Introduction to Order of Operations

Order of Operations

In this video we are gonna talk a little bit

about order of operations.

And I want you to pay close attention

because, really, EVERYTHING else that

you are gonna do in mathematics is going to be based on

your having a solid grounding in Order of Operations.

So, what are we even talk...mean,

when we say Order of Operations?

So let me give you an example.

The whole point, is so we have one way

to interpret a mathematical statment.

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 2 ways of interpreting this statement.

You could just read it left-to-right.

So you could say "well, let me just take 7 + 3."

You could say 7 + 3 and then multiply that times 5 -

and 7+3 is 10.

and then you multiply that by 5.

10 x 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

"oh, I like to do multiplication before I do addition"

so you might interpret it as - I'll try to color code it -

7+ ... and you do the 3x5 first

7 + 3x5 which would be 7+

3x5 is 15 ... and 7+15 is 22.

So notice we interpreted this statement in two different ways

this was just straight left-to-right,

doing the addition, then the multiplication.

This way, we did the multiplication first,

then the addition. We get 2 different answers.

That's just not cool in mathematics.

If this was part of some effort to send something to the Moon

because 2 people interpreted it a different way

or 1 computer interpreted it 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 --

'parentheses' first. 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 exponents in our examples.

So you don't really have to worry about it for this video.

Then you do multiplication -

I'll just write "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 is this order of operations?

Let me label it - this right here is,

that is the agreed upon order of operations

and 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,

these 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 the order of operations say

'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 three times five,

which is fifteen, and then we add the seven.

The addition or subtraction - I'll do it here

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+3

- put some parentheses there -

x 4 divided by 2 - 5 x 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+3.

So it says, "lets do that first". So 7+3 is 10.

So this we can simplify

- just looking at this order of operations -

to 10 times all of that. Let me copy and paste that,

so I don't have to keep rewriting it.

So, let me copy. Let me paste it.

So that simplifies to ten 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 just so if you are curious what exponents would look like

an exponent would look like

- you know, seven 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

and in our order of operations, multiplication and division

are at 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'll do that multiplication first -

we could simultaneously do this multiplication

cause 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 10x4.

10x4 is 40

Then you have 40 divided by 2

- let me copy and paste all of that again -

Then it simplifies to that right there.

Remember multiplication and division,

they are at the exact same level -

so we're going to do it left-to-right.

You xould also express this as multiplying by one-half

and then it wouldn't matter the order. But for simplicity

multiplication / division go left-to-right.

So then you have 40 divided by 2 minus 5 times 6.

So, division - you just have 1 division here -

You want to do that. This is going to take...

You have this division and this multiplication.

They are 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/division take priority over addition/subtraction

we can put parentheses around them.

Just say "look, we're gonna do that and that first,

BEFORE I do that subtraction"

because multiplication / division have priority.

So 40 divided by 2 is 20.

We're going to have that minus sign.

-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 + 2 - 3 + 4 - 1

so addition and subtraction are at the same level

in order of operations - you should go left-to-right.

You should interpret this as 1+2 is 3.

So this is the same thing as 3 - 3 + 4 - 1.

Then you do 3 - 3 is 0, + 4, - 1.

OR this is the same thing as 4 - 1

which is the same thing as 3 - you just go left to right.

Same thing if you have multiplication and division

all on the same level.

So if you have 4x2, divided by 3, times 2,

you do 4x2 is 8, divided by 3, times 2

and you say 8 divided by 3 is - well you get a fraction there -

it would be 8/3. So this would be 8/3 times 2.

And 8/3 times 2 is equal to 16/3.

THAT's how you interpret it - you don't do this multiplication first,

and then divide the 2 by that, and all of that.

Now the one time you can be loosey-goosey with order of operations

is if you have ALL addition or ALL multiplication.

So if you have 1+5+7+3+2

does not matter what order you do it in.

You could do the two plus three;

you could go from the right to the left;

you could 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 mutliplication -

if it's 1 times 5 , times 7, times 3, times 2

does not matter what order you are doing it in.

That's only with all multiplication OR all addition.

If there's some division in here or some subtraction in here,

you're best off just going left-to-right.