Professor Dave here, let’s learn the order of arithmetic operations.
We’ve learned about addition, subtraction, multiplication, division, and exponentiation.
That’s quite a few operations.
So let’s look at cases where multiple operations are being used at once.
How about this: five times three plus four minus two times six.
What’s the best way to do this?
Let’s start from the left.
Five times three is fifteen, plus four is nineteen, minus two is seventeen, times six
is a hundred and two.
Out of curiosity, what if we start from the right?
Two times six is twelve.
Four minus twelve is negative eight.
Three plus negative eight is negative five.
And five times negative five is negative twenty-five.
Well that’s strange, that’s a totally different number.
Actually, there are plenty of other numbers we could get too; depending on which operations
we do first.
But shouldn’t we all be doing the same math and getting the same answer?
Since it’s arbitrary where to start, we have come up with a convention that we all
follow so that we all get the same answer.
This convention for the correct order of operations is called PEMDAS.
It stands for parentheses, exponents, multiplication, division, addition, and subtraction.
This tells us the order that we do things, from left to right.
First we perform any operations found within parentheses.
Then we apply any exponents that are present.
Then we do any multiplication and division, and lastly we do addition and subtraction.
Let’s apply this method to our previous example.
There are no parenthetical terms or exponents, so let’s jump straight to the multiplication.
Five times three is fifteen, and two times six is twelve.
We must do this before doing any addition.
Now this reads fifteen plus four minus twelve.
Now it’s all addition, so we just go left to right.
Fifteen plus four is nineteen, minus twelve is seven, and according to PEMDAS, this is
the only correct answer.
Now let’s try a trickier one.
Fifteen plus three plus two quantity squared.
Notice that this is how we refer to this parenthetical term verbally; it implies that three plus
two lies within parentheses.
Continuing with the expression we have minus nine times six plus two cubed.
So let’s go through the order of operations.
First we do the parenthetical term.
Three plus two is five.
Then we do any exponents.
Five squared is twenty-five, and two cubed is eight.
Now we have to do the multiplication before we do any addition, so nine times six is fifty-four.
Now we can just add and subtract across.
Fifteen plus twenty-five minus fifty-four plus eight equals negative six.
This is the only correct evaluation of this expression when we follow the order of operations.
If you want to make sure you can do this on your own, let’s check comprehension.