Order of operations introduction
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- Every few months, you'll see an expression like this go viral on social media because it looks simple but depending on how people interpret this expression, they often get different answers. So first, why don't you pause this video and think about what you think three plus two times five is equal to. All right, now let's work through this together. Now some of you might have interpreted this as we should just go from left to right. So you might've said, all right, let me first add three plus two. So I can do that part first, which is of course equal to five. And then I could multiply that times five. And that is going to get me to 25. Now others of you would have said no, that makes no sense at all. We know that we should do some operations before others. For example, multiplication should come before addition. And so if you follow that, you would do the two times five first to get 10. And so this would become three plus 10, which is of course, going to be equal to 13. Who do you think would be right? Well, it turns out that the second way of doing it is the correct way and that's because we have something known as order of operations. And the order of operations is the convention that mathematicians have decided to use in order for us to have one way to interpret an expression like this and the order of operations are to do parentheses first. So, for example, if you really wanted to add the three and the two first, you should put parentheses around it to say, hey, that's what you've got to do first. But then after parentheses, do exponents, which is really, you could view as repeated multiplication, then do multiplication and division. Then do addition and subtraction. Now, some people might just memorize this as PEMDAS, or PEDMAS or something like that. And you can do that, but there's a rationale to this. Parentheses are just, the person who writing the expression saying do this for sure first. But then if you think about it, exponents are repeated multiplication and multiplication is repeated addition. So you're doing the most repeated things first, then next repeated things like multiplication. And then you go straight to the addition. The reason why multiplication and division are on the same line and addition and subtraction on the same line is the convention there is just go left to right. So now that we are armed with our order of operations, let's tackle another slightly hairier expression. And I use the term hairy for a little bit more complex. So two times 10, minus eight, divided by four, plus one. Pause the video and see how you could evaluate this now that you know about order of operations. All right, now let's do this together. We don't see any parentheses here. We don't see any exponents here but we do see some multiplication and division here. So we could see it over here. This two times 10. And we also have this eight divided by four. So that's what we're going to want to do first. So that two times 10, that is going to be equal to 20. The eight divided by four, is of course, equal to two. And then we have 20 minus two plus one. And this is now going to get us, if we go left to right, which we should do when we're just adding and subtracting, is we get 20 minus two is 18, plus one, which is going to be equal to 19. And we're done. Now, you might say, well, what if we wanted someone to add the four and the one first? Well, one way to do that is you could have added a parentheses there to force it or you could have even written your division differently. For example, you could use a fraction sign to make things clearer. If we wanted to add the four plus one first, instead of writing eight divided by four plus one like that, you could write eight over four plus one. Now of course, this is fundamentally different than this right over here. 'Cause here we would do the eight divided by four first, then add the plus one, while here, you would add the four plus one and then divide eight by that.