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so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | the only thing that i can give up is this right over here . and i 'll say , well , i guess k will have to stay undefined . this whole contradiction happened because i attempted to define what x/0 is . | is there a difference between undefined and indeterminate ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | now that out of the way ... ok ... this was a situation when x does not equal zero . but what about when x does equal zero . so let 's think about that a little bit . | if zero divided by zero can be anything why not just say it is equal to all real numbers ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | the only thing that i can give up is this right over here . and i 'll say , well , i guess k will have to stay undefined . this whole contradiction happened because i attempted to define what x/0 is . | is there any difference between indeterminate and undefined or they are same ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but once again , anything i do to once side of the equation , inorder for the equation to hold true , i need to the other side of the equation . and these two were equal beforehand . any operation i do to this inorder for it to still be equal , i need to do to that . | what 's the difference between the two terms `` undefined '' and `` indeterminate '' ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and once again , i will try to define it . so i will assume ... that 0 divided by 0 is equal to some number . well once again , so let 's say it is equal to k again . | wait a second , if x/0 is undefined , 0/0 is 0 right ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but i just hit a contradiction ! i assume that x does not equal to 0 , and now i am being forced to say that x=0 . and i am not willing to give up the idea , i am not willing to give up either of these ideas . | if there was an equation 0/0=x , we ca n't cancel out the 0s , so i would multiply 0 by 0 , which would cancel it out , so would n't x=0 ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and once again , i will try to define it . so i will assume ... that 0 divided by 0 is equal to some number . well once again , so let 's say it is equal to k again . | lets say : x*0 = 0 for any number x x = 0 and now let 's treat 0 like every other number and just manipulate algebraically 0*0 = 0 0*0 / 0 = 0/0 0 = 0/0 so shouldnt zero devided by zero just equal 0 ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | this actually is true for any k , this is one of the core assumptions that i 've made in my mathematics that i am not willing to give up . so this is true true for any k. it 's not a contradiction . but the problem here is i wan na come up with the k , i 'd like a resolve for a k. it would not be nice if this turned ou... | is there any true solution for undefined expressions ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but i just hit a contradiction ! i assume that x does not equal to 0 , and now i am being forced to say that x=0 . and i am not willing to give up the idea , i am not willing to give up either of these ideas . | if x * 0 = 0 and 0 / x = 0 then 0 / 0 = x ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | the only thing that i can give up is this right over here . and i 'll say , well , i guess k will have to stay undefined . this whole contradiction happened because i attempted to define what x/0 is . | what is the difference between undefined and indeterminate ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but i just hit a contradiction ! i assume that x does not equal to 0 , and now i am being forced to say that x=0 . and i am not willing to give up the idea , i am not willing to give up either of these ideas . | if we take x as 0 , then would n't 0/0 = x/x = 1 ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but i just hit a contradiction ! i assume that x does not equal to 0 , and now i am being forced to say that x=0 . and i am not willing to give up the idea , i am not willing to give up either of these ideas . | is n't x/0 * 0 a contradiction since x/0 * 0 must be equal to 0 by the 2nd rule and x by the first ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and once again , i will try to define it . so i will assume ... that 0 divided by 0 is equal to some number . well once again , so let 's say it is equal to k again . | should n't 0 be null in the expression ( x/0 ) .0 since that the zeros cancel each other ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but i just hit a contradiction ! i assume that x does not equal to 0 , and now i am being forced to say that x=0 . and i am not willing to give up the idea , i am not willing to give up either of these ideas . | but ... x/0 isnt't an infinity ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but i just hit a contradiction ! i assume that x does not equal to 0 , and now i am being forced to say that x=0 . and i am not willing to give up the idea , i am not willing to give up either of these ideas . | does n't the x/y*y=x rule also work for multiplication ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but once again , anything i do to once side of the equation , inorder for the equation to hold true , i need to the other side of the equation . and these two were equal beforehand . any operation i do to this inorder for it to still be equal , i need to do to that . | if 1/1=1 then what does -1/1 equal ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and once again , i will try to define it . so i will assume ... that 0 divided by 0 is equal to some number . well once again , so let 's say it is equal to k again . | is 0 divided by 0 is undefined ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but i just hit a contradiction ! i assume that x does not equal to 0 , and now i am being forced to say that x=0 . and i am not willing to give up the idea , i am not willing to