You're out of order! The whole operation's out of order! | James Howard You’re out of order! The whole operation’s out of order! | James Howard

### Eur Ing Dr James P. Howard, IIA Mathematician, a Different Kind of Mathematician, and a Statistician # You're out of order! The whole operation's out of order!

It’s ambiguous. There are two problems here. First, PEMDAS doesn’t necessarily include all of the things in this example problem. The implicit multiplication is one. That is, which rule governs 2(? Some argue it is covered by the parenthesis rule. Others the multiplication rule. When pressed, many mathematicians resolve the ambiguity by introducing a new rule that places implicit multiplication ahead of explicit. But others, who really fail to understand PEMDAS, argue implicit multiplication is the same as explicit. This is not obviously incorrect, but neither is it obviously correct. That’s why these two calculators give different answers.

$\displaystyle\frac{4}{4 - 2 * 2}\text{.}$
This is clearly the same as $4 / (4 - 2 * 2) = 4 / (4 - 4) = 4 / 0 = \text{undefined}$. But if you run PEMDAS over it naïvely, you get $4 / 4 - 2 * 2 = 1 - 4 = -3$, which is really, really, wrong. But I think the implicit division rule is much clearer, in this example, than the implicit multiplication rule in the source problem.