Order of operation in math

Question

Who decides order of operation in any math calculation. Is it scientific or arbitrary? e.g. 1+4-6x7+7

Answer 1

Originally we used numbers to count; 1 apple, 2 apples, 3 apples... Adding was a natural action, and we chose to use "+" to denote addition.

If we buy 4 apples and 5 oranges, we could represent that as "4 apples + 5 oranges".

Now "4 apples" is the same as "apple + apple + apple + apple", and so we require that:

"4 apples + 5 oranges" is read as "( 4 apples ) + ( 5 oranges )"

and not for example "4 ( apples + 5 ) oranges" (what on earth would that mean anyway?).

It so happens that we might like to view "4 apples" as "4 times of an apple", and so:

"$4 \times \text{apple} + 5 \times \text{orange}$" is read as "$( 4 \times \text{apple} ) + ( 5 \times \text{orange} )$".

Now this idea extends to multiples of numbers themselves.

When we have 2 dozen apples, we have "2 times of a dozen apples" which is "2 times of 12 times of an apple", or in symbols "$2 \times ( 12 \times \text{apple} )$".

Notice that the result of such two-step multiplication is always the same in some sense, namely that it does not matter what was multiplied (in this case an apple). In general $2 \times ( 12 \times \text{X} ) = 24 \times \text{X}$ for any object $X$. So we define $2 \times 12$ to be $24$ precisely so that:

$2 \times ( 12 \times \text{X} ) = ( 2 \times 12 ) \times \text{X}$.

This allows us to manipulate the numbers separate from the objects they are used to count. This also means that it would be most convenient to have the familiar precedence rules (default order of operations). Consider:

"2 dozen and 3 apples" means "$( ( 2 \times 12 ) + 3 ) \times \text{apple}$.

Clearly by default in our natural language multiplication binds tighter than addition (we interpret "2 dozen and 3" to mean "( 2 dozen ) and 3". Similarly in the symbolic notation it is most convenient to adopt the same convention and so we have:

$2 \times 12 + 3 = ( 2 \times 12 ) + 3$.

Similarly there is usually reasonable justification for each choice of order of operations in general beyond arithmetic, although in some cases different people choose different conventions. For example, in most programming languages "AND" has higher precedence than "OR", but in some logic textbooks they are not allowed to occur consecutively without using brackets to indicate the order.