give up either of these ideas . | so is another way of seeing it is that due to the inconsistency of the graphs y=0/x and y=x/0 0/0 is indeterminant ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and once again , i will try to define it . so i will assume ... that 0 divided by 0 is equal to some number . well once again , so let 's say it is equal to k again . | would something to the zeroeth power be 0 ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and once again , i will try to define it . so i will assume ... that 0 divided by 0 is equal to some number . well once again , so let 's say it is equal to k again . | on a number line how many spaces would be found going from +1 to -1 if we measured between =1 , through 0 , onto -1 ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and this happens when we just multiply and divide with regular numbers . if i get 3 divided by 2 times 2 , that 's gon na get me 3 . if i say 10 divided by 5 times 5 , that 's going to get me 10 . | if each dimension increases at the rate of 2 inches per second then how long will it take the volume of the box to be at least 6 ties its original size ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but once again , anything i do to once side of the equation , inorder for the equation to hold true , i need to the other side of the equation . and these two were equal beforehand . any operation i do to this inorder for it to still be equal , i need to do to that . | are two quantities which are undefined equal to each other ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | now that out of the way ... ok ... this was a situation when x does not equal zero . but what about when x does equal zero . so let 's think about that a little bit . | why can we not say x divided by zero equals plus or minus infinity ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | the only thing that i can give up is this right over here . and i 'll say , well , i guess k will have to stay undefined . this whole contradiction happened because i attempted to define what x/0 is . | what is the difference between undefined and indeterminate ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and once again , i will try to define it . so i will assume ... that 0 divided by 0 is equal to some number . well once again , so let 's say it is equal to k again . | ca n't r/0 = +-infinity ( r is any real # not equal to 0 ) because of the graph , not undefined ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | why does the = sign have a line through it ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | the only thing that i can give up is this right over here . and i 'll say , well , i guess k will have to stay undefined . this whole contradiction happened because i attempted to define what x/0 is . | how do i determine when a numerical expression is undefined ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and so , once again ... we are trying to do the same logic , so we 'll write 0/0 is equal to k. actually , let me colourcode these zeros . this will be a magenta zero and this is a blue zero right over here . and once again , i am not willing to give up the idea that if i start with a number x , i divide it by somethin... | what possible value can be derived form any operation dividing zero ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and once again , i will try to define it . so i will assume ... that 0 divided by 0 is equal to some number . well once again , so let 's say it is equal to k again . | why is 0/0 not just 0 ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but i just hit a contradiction ! i assume that x does not equal to 0 , and now i am being forced to say that x=0 . and i am not willing to give up the idea , i am not willing to give up either of these ideas . | since -x + x = 0 would ( - infinity ) + ( + infinity ) = 0 ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and once again , i will try to define it . so i will assume ... that 0 divided by 0 is equal to some number . well once again , so let 's say it is equal to k again . | what is the value of 0/0 and 1/0 ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but i just hit a contradiction ! i assume that x does not equal to 0 , and now i am being forced to say that x=0 . and i am not willing to give up the idea , i am not willing to give up either of these ideas . | is n/0 the same thing as n x 1/0 ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | now , that out of the way . you wan na start exploring the divide-by-0 question . so the first thing that you say : `` well , let me just try to define it . '' | what does it really mean to divide by 0 ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | so this is true true for any k. it 's not a contradiction . but the problem here is i wan na come up with the k , i 'd like a resolve for a k. it would not be nice if this turned out to be 0. if this turned out to be one , or if this turned out to be negative one but now i see , given the assumptions right here this co... | i is the symbol for imaginary numbers , so maybe since we do n't have a 'real ' answer , could it be i ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and this happens when we just multiply and divide with regular numbers . if i get 3 divided by 2 times 2 , that 's gon na get me 3 . if i say 10 divided by 5 times 5 , that 's going to get me 10 . | if you have 1.1 divided by 1.2 =t what would be your awnser ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | so the left i get 0 , i just get this magenta 0 , and on the right i could just write the zero here , but i wo n't simplify it . i get k times 0 . well , this i see right over here ... | if we have no definition for 0/0 , why do n't we do what we did with complex numbers and call it some number k ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | so the left i get 0 , i just get this magenta 0 , and on the right i could just write the zero here , but i wo n't simplify