convention developed historically, but as it's a question about history of mathematics, not about mathematics, this is not the best forum for such a question. – Alex Kruckman Apr 12 '16 at 00:04
  • This comes down to multiplication is invented as a shorthand of repeated addition. Suppose you have $42$ dollar bills in your wallet and you want to count them. After you counted $10$ dollar bills, you will fold them inside a single bundle. At the end, you get 4 bundles, each worth $10$ dollars together with $2$ extra dollar bills. Conceptually, one can view the $42$ dollar bills as $4 \times 10 + 2$. For daily life, this sort of situation is simply much more common than one need to interpret $2 + 4\times 10$ as $(2+4) \times 10$. – achille hui Apr 12 '16 at 00:19

3 Answers3

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Multiplication has higher priority than addition because first of all, multiplication is indeed repeated addition. Take the expression $4 + 3 \times 9$. This is 4 plus 3 groups of 9. So the mutliplication can be dismantled as $4 + 9 + 9 + 9 = 31$. It isn't $7 \times 9 = 63$. When you simplify an expression, you start with the parentheses, which are the highest and the most complex parts and evaluate what's in there and then combine the result to what else is there. Then, the middle parts come, which are exponents, which is repeated multiplication, then multiplication, which is repeated addition; and finally, we add everything up.

Also, this draws to the distributive property. The distributive property states: $$a(b+c) = ab+ac$$ If addition had higher priority, the property would look awkward: $$a \cdot b + c = (ab) + (ac)$$ It is worse with implied multiplication: $$ab+c=(ab)+(ac)$$ With multiplication having higher priority, you don't have to add parentheses. That's why PEMDAS is so useful. Without PEMDAS, $P(x) = ax^4 + bx^3 + cx^2 + dx + f$ would have to be written as $P(x) = (a(x^4)) + (b(x^3)) + (c(x^2)) + (dx) + f$ for us to understand.

In short, multiplication has higher priority because of the distributive law of multiplication over addition as well as it being repeated addition.

Resources:

Obinna Nwakwue
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PEMDAS

We have parentheses because it gives clarity and grouping to operations and terms

Exponentiation because it is repeated multiplication

Multiplication/Divison are the same thing if you think about it, multiplication is repeated addition

$\displaystyle A \div B = A \times \frac{1}{B}$, so division is multiplying and has the same priority.

Addition/Subtraction are also the same thing

$A-B=A+(-B)$, so subtraction is addition and has the same priority

And yes, you could TOTALLY redefine the order and priority of operations. Seriously.

It's just that we have invented a convention, so that each expression has a single and defined value, that doesn't vary based on who's performing it.

Saketh Malyala
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Certainly we could define addition/subtraction to have higher priority than multiplication/division. If we wanted to make a rational choice we would review many mathematical publications to see which version would let us write the expressions with fewer parentheses, then mandate that precedence. History is more random than that, respecting the views of certain people at the time the tradition is being laid down. Now the tradition is too strong to change.

Ross Millikan
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