Question

How can you have different answers to the same expressions with or without the application of the PEMDAS rule?

Answer 1

PEMDAS is a guide to attempt to help reduce ambiguity, but it can result in unintentional interpretations.

Explanation:

If the order of operations is not clearly indicated then without the guidance of PEMDAS they are ambiguous.

Consider a popular example:

`9-:3(1+2)`

If we follow the rules of PEMDAS strictly without embellishment then this expression is the same as:

`9/3xx(1+2)`

According to PEMDAS we evaluate the parentheses first, then the division and multiplication from left to right:

`9/3xx(1+2) = 9/3xx3 = 3xx3 = 9`

Note however that historically the `-:` sign was used to express dividing one complete expression by another complete expression. So with that interpretation we would have:

`9-:3(1+2) = 9/(3(1+2)) = 9/(3xx3) = 9/9 = 1`

The purpose of rules like PEMDAS is to try to reduce the ambiguity, but if you want your expressions to be unambiguous it is often helpful to add some parentheses.