“Please Excuse My Dear Aunt Sally,” I will never un-hear my college professor saying this over and over. I had already become very acquainted with the order of operations from computer programming so it was old news to me. Although sometimes I have to remind myself when evaluating complex computer logic, and still to this day hear my college professor.
What are Operators?
Operators are simply calculations, or mathematical processes. Addition, subtraction, multiplication, division, squaring, etc., are all operations. We assign symbols to the operators such as + and – for addition and subtraction respectively. Don’t worry, this is not a math class and you will not be graded after, at least not by me. I will not be diving into complex math, linear algebra, calculus, relativity, or theoretical quantum silly string theory. I am only going to describe the basic math operations.
The order of operations are easily remembered with the acronym PEMDAS which is how I remembered them from computer programming. My preferred method for remember them is the same as my college professor drilled into my head so long ago. Some of my other favorite sayings are:
- Peanuts Ended My Doughnut Addiction Saturday
- People Everywhere Must Devour Additional Seafood
- Please End My Depression and Suffering
- Please End My Day At School
- Pancake Explosion Many Deaths Are Suspected
- Please Eat My Dad and Sister
- Please Excuse My Dumb Ass Students
I especially like the last one. I’m sure there are many more and would love to hear them in the comments!
Doing Things Right (or at Least Pretending)
The order of operations are:
- P = Parentheses – groupings like parentheses ( ), brackets [ ], curly braces { }, etc.
- E = Exponents or powers (^ or floaty numbers by other numbers or variables): (x2)
- D = Division from left to right: (÷, /, — (horizontal line))
- M = Multiplication same precedence as division, left to right: (*, x, ⋅, or variables side by side xy)
- A = Addition, left to right: (+)
- S = Subtraction, same precedence as addition and also left to right: (-)
That’s it. Not really, there are tons of operators I will not be covering here. But if you are like me and just really curious, feel free to go on a deep dive on the Basic Math Symbols. For the sake of sanity, and my lousy math skills, I will stick to the pre-K basics!
Examples
I promised this wouldn’t be a math class. So I am not going to show excessive amounts of examples since there are far too many out there already like Math Is Fun. Let’s walk through a few examples of each.
Subtraction (Opposite of Addition)
Ex 1: 3 – 1 = 2
Ex 2: 3 – 1 – 1 = 1 (left to right)
Ex 3: 1 – 1 – 3 = -3 (still left to right)
One caveat about subtraction is when you are subtracting a negative number.
Ex 4: 1 – -2 = 3 (commonly written: 1 – (-2) = 3
This causes a double negative which creates a positive and is the same as adding. The statement actually becomes 1 + 2. Confusing? Yes, it’s math, it’s supposed to be. Read more about it here.
Addition (Opposite of Subtraction)
Ex 1: 1 + 1 = 2
Ex 2: 1 + 2 + 3 = 6 (left to right)
Ex 3: 1 + 3 + 2 = 6 (still…left to right)
Adding a negative number is the same as subtraction
Ex 4: 1 + -2 = -1 (commonly written as 1 + (-2) = -1)
Multiplication (Opposite of Division)
Ex 1: 2 * 3 = 6
Ex 2: 3 * 2 = 6
Ex 3: 2,000,000 * 1 = 2,000,000 (any number times 1 is itself)
Ex 4: 2,000,000 * -1 = -2,000,000 (switch the sign)
Ex 5: 39,000,000,000,000 * 0 = 0 (any number times zero is zero including the national debt)
Division (Opposite of Multiplication)
Ex 1: 6 / 2 = 3
Ex 2: 6 / 3 = 2
Ex 3: 10 / 100 = .1
Ex 4: 100 / 10 = 10
Ex 5: 100 / -1 = -100 (also switches the sign)
Ex 6: 1 / 0 = Just don’t. This is undefined and would result in infinity or a paradox that would desroy the galaxy.
Exponents
Ex 1: 2^ 2 or 22 = 4
Ex 2: 3 ^ 3 or 33 = 9
Ex 5 ^ 5 or 55 = 25
Raising to a power is the same as multiplying a number by the exponent such as 44 would be the same as 4 * 4 * 4 * 4 = 256. Remember, go left to right:
4 * 4 * 4 * 4 =
16 * 4 * 4 =
64 * 4 = 256
Parentheses (Grouping)
Ex 1: 2 + (4 * 2) = 10 (unnecessary since multiplication already goes before addition)
Ex 2: (2 + 4) * 2 = 16
Ex 3: 14 – (2 + 4) * (3 – 1) = 2
Ex 4: (14 – 2 + 4) * 3 – 2 = 46
Ex 5*: 30 * (9/5) + 32 = 86
If your math instructor was a sadist and taught you to calculate Celsius to Fahrenheit this way like mine did: stop it! (9/5) = 1.8 so plug that in instead of that stupid fraction:
30 * 1.8 + 32 = 86. Follow me for more tips!
Finally, how do you handle exponents on the outside of parentheses? What do you do if you have exponents inside and outside the groupings?
Ex 1: (4-2)2
First you do what’s inside the grouping: 4-2 = 2. Now the problem is simply 22 = 4
Ex 2: (42)2. In this case, you have to simplify and add (or subtract depending on the exponents sign). The problem then becomes 42+2 = 44 = 256.
For more information, more than you would like, check out this great article by Purple Math.
Yes, I know I listed the order of operations in reverse order from PEDMAS, that was on purpose.
Conclusion
Thank God that’s over! I am not going to go deep into math. I am going to be writing about money and finances, not how to build a rocket and slingshot around the moon and return home. There’s no need to make calculations harder than necessary and you’ll be happy to know that most are incredibly simple. Also, the best part is if something is complex, you can, and should, break it down step-by-step until you understand it. Learn the order of operations and you’ll be off to a good start on calculating all kinds of things.
If you made it through this entire train-wreck, this gold star is for you: 🌟