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