How did the current mathematical order of operations (PEMDAS) arise?  

 

Why does it always seem that the more powerful operators also have the
higher precedence?  Is it so that expressions don't explode by default (e.g.
3*4^5*6 would have a very different value if * was higher precedence than
^).  If we introduce a new operator in the chain + * ^, would it necessarily
also have the highest precedence?

 

-Dan

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