What order of operation should I use calculating $2^3 \times 4 - 6 \times 3 ÷ (4 + 3) - 16 ÷ 4$?

Question

I am struggling if I should use PEMDAS or BODMAS in this equation. What is the right method to get around these type of equations?

$$2^3 \times 4 - 6 \times 3 ÷ (4 + 3) - 16 ÷ 4 = \ ?$$

Answer 1

You silly humans with your mnemonics that just wind up confusing you even more! Mwahahaha! Mwahahahahahahahahaha!

Now, let's see: PEMDAS is Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. And BODMAS is Brackets, Orders, Division, Multiplication, Addition, Subtraction.

Parentheses are pretty much the same thing as brackets. Exponents and orders might be different things, in that all exponents are orders but not all orders are exponents. Or am I trying to confuse you further? Behold: $$\sqrt x = x^{\frac{1}{2}}, \root 3 \of x = x^{\frac{1}{3}}, \root 4 \of x = x^{\frac{1}{4}}, \root 5 \of x = x^{\frac{1}{5}}, \ldots$$

After that, the two mnemonics are the same except that they switch the order of multiplication and division. This was on purpose, to create further confusion, and obfuscate the fact that multiplication and division are basically the same thing, just notated differently, e.g., $$\frac{x}{2} = x \times \frac{1}{2}, \frac{x}{3} = x \times \frac{1}{3}, \frac{x}{4} = x \times \frac{1}{4}, \ldots$$

Then $$2^3 \times 4 - 6 \times 3 ÷ (4 + 3) - 16 ÷ 4$$ $$= 2^3 \times 4 - 6 \times 3 ÷ 7 - 16 ÷ 4$$ $$= 8 \times 4 - 6 \times 3 ÷ 7 - 16 ÷ 4$$ $$= 8 \times 4 - 6 \times \frac{3}{7} - 4 \textrm{ or } 32 - 18 ÷ 7 - 16 ÷ 4$$ $$= 32 - \frac{18}{7} - 4 \textrm{ or } 32 - \frac{18}{7} - 4$$

You see, they're the same.

Answer 2

In this case both PEMDAS and BODMAS give the same result:

$2^3 \times 4 - 6 \times 3 ÷ (4 + 3) - 16 ÷ 4 =$

$8\times 4 - 6\times 3\div 7 - 16\div 4 = $

$32 - 18\div 7 - 16\div 4 = $(PEMDAS) or $8\times 4 - 6\times \frac 37 - 4$ (BODMAS)

$32 - \frac {18}7 - 4=$

$\frac {206}7 - 4 = \frac {178}7$.

The ambiguity lies when you have something like $16\div 4 \times 2$.

Whether that is supposed to be $(16\div 4)\times 2$ or $16\div (4\times 2)$.

I would say treat Multiplication and Division with equal weight and go from left to right and do $16\div 4 \times 2 = 4\times 2 = 8$, just as you'd do $16 - 4 + 2 = 12 +2 = 14$ and you'd NEVER do $16 - 4 + 2 = 16 - 6 = 10$. EVER!

Except.... many text do give multiplication when written like this heavier weight. So SOME text will do $16\div 4\times 2 = 16\div (4\times 2) = 16\div 8 = 2$.

But those texts are rare and it's usually easy to tell from context what to do.

Answer 3

Wolfram Alpha is a website I strongly recommend for anyone interested in math, whether a serious student or a casual dilettante. Go to Wolfram Alpha and put in 2^3 * 4 - 6 * 3 / (4 + 3) - 16 / 4. Next, try 2^3 * 4 - 6 * (3 / (4 + 3)) - (16 / 4). The results should be the same.

The important things to remember are that multiplication and addition are both commutative, but multiplication has higher precedence than addition in the absence of parentheses.

U. S. President Abraham Lincoln (R) was intelligent, eloquent and honorable, qualities thoroughly missing from today's Republicans. We should rightly admire the eloquence of his Gettysburg address.

However, "four score and seven years ago" instead of "eighty-seven years ago" is not so much a poetic turn of phrase as it is a demonstration that the rules of operator precedence are derived from our natural understanding of numbers.

Lincoln could just as easily have said "seven and four score years ago" and everyone would have understood he meant 87. It's $4 \times 20 + 7 = 7 + 4 \times 20$. This sort of thing persists in French, e.g., "sette e quatre-vingtes".

Of course things get a lot hairier when you also have subtractions and divisions in the mix. As a sanity check, try rewriting the formula with multiplications and additions only. So $$2^3 \times 4 - 6 \times \frac{3}{4 + 3} - \frac{16}{4}$$ becomes $$2 \times 2 \times 2 \times 4 + (-6) \times 3 \times \frac{1}{7} + \left(-16 \times \frac{1}{4}\right).$$

Performing the multiplications in whatever order you like, you should now get $$32 + \left(-\frac{18}{7}\right) + (-4),$$ which looks an awful lot like the other two answers. At this point you can now perform the additions in whatever order you like.

Hmm... I've gotten this far without mentioning either of the two mnemonics you mentioned. As it turns out, there's a variant of PEMDAS, PFEMDAS, in which the F stands for "factorial." In his answer to Which binds first, product or factorial?, Markus Scheuer cites the OEIS page on operator precedence.

There's probably also BFODMAS, but Google insists I want BODMAS.