it . i get k times 0 . well , this i see right over here ... | 2'28 '' why did sal use k instead of z ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | so the left i get 0 , i just get this magenta 0 , and on the right i could just write the zero here , but i wo n't simplify it . i get k times 0 . well , this i see right over here ... | 0why did sal use k instead of z ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and once again , i will try to define it . so i will assume ... that 0 divided by 0 is equal to some number . well once again , so let 's say it is equal to k again . | wound n't 0/0 by definition be every number in existence ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and so , once again ... we are trying to do the same logic , so we 'll write 0/0 is equal to k. actually , let me colourcode these zeros . this will be a magenta zero and this is a blue zero right over here . and once again , i am not willing to give up the idea that if i start with a number x , i divide it by somethin... | what are the factors of zero ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and so , once again ... we are trying to do the same logic , so we 'll write 0/0 is equal to k. actually , let me colourcode these zeros . this will be a magenta zero and this is a blue zero right over here . and once again , i am not willing to give up the idea that if i start with a number x , i divide it by somethin... | why are the zero 's color coded differently in this video ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | so let 's start , let 's assume that i have , so this is ... so let 's make a further assumption ... that x is some non-zero number . let 's just say , well , maybe the best way of finding out what x divided by 0 should be , how i should divide it , let 's just assume there is define , and then come up with any results... | i realize that this is more of a philosophical question rather than mathematical , in that i am suggesting no mathematical proof , but should n't zero , even on the number line , be infinite in its mathematical understanding ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and once again , i will try to define it . so i will assume ... that 0 divided by 0 is equal to some number . well once again , so let 's say it is equal to k again . | just wondering ... if 0/0=0 shouldnt 0/x where x is anything equal 0 ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but i just hit a contradiction ! i assume that x does not equal to 0 , and now i am being forced to say that x=0 . and i am not willing to give up the idea , i am not willing to give up either of these ideas . | so how can x*0=0 for any x ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | so the left i get 0 , i just get this magenta 0 , and on the right i could just write the zero here , but i wo n't simplify it . i get k times 0 . well , this i see right over here ... | at time says it would of been nice if this turned out to be 0 ( that 's what the audio states even though the text says it would not be nice ) , so when simplified , then 0 would of equaled 0 ( 0=0 ) so k=0 ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but once again , anything i do to once side of the equation , inorder for the equation to hold true , i need to the other side of the equation . and these two were equal beforehand . any operation i do to this inorder for it to still be equal , i need to do to that . | what does positive infinity over negative infinity equal ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but i just hit a contradiction ! i assume that x does not equal to 0 , and now i am being forced to say that x=0 . and i am not willing to give up the idea , i am not willing to give up either of these ideas . | x/0= all numbers real and imaginary ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | so the left i get 0 , i just get this magenta 0 , and on the right i could just write the zero here , but i wo n't simplify it . i get k times 0 . well , this i see right over here ... | but , since we are multiplying k by 0 , would n't it always come out to be 0 no matter what the value of k is ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | so let 's start , let 's assume that i have , so this is ... so let 's make a further assumption ... that x is some non-zero number . let 's just say , well , maybe the best way of finding out what x divided by 0 should be , how i should divide it , let 's just assume there is define , and then come up with any results... | why not do what we do with complex numbers and make a quick patch for dividing by zero ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | i get k times 0 . well , this i see right over here ... this actually is not a contradiction . | can infinity divided by infinity be also indeterminate as well ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | so these are close to my heart . i wan na extend mathematics . these two things are things that can not be contradicted , can not be untrue . | can the word `` indeterminate '' be used throughout other mathematics subdivisions , such as calculus ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | if i get 3 divided by 2 times 2 , that 's gon na get me 3 . if i say 10 divided by 5 times 5 , that 's going to get me 10 . the other things that i want to assume ... - and this is very close to my heart - i feel that any type of definitions i make have to be constant with the idea x*0 has to be 0 or any x . | 5 , how can 10 divided by 5 times 5 equal 10 ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | so the left i get 0 , i just get this magenta 0 , and on the right i could just write the zero here , but i wo n't simplify it . i get k times 0 . well , this i see right over here ... | if 0/0 is true for any k , then why do n't you take any k as the solution ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but i just hit a contradiction ! i assume that x does not equal to 0 , and now i am being forced to say that x=0 . and i am not willing to give up the idea , i am not willing to give up either of these ideas . | why can sal assume that x can not equal zero 0 in the video when he is proving that x/0 is undefined ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | so the left i get 0 , i just get this magenta 0 , and on the right i could just write the zero here , but i wo n't simplify it . i get k times 0 . well , this i see right over here ... | ( x/0 ) 0 = k ( 0 ) = > 0 = 0 therefore , x/0 is defined ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | so the left i get 0 , i just get this magenta 0 , and on the right i could just write the zero here , but i wo n't simplify it . i get k times 0 . well , this i see right over here ... | 0/0 = 5 0*5 = 0 denominator times answer is numerator ... 