Applying the order of operations is not as intuitive as some might think. I found myself doing a lot of math problems on X, formerly Twitter. And on Facebook, before I decided to focus my attention on X,. It was on Facebook that I was applying the Order of Operations incorrectly.

## PEMDAS: How I was taught

If you are a US citizen, you might recall the acronym PEMDAS. It stands for:

Parentheses

Exponents

Multiplication

Division

Addition

Subtraction

You may have even been taught this little mnemonic to remember it: “Please Excuse My Dear Aunt Sally.” My father used this trick to help me remember how to spell geography for my spelling test. Thanks, dad; that was awesome. For the curious, the mnemonic he taught me was, “George Eliot’s Old Grandfather Rode a Pig Home Yesterday.”

But I digress. The way I was taught was to perform each operation in order from top to bottom. Which meant that you are supposed to evaluate the operation in parentheses, then exponents, then multiply, then divide, then add, and finally subtract. This may come as a surprise to some of you who were taught the same way, but this is entirely wrong.

## But wait, there’s more

In my attempts to evaluate math problems posted on X, I found that some people ignore the order of operations when the problem does not contain any parentheses and just evaluate the problem from left to right. Take this problem, for example:

https://twitter.com/CordeiroRick/status/1743047794452042233

## The Wrong Way

The stated problem:

13 + 169 ÷ 13 – 5 x 4 =?

At the time of writing this article, everyone who replied got the answer, which is 6. In this particular problem, doing the multiplication first isn’t going to affect the outcome. But if you ignore the order of operation and evaluate from left to right, you should arrive at 36 for your answer. This, of course, is the wrong answer.

## Parentheses

Some of you need to hear this: “The absence of parentheses does not negate the order of operations.” Let me say that again: “The absence of parentheses does not negate the order of operations.” All that parentheses are there for is to ensure that the operations within those parentheses are done first and sometimes to provide clarity. But in the absence of parentheses, the order of operations does not change. So in the absence of parentheses, operations with a higher level of precedence are performed first, regardless of the order in which they appear in the problem.

## Calculators Can be Wrong Too

Just because a calculator agrees with your answer does not mean that your answer is correct. But a calculator cannot make a mistake! That is true; a calculator cannot make a mistake, but the humans that program the calculator’s logic do. As do the people who use them. If you bother to read the owner’s manual, you may just find out that you have to insert the parentheses in order to make sure that the calculator evaluates a problem correctly. Some calculators can actually evaluate a problem without adding the absent parentheses. Some cannot. So when a calculator outputs a wrong answer, the problem is always human error. Either on the user’s end or on the manufacturer’s end.

## The Correct Way

To apply the order of operations correctly, you need to understand that multiplication and division have the same level of precedence, as do addition and subtraction, although they have a lower level of precedence than the former. This means that 5 x 2 ÷ 3 and 3 ÷ 5 x 2 are not the same. This is true for 5 + 2 – 1 and 1 – 2 + 5 as well. They are all evaluated in order from left to right. The answers for these expressions are 3.333…, 1.2, 6, and 4 respectively.

## PEMDAS revised

PEMDAS can still be used, but I would recommend writing it like this: PE(M|D)(A|S). The pipe symbol means or, and the parentheses show the same level of precedence for those operations. The mnemonic can be used as well with the following modifications:

Please

Excuse

Mom or Dad

Adam or Sam

## One Final Word

Another mistake that I have found someone making is confusing a fraction bar with a division operation. While converting a fraction such as 1 ⁄ 2 involves division, 1 ÷ 2 = 0.5. This 1 ⁄ 2 and this 1 ÷ 2 are not the same expression, especially when presented with a math problem. So when you are presented with a problem such as this: 13 + 169 ÷ 13 – 5 x 4. What they really mean is 13 +(169 ÷ 13) – (5 x 4) and not (3 + 169) ÷ (13 – (5 x 4)), because ÷ ≠ ⁄ in every context.