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We're all used to the traditional operators like addition and subtraction

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and multiplication and division, and we've seen there's multiple ways to represent this

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but what we are going to do in this video is a little bit of fun.

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We are actually going to define our own operators.

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And what's neat about this it kind of shows how broad

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mathematics can be. And in a more practical sense it is something

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that you actually might see on some standardized test.

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And the reason why they do that is so that you can appreciate

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that these are not the only operators out there, plus

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exponentiation and all those, that in mathematics you can define whole

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new set of operators. Let's just do that. So let me just define x diamond

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x diamond y and I'm going to define that as

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I'm just gonna define that as 5x minus y. So you

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could view this kind of defining a function

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but we are defining it using an operator

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so if i have x diamond y by definition we've defined

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this operator, that means that's going to be equal to 5x minus y.

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So given that definition, what would-what would

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7 diamond 11 be?

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Well, you just go to the definition. 7 diamond 11

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for instead of a x we have 7, so it's going to be

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5 times 7. Just let me do 5 times 7 minus, and instead

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of a y, we have an 11.

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So one way to think about it is every--in our definition

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every place you saw a x you can replace with 7

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and every place you saw y, you replace with 11.

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So you have minus 11 over here. So this is the 7

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This 7 is this 7

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and this 11--this 11 is this 11 right over here.

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And then we just evaluate that. So 5 times 7 is 35,

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So this is equal to 35, minus 11, which is equal to 24

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So 7 star--or not, this isn't a star--7 diamond 11 is equal to 24.

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We can define other things.

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We can define something crazy like--let me define

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a--well I mentioned a star, let me use a star

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a star-- a -- write it this way

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a star b--let's say that is the same thing as-- I don't know

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a over a plus b. So it's the same idea

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What would,

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5 star --5 star 6 be?

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Well you go back to the definition. By definition

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every place you see a you would now replace with 5

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every time you'd see b, you'd now replace with 6

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this is going to be equal to 5 over 5 plus 6

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a plus b

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a is 5, b is 6

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over 5 plus 6

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so this would be 5/11

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5/11. And then you can compound them.

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We haven't defined and order of operations for these particular

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operations we just have defined.

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So we are going to be careful to use parentheses

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when we do, when we put some of these together.

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But you could do something like, something interesting

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like -1 diamond--negative 1 diamond 0 star--0 star 5

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and once again we just focus on parentheses, because

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that is the only thing that is telling us what to start on first

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because we haven't, we haven't figured out

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we haven't defined whether diamond takes precedence over

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star or star takes precedence over diamond.

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The way we have that thing saying hey, you do multiplication before you do addition

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we haven't defined it for those operations.

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But that's what parentheses help us do.

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So we want to evaluate the parentheses first.

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0 star 5, that is 0, because so now this you can zero as the a

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and the 5 as the b

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this is going to be 0 over 0 plus 5. Over 0 plus 5.

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which is just going to be 0. So this is just goes to 0.

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So this whole expression simplifies to negative 1 diamond

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this diamond right over here

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diamond 0. And now we go to the definition of the

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diamond operator. Well that's five times first, the first number

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in our operator. The first term we are giving the operator

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I guess you can think of it that way.

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So 5 times that, so that will be 5 times negative 1

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x is negative 1

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minus y. Well y here is the zero. Minus 0

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So 5 times negative 1. 5 times negative 1 is negative 5

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And you will see and the idea is just to make you feel comfortable

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defining new operators like this, and not feel being daunted

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if you all of a sudden see a diamond. And they are

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defined the diamond for you

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and you are like you never saw a diamond. They are actually defining it for you

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So you shouldn't say I never saw a diamond.

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You'll just say, oh well they have defined diamond for me

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this is how I use that operator. And sometimes you'll see even wackier things.

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You'll see things like, let me draw

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So they'll define. So this is, I don't know if you'd even consider this as an operator.

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By definiton, if somebody writes a symbol like this

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and they put an a b c--let me write this way

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a, b, c, d. They'll say this is the same thing as

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ad minus b, all of that over c.

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And once again, this is just a definition.

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They have this weird symbolic way of representing these variables

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but they are just defining how (do) you evaluate this crazy expression.

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And so if someone were to give you, say evaluate this diamond

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evaluate this diamond--let me evaluate the diamond

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so evaluate the diamond where in my little sections of the diamond I have a

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negative 1, a five, a three and a two. And we would just use the definition of how we evaluate the diamond.

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And we'd say every time we see an a, we say that's negative 1

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so we have negative 1 times d, well d. Well d is whatever is in the bottom right section of this

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diamond or this kite. So d is going to be 2. Let me write this this way.

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This is a.

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This is b.

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This is c.

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and this is d.

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So this is going to be negative 1

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times 2, minus b

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Well b is 5

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minus five

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all of that over c,

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which is 3.

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So this is going to be equal to negative 2

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minus 5. So that is negative 7. Negative 7 over 3.

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And you can go crazy like this. And it might be a fun thing actually

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if you have some spare time.

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Define your own operators and see how creative you can be with those operators.