How and why the BODMAS rule was formed ?
Some where I have read that the Order of Hierarchy emerges from a natural occurance, perhaps ingrained in human brain as a part of our psychology by the mother nature. Probably the emergence of the Order of Hierarchy is from the same centers of our brain where the Rationalization and the Justice rest.
Addition and Multiplication have a commutative property which is why we like to give Addition and Multiplication a higher preference to Subtraction and Division.
At the same time, the strength of the operator is also accounted in to assign the order of preference. Thus Exponent and Roots are given preference over Multiply, Divide, Add or Subtract.
Between Multiply and Divide , Multiplication of OF form is given precedence due to Commutative Property. And Addition is given preference over Subtraction due same Commutative Property.
Commutative Property is when terms being interchanged do not affect the answer. That is, a+b and a×b are same as b+a and b×a respectively, whereas a-b and a÷b are not same as b-a and b÷a. Thus, Addition and Multiplication are Commutative while Subtraction and Division are non-commutative.
In between, owing to the strength of operator, Multiplication and Division take precedence over Addition and Subtraction.
Brackets or Parenthesis takes precedence over the rest due to ingrained psychology to give preference to internal resolution before interaction with the outer operator.
One question is why Division is given preference over Multiplication of Direct Form (x) ?
This is possibly because Division of (/) slash form is seen as parenthesis operation. Similarly, Division of ÷ operator is seen inferior to all other operator signs of Multiplication and Division.