What is the logic of priority of operations? 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 $$a+b\times c \: : \: (a,b,c) \to (a+b) \times c.$$
 A: 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.
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*Why does PEMDAS do multiplication and division before addition and subtraction?
A: 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. 
A: 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.
