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

Apr 8, 2016

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:

$9 \div 3 \left(1 + 2\right)$

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

$\frac{9}{3} \times \left(1 + 2\right)$

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

$\frac{9}{3} \times \left(1 + 2\right) = \frac{9}{3} \times 3 = 3 \times 3 = 9$

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

$9 \div 3 \left(1 + 2\right) = \frac{9}{3 \left(1 + 2\right)} = \frac{9}{3 \times 3} = \frac{9}{9} = 1$

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.