In this video we're going to talk a little bit about
order of operations.
And I want you to pay close attention because really
everything else that you're going to do in mathematics
is going to be based on you having a solid grounding
in order of operations.
So what do we even mean when we say order of operations?
So let me give you an example.
The whole point is so that we have one way to interpret
a mathematical statement.
So let's say I have the mathematical statement
7 plus 3 times 5.
Now if we didn't all agree on order of operations, there
would be two ways of interpreting this statement.
You could just read it left to right, so you could say well,
let me just take 7 plus 3, you could say 7 plus 3 and then
multiply that times 5.
And 7 plus 3 is 10, and then you multiply that by 5.
10 times 5, it would get you 50.
So that's one way you would interpret it if we didn't agree
on an order of operations.
Maybe it's a natural way.
You just go left to right.
Another way you could interpret it you say, I like to do
multiplication before I do addition.
So you might interpret it as -- I'll try to color code it --
7 plus -- and you do the 3 times 5 first.
7 plus 3 times 5, which would be 7 plus 3 times 5 is
15, and 7 plus 15 is 22.
So notice, we interpreted this statement in
two different ways.
This was just straight left to right doing addition
then the multiplication.
This way we did the multiplication first then the
addition, we got two different answers, and that's just not
cool in mathematics.
If this was part of some effort to send something to the moon
because two people interpreted it a different way or another
one computer interpreted one way and another computer
interpreted it another way, the satellite might go to mars.
So this is just completely unacceptable, and that's why
we have to have an agreed upon order of operations.
An agreed upon way to interpret this statement.
So the agreed upon order of operations is to do parentheses
first -- let me write it over here -- then do exponents.
If you don't know what exponents are don't worry
about it right now.
In this video we're not going to have any exponents in our
examples, so you don't really have to worry about
them for this video.
Then you do multiplication -- I'll just right mult, short for
multiplication -- then you do multiplication and division
next, they kind of have the same level of priority.
And then finally you do addition and subtraction.
So what does this order of operations -- let me label
it -- this right here, that is the agreed upon
order of operations.
If we follow these order of operations we should always
get to the same answer for a given statement.
So what does this tell us?
What is the best way to interpret this up here?
Well we have no parentheses -- parentheses look like that.
Those little curly things around numbers.
We don't have any parentheses here.
I'll do some examples that do have parentheses.
We don't have any exponents here.
But we do have some multiplication and division
or we actually just have some multiplication.
So we'll order of operations, do the multiplication
and division first.
So it says do the multiplication first.
That's a multiplication.
So it says do this operation first.
It gets priority over addition or subtraction.
So if we do this first we get the 3 times 5, which is
15, and then we add the 7.
The addition or subtraction -- I'll do it here, addition,
we just have addition.
Just like that.
So we do the multiplication first, get 15, then
add the 7, 22.
So based upon the agreed order of operations, this right
here is the correct answer.
The correct way to interpret this statement.
Let's do another example.
I think it'll make things a little bit more clear, and
I'll do the example in pink.
So let's say I have 7 plus 3 -- I'll put some parentheses there
-- times 4 divided by 2 minus 5 times 6.
So there's all sorts of crazy things here, but if you just
follow the order of operations you'll simplify it in a very
clean way and hopefully we'll all get the same answer.
So let's just follow the order of operations.
The first thing we have to do is look for parentheses.
Are there parentheses here?
Yes, there are.
There's parentheses around the 7 plus 3.
So it says let's do that first.
So 7 plus 3 is 10.
So this we can simplify, just looking at this
order operations, to 10 times all of that.
Let me copy and paste that so I don't have
to keep re-writing it.
So that simplifies to 10 times all of that.
We did our parentheses first.
Then what do we do?
There are no more parentheses in this expression.
Then we should do exponents.
I don't see any exponents here, and if you're curious what
exponents look like, an exponent would look
like 7 squared.
You'd see these little small numbers up in the top right.
We don't have any exponents here so we don't have
to worry about it.
Then it says to do multiplication and
division next.
So where do we see multiplication?
We have a multiplication, a division, a
multiplication again.
Now, when you have multiple operations at the same level,
when our order of operations, multiplication and division
are the same level, then you do left to right.
So in this situation you're going to multiply by 4
and then divide by 2.
You won't multiply by 4 divided by 2.
Then we'll do the 5 times 6 before we do the
subtraction right here.
So let's figure out what this is.
So we'll do this multiplication first.
We could simultaneously do this multiplication because it's
not going to change things.
But I'll do things one step at a time.
So the next step we're going to do is this 10 times 4.
10 times 4 is 40.
10 times 4 is 40, then you have 40 divided by 2 and it
simplifies to that right there.
Remember, multiplication and division, they're at the exact
same level so we're going to do it left to right.
You could also express this as multiplying by 1/2 and then it
wouldn't matter the order.
But for simplicity, multiplication and division
go left to right.
So then you have 40 divided by 2 minus 5 times 6.
So, division, you just have one division here,
you want to do that.
You have this division and you have this multiplication,
they're not together so you can actually kind of do
them simultaneously.
And to make it clear that you do this before you do the
subtraction because multiplication and division
take priority over addition and subtraction, we could put
parentheses around them to say look, we're going to do that
and that first before I do that subtraction, because
multiplication and division have priority.
So 40 divided by 2 is 20.
We're going to have that minus sign, minus 5 times 6 is 30.
20 minus 30 is equal to negative 10.
And that is the correct interpretation of that.
So I want to make something very, very, very clear.
If you have things at the same level, so if you have 1 plus
2 minus 3 plus 4 minus 1.
So addition and subtraction are all the same level in order of
operations, you should go left to right.
So you should interpret this as 1 plus 2 is 3, so this is the
same thing as 3 minus 3 plus 4 minus 1.
Then you do 3 minus 3 is 0 plus 4 minus 1.
Or this is the same thing as 4 minus 1, which is
the same thing as 3.
You just go left to right.
Same thing if you have multiplication and division,
they're at the same level.
So if you have 4 times 2 divided by 3 times 2,
you do 4 times 2 is 8 divided by 3 times 2.
And you say 8 divided by 3 is, well, we got a fraction there.
It would be 8/3.
So this would be 8/3 times 2.
And then 8/3 times to is equal to 16 over 3.
That's how you interpret it.
You don't do this multiplication first or divide
the 2 by that and all of that.
Now the one time where you can be loosey-goosey with order of
operations, if you have all addition or all multiplication.
So if you have 1 plus 5 plus 7 plus 3 plus 2, it does not
matter what order you do it in.
You can do the 2 plus 3, you can go from the right to the
left, you can go from the left to the right, you could
start some place in between.
If it's only all addition.
And the same thing is true if you have all multiplication.
It's 1 times 5 times 7 times 3 times 2.
It does not matter what order you're doing it.
But it's only with all multiplication or all addition.
If there was some division in here, if there's some
subtraction in here, you're best off just going
left to right.