New operator definitions 1

we're all used to the traditional operators like addition and subtraction and multiplication and division and we've seen there's multiple ways to represent this but what we're going to do in this video is is a little bit of this a little fun we're actually going to define our own operators and what's neat about this is it kind of shows how broad mathematics can be and on a more practical sense it's actually something that you might see on some standardized tests and the reason why they do that is so that you can appreciate that these aren't the only operators out there plus exponentiation all those that in mathematics you can define a whole new set of operators so let's just do that so let me just define X diamond X diamond Y and I'm going to define that as I'm just going to find that is 5x minus y so you could view this as kind of defining it a function but we're defining it using an operator so if I have X diamond Y by definition we have defined this operator that means that's going to be equal to 5x minus y so given that definition what would what would seven diamond 11 be well you just go to the definition seven diamond 11 for instead of an X we have a 7 so it's going to be 5 times 7 so let me do it 5 times 7 minus instead of Y we have an 11 so one way to think about it is every in our definition every place you saw an X you can replace it with a 7 every place you saw Y you replace with 11 so you have minus 11 over here let me make the number so this is the 7 this 7 is this 7 and this 11 this 11 is this 11 right over here and then we just evaluate that so 5 times 7 is 35 so this is equal to 35 minus 11 which is equal to 24 so 7 star or not this isn't a star seven diamond 11 is equal to 24 we can define other things we can define something crazy like let me define a well I mentioned a star let me use a star a star a righty this way a star B let's say that that is the same thing as I don't know a over a plus B and so the same idea what would what would what would five star five star six be well you go back to the definition by definition every place where you see the a you would now replace with the five every time you saw the B you would now replace with the six so this is going to be equal to 5 over 5 plus six a plus B a is 5 B is 6 over 5 plus 6 so this would be 5 11 s 5 11 s and then you can compound them and we haven't defined any order of operations for these particular operations that we've just defined so we're going to be careful to use parentheses when we do when we put some of these together but you could do something like something interesting like negative 1 star diamond negative 1 diamond 0 0 star 0 star 5 and once again we just focus on parentheses because that's the only thing that's telling us what to start on first because we haven't we haven't figured out we haven't defined whether diamond takes precedence over star or star takes precedence over diamonds the way that we have that saying that hey you do multiplication before you do addition we haven't defined it for those operations but that's what the parentheses helps us do so we want to evaluate this parentheses first 0 star 5 that is 0 because so now this you could view this as 0 as the a and the 5 as the B so it's going to be 0 over 0 plus 5 over 0 plus 5 which is just going to be 0 so this over here is 0 or 5 or just goes to 0 so this whole expression simplifies to negative 1 diamond this diamond right over here diamond 0 and now we go to the definition of the diamond operator well that's five times the first the first number in our operator the first the first term that we're giving the operator I guess you could think of it that way so five times that so it'll be five times negative one X is negative 1 minus y well Y here is the 0 minus 0 so 5 times negative 1 5 times negative 1 is negative 5 and you will see and the idea here is just to make you feel comfortable defining new operators like this and not being daunted if all of a sudden you see a diamond and they're defining the diamond 4 and you're like wait I never saw a diamond and the reason they're actually defining it for you so you shouldn't say I never saw a diamond you'll say oh well they've defined a diamond for me I this is how I use that operator and sometimes you'll see even wackier things you'll see things like you'll see things like it let me draw so they'll define so this is you know I don't know if you would even consider this an operator but you'll see you'll see something like this that by definition if someone writes a symbol like this and they put in a b c let me write it this way a b c d they'll say this is the same thing as a d minus B all of that over C and once again this is just a definition they have this weird symbolic way of representing these variables and all this but they're just they're defining how do you evaluate this crazy expression and so if you if someone were to give you where to say evaluate this diamond evaluate let me evaluate the diamond so evaluate the diamond where in my little sections of the diamond I have a negative 1 of 5 a 3 and a 2 we would just use the definition of how to evaluate this diamond and we would say okay every time we see an a that is going to be negative 1 so we have a negative 1 times every 10 times D well D is whatever is in the bottom right section of this diamond or this kite so D is going to be 2 let me write it this way this is a this is B this is C and this is D so it's going to be negative 1 times 2 minus B well B is 5 minus 5 all of that over C which is 3 so this is going to be equal to negative 2 minus 5 so that is negative 7 negative 7 over and you could go crazy like this and it might be a fun thing actually if you have some spare time define your own operators and see how creative you can get with those operators