Not everyone is taught PEMDAS, and it's not univerally accepted, so there is a lot of confusion around seemingly simple math problems where there is an ambiguity in the mathematical format.

What is PEMDAS?

PEMDAS is a mathematical acronym that specifies the order of operations. It stands for: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. It seems straight forward, yet there is a slight vagueness that can be problematic. This is because PEMDAS alone doesn't specify the entire rule. The proper order of operations is as follows: Inside Parentheses, Exponents, Multiplication and Division, then Addition and subtraction. Notice that multiplication and division are together, as are addition and subtraction. These two pairs need to be done in a left-to-right order. For example, given the math problem \(5\times 2\div 3\times7\), we don't do both multiplies first and then do the divide; instead, we work the entire equation left-to-right. This, most people are clearly taught.

The problem arises because PEMDAS seems to violate another rule of math that we are taught in school: that of the distributive property of multiplication. I say seems, because it doesn't really violate any rules. The distributive property of multiplication generally arises in the form \(5(a+b)\). Because we can't add \(a\) and \(b\), we distribute the 5 to get \(5a+5b\). In the similar problem \(a(5+3)\), we will get the result of \(8a\) if we distribute first then add \((5a+3a=8a)\) as we would if we add then multiply \((a(8)=8a)\). Now we throw a monkey wrench into the mix, and we get a problem like \(6\div 2(1+2)\). Because we're taught to do parentheses first, and we are also taught that the distributive property of multiplication works the same as doing the inside of the parentheses before mutiplying, we make the erroneous assumption that it will always work that way. In this case, the problem is that the divide must occur before the implied multiply, so distribution isn't possible. The correct answer is 9, not 1.

I first encountered a problem like this on my college placement test, and even I was fooled by the ambiguity. Not only that, the calculator I had got it wrong! Texas Instruments calculators, I know, always get it right, but a lot of the cheap scientific calculators on the market do not. In truth, nobody would ever write a problem like this except to test your knowledge of the proper order of operations. In real-life situations, a mathematical equation will be written clearly. For example, the problem \(6\div 2(1+2)\) would be written as \(6\div 2\times(1+2)\), \((6\div 2)(1+2)\), \(\frac{6}{2}(1+2)\), or \(\frac{6(1+2)}{2}\) in order to be clear.

I find the topic of the order of operations and PEMDAS to be interesting, because so many people (including myself) are or have been confused by such seemingly simple problems. In fact, the internet is full of erroneous "proofs" written using a flawed order of operations. The problem is so big, in fact, that the issue of PEMDAS has reached viral status. In case you're interested, here's a link to one of the recent articles on the subject of PEMDAS: Thousands Stumped by Simple Math Problem, Goes Viral.

The truth of the matter is that the PEMDAS issue reflects the broken state of our modern educational systems. Not only are there differences in what people are taught (for example, in some regions of the world, they are taught PODMAS or PEDMAS, which places the division before the multiplication), people aren't always properly taught that the multiplication and division must always be done in a left-to-right order if at the same level of the mathematical expression, regardless of the acronym used. After all, a division is really just a multiplication of an inverse.