Multiplication, Division and Parenthesis

Question

Considering the arithmetic expression: $8/2(4)$

Is the answer simplified to be:

$(8) / (2*4) = 1$

or is it:

$(8/2) * (4) = 16$?

Please, explain why with your answer.

Thanks!

Answer 1

$8/2(4)$

Considering BODMAS, lets begin with the Bracket (B). That is,

$8/2(4) = 8/2 × 4$ -- (1)

From the RHS of (1), we evaluate further via Division (D). That is,

$8/2 × 4 = 4 × 4$ -- (2)

From the RHS of (2), we conclude the simplification via Multiplication (M). That is,

$4 × 4 = 16$ ...Ans.

Answer 2

This is a rather weird way to express $\frac{8}{2} \cdot 4.$ As it is written, it represents the second option (which is 16).

ANSWER TO COMMENT: The parentesis takes precedence, but there are no operations to do in it! It's just a 4. So after you're done with the operations in the parenthesis, you begin to make the division/multiplication from left to right.

Answer 3

This is one of those cases that go viral on the Internet because people can't agree on a convention. The reason why we need a convention is because division is not associative, in particular division and multiplication are not compatible associatively. In the language of abstract algebra, the taking inverse and multiplication by an element functions do not commute.

As such, an epression of the form: $$ x = x_1 \circ_1 x_2 \circ_2 \ldots \circ_{n-1} x_n $$ Where any of the $\circ_i$ operations are non-associative, is not well defined for general operations $\circ_i$ unless we use parenthesis.

The problem is how we are going to group a series of non-associative operations. For this example I will only use one non-associative operation but the same applies to many. If your convention is: $$ \prod_i x_i = (((x_1 \circ x_2)x_3)\ldots)x_n $$ Then the answer is $16$. If the convention is: $$ \prod_i x_i = x_1(x_2(x_3\ldots(x_{n-1} \circ x_n))) $$ Then the answer is $1$. In standard mathematics we went with the first convention. So the answer that makes most sense in this context is $16$.

Also because starting with the last operation in a string of operations that, for example, might be the result of a limiting process (i.e. infinite series, the string is infinite), is kinda convoluted.