What is $48\div2(9+3)$? [duplicate]

There is a huge debate on the internet on the value of $48\div2(9+3)$.

I believe the answer $2$ as I believe it is part of the bracket operation in BEDMAS. Mathway yields the same answer. I also believe that if $48\div2\times(9+3)$ was asked it would be $288$ which Mathway agrees with as well.

However, WolframAlpha says it is $288$ either way.

A friend of mine (who is better at math) told me that there is no such thing as 'implicit multiplication', only shorthand so that is in fact done after the division (going left to right, not necessarily because division occurs before multiplication. But he didn't explicitly give a reason)

What is the answer and why?

Solution 1:

There is no Supreme Court for mathematical notation; there were no commandments handed down on Sinai concerning operational precedence; all there is, is convention, and different people are free to adhere to different conventions. Wise people will stick in enough parentheses to make it impossible for anyone to mistake the meaning. If they mean, $(48\div2)(9+3)$, they'll write it that way; if they mean $48\div\bigl(2(9+3)\bigr)$, they'll write it that way.

Solution 2:

It's ambiguous, there is not one right answer in this case, other than possibly that it is undefined. You may have $$\frac{48}{2(9+3)} = 2$$
or
$$\frac{48}{2}(9+3) = 288$$

Therefore, there is no point in debating this.

Note that the reason you are get different answers from mathway and google calculator is that the algorithms they use for parsing input are different. These algorithms apparently (and understandably) leave it up to the user in this case to give input that can only be interpreted in one way. This is not the case, and is therefore why the two's answers differ.

Solution 3:

I would say it isn't even well-defined. In Group Theory or such, you usually pass by a statement that says "associativity means that $(1 + 2) + 3$ is the same as $1 + (2+3)$, so we can write $1 + 2 + 3$ without ambiguity." $\div$ doesn't have this property of being unambiguous.

This is one of the advantages of using $\frac{48}{2}(9+3)$ or $\frac{48}{2(9+3)}$ - it's not quite associative, but it isn't ambiguous. I haven't seen $\div$ since elementary school probably for this very reason.

Solution 4:

There is no order difference between implicit and explicit multiplications. Purplemath suggests that implied multiplication outside of parentheses also gets parenthetical order priority over all other multiplication(division). So they would interpret the implied multiplied parenthetical as $48\div(2\times(9+3)) = 2$.

Alternatively, all implicit-capable calculators I've tried give the same results as Wolfram Alpha which interprets the implication as $(48\div2)\cdot(9+3) = 288$

Some confusion also seems to be the $\div$ division symbol itself as $48/2(9+3)$ visually supports the mutual parenthetical implied multiplication of Wolfram Alpha: $\frac{48}{2}\cdot(9+3) = 288$

The short answer is that the formula as written is too easily misinterpreted and the author should clarify it to ensure its proper calculation.

PS: to further furrow your brow - employing the parenthetical as a variable yields different results. So, $48\div2c$ where $c=9+3$ yields $2$ - but this conflates the distinction from parenthetical to coefficient-variable syntax. $48\div2\cdot c$ where $c=9+3$ yields $288$.