 Alright, so now we are talking about the combination of errors we have all we have just now introduced what will happen to errors if two major quantities are added or subtracted okay so it does not matter whether they are added or subtracted now there are a lot of formulas in physics where you multiply the two quantities what then is force into displacement in the direction of force okay force is equal to mass time acceleration v is equal to u plus a into t that is equal to ut plus half a into t into t okay so multiplication is a common thing in physics so we should also know what will happen to error if you multiply the two quantities which themselves have the errors okay so write down multiplication of errors multiplication or division both you write multiplication and division now suppose you have this formula y is equal to by the way we whenever we have deriving the equation or formula for the error we just use a and b some a and b it could be a plus b plus c then also errors of a and errors of b and c will get added up so it can go on tell whatever number of quantities you are adding okay similarly product also I am just multiplying two quantities it can be a multiplication of three quantities or four quantities okay so whatever formula that comes out is applicable of any number of quantities getting multiplied okay so let's say y is equal to a into b okay can you similarly the way you derive the previous one can you derive what will happen to the error in y substitute a as a mean plus minus delta a mean b as b mean plus minus delta b mean and y as y mean plus minus delta y mean put it and try to simplify it derived okay so y mean plus minus delta y mean will be equal to what multiplication of a mean plus minus delta a mean with b mean plus minus delta b mean just open the brackets okay so this will come out to be a mean b mean plus minus a mean delta b mean plus minus b mean delta a mean okay so this is y mean plus minus delta y I can say that y mean is equal to a mean into b mean so delta y mean is the rest of the expression so it's a mean plus delta b mean plus this is equal to a mean delta b mean I am trying to find a maximum possible error like this okay now this are less as in less than the measured quantity yes or no if doesn't make sense to have error which is more than the measured quantity you measure 5 centimeter and your error is 10 centimeter something is wrong with it so the the error is much less than the measured quantity okay so that's why delta a m into delta b m will be very less compared to these two quantities yes or no right so you can ignore this put this expression like this because the errors are much less compared to the a m and b m multiplication of error is much less compared to multiplication of reading with error it should represent maximum but we like to have a better looking expression as a formula okay but if you want to be very very accurate you want to be very very you want to be very sure about it you add that also okay but in our syllabus the way we are deriving it we are ignoring it because it tends to be very less okay now can you see a pattern over here can you do something and try to write a error in a better manner what can you say can you say that the error in the reading of y is a sum of the errors you can't say that yes or no if b m it is not delta b m plus delta a m can I say suppose I have to explain what will happen to the error if you multiply suppose you divide by a m into b m all of you do this you add the okay sorry why so I can say that delta y by y is equal to delta a by a plus delta b by b yes or no yeah y m is the actual value so if I just write y it means actually understood so the correct way of saying is that if you if the two quantities are multiplied they are related absolute error gets added up if they are multiplied their relative error gets added up okay same can be proved for that division as well if y is equal to a by b can you prove similarly yeah now you said yes good what happens to the error I hope you have an approximation binomial approximation is something which you'll be using again and again so get familiar with it this is what it is 1 plus x raise to power y is equal to 1 plus x y this is approximate value if is very less than one so if x is like point zero zero zero one or point zero minus or point zero zero one then you can approximate this as that doesn't matter the sign of y it could be negative or positive both it could be fraction also understood so use this while deriving it's a common approximation you can you know check also here suppose it is one plus point zero one whole square if you do this I am saying that this is roughly equal to one plus two into point zero one does it make sense a square plus b square b square is very less plus two a b which is two into point zero one okay similarly there is an expansion of one plus let's say point zero one raise to power three this will also be equal to roughly one plus three into point zero one there will be square or cube terms of point zero one in further expansion okay so just think it for the first time but this we are going to use again and again in that sense okay comfortable with it there is a way to do it you said yeah right I didn't think it through that so why m plus delta ym is equal to multiply with bm plus minus delta bm to the power minus one then delta bm becomes okay yeah that one works power minus one understood now if I take bm outside it will become a m plus minus delta a m outside you become one plus minus delta bm by bm raise to power minus one then this delta bm by bm is very less than one so I can take minus one inside and if minus one comes inside with plus and minus it remains plus and minus itself so this thing will become a m plus minus delta a m multiplied with bm one plus minus delta bm by bm which again is same a m plus minus delta a m minus one will be there by the way okay you are taking bm outside bm raise to power minus one will be there this divided by bm one plus minus delta bm by bm simplify it further same thing you will get absolute errors will get like that you can do this in case of division also in case of division also the relative error get paradox what will happen to this if y is equal to take you then what will happen it's multiplication of a three times so the relative error in y delta y by y is equal to delta a by a plus delta a by a plus delta a by a okay so it become three times delta a by exponent will come in front okay but even if it is a fraction then also it comes front for example if you have this expression y is equal to b raise to power three by two delta y by delta b by b remember these things but this can only be for constant what constant the power should be power should be constant it can be any real number okay so what if you had y is equal to a to the power of b there's a different method to deal with but in our curriculum we are only dealing with addition subtraction division multiplication and the exponents which are constant okay there are ways I mean I'll be talking about it in some time right now please focus here okay can you quickly write for this what if y is equal to a cube b raise to power three by two divided by c raise to power seven by two quickly write down what is the error in y what it will be three times delta a by a plus three by two b by b minus even if it is divided the errors get added up you have to bring right hand side you have to take it on the left hand side then also you should this plus if suppose this error I have to move it to the left hand side just never get subtracted you'll write it in boards you'll never subtract error