0/0 = 987,654,321 987654321*0 = 0 denominator times answer is numerator ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but i just hit a contradiction ! i assume that x does not equal to 0 , and now i am being forced to say that x=0 . and i am not willing to give up the idea , i am not willing to give up either of these ideas . | if 0 / x = 0 , and 0 / x is the inverse of x / 0 , then x / 0 = 1 / 0 because the inverse of a number n is 1 / n , and 0 / x = 0. so x = 1 , so all numbers are 1 ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | so the left i get 0 , i just get this magenta 0 , and on the right i could just write the zero here , but i wo n't simplify it . i get k times 0 . well , this i see right over here ... | cound n't you take assumption # 2 and get 0 = 0 , or x/0 = indeterminate ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | i ca n't give this up . otherwise it does n't seem like a good definition for the division . so what i am gon na do - i am gon na multiply the left-hand side times 0 and by this property that i am not willing to give up , the left-hand side should simplify to this magenta zero . | does sal draw the sketches like the one in this video ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | but i just hit a contradiction ! i assume that x does not equal to 0 , and now i am being forced to say that x=0 . and i am not willing to give up the idea , i am not willing to give up either of these ideas . | why x*0=0 has to be indisputable ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and this happens when we just multiply and divide with regular numbers . if i get 3 divided by 2 times 2 , that 's gon na get me 3 . if i say 10 divided by 5 times 5 , that 's going to get me 10 . | let take the case 1/3*3= ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | so the left i get 0 , i just get this magenta 0 , and on the right i could just write the zero here , but i wo n't simplify it . i get k times 0 . well , this i see right over here ... | i assume that 0/0 = k now this can be written as 0^1 / 0^1 = k that would give me : 0^1-1 = k which is 0^0 = k and many mathematicians now have agreed to take 0^0 as 1 so what does this mean is 0/0 = k = 1 where * k is indeed defined as 1 ? |
so , once again let 's think of yourself as some type of ancient philosopher/mathematician , who is trying to extend mathematics as much as possible and try to make sure that you 're not being lazy and leaving things undefined , when you might be able to define them . whenever you start extending mathematics , especial... | and once again , i will try to define it . so i will assume ... that 0 divided by 0 is equal to some number . well once again , so let 's say it is equal to k again . | if division is basicly multiplication backwards would n't 0/0 = 0 because 0*0=0 ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | 60 plus 36 . it 's equal to 96/16 . so it 's 9 minus 96 over 16z . | can someone explain to me how sal got the 96/16 ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | so let 's solve this equation . we get 5x minus 3 is equal to 3 times x plus 5 . a good place for me -- i like to distribute out this 3 , so that is equal to 3x plus 15 . | when you simplify ( x * 5 ) -3 = 3 ( x+5 ) you reach this stage : 5x - 3 = 3x + 15 how do you know which order to subtract the x 's ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | you 're just left with z is equal to 2 times this thing . so 2 times negative 16 is negative 32/87 . and we are done . | what if you get something like this : 3b-4=1- ( 2b+5 ) , what do you do if there is a negative or positive sign before the brackets ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | like the last video , i want to start with two warm up problems . and then we 'll do an actual word . | how exactly does the distributive property work ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | we get 9 over 16z is equal to -- this 9 and that 9 will cancel out -- 2 times 3z plus 1 . now let 's distribute this 2 . so we get 9 over 16z is equal to 2 times 3z is 6z plus 2 . | why does sal distribute the 3 and 2 ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | you could imagine this is being adding this equation to the equation 3 is equal to 3 . 3 is , obviously , equal to 3 . negative 2x is , obviously , equal to negative 2x . | in why 2x-3=15+3 is 18 ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | that means that these two are equals . so let 's solve this equation . we get 5x minus 3 is equal to 3 times x plus 5 . a good place for me -- i like to distribute out this 3 , so that is equal to 3x plus 15 . | i can manage to solve most basic equations but not ones like these ones : > y-7=2y+3 > 2z+5=z-1 can someone help ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | these cancel out . 87 , 87 , 16 and 16 . the negative signs plus . | are you sure -32/87 can not be reduced ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | so let 's solve this equation . we get 5x minus 3 is equal to 3 times x plus 5 . a good place for me -- i like to distribute out this 3 , so that is equal to 3x plus 15 . | how do you rewrite 3 ( x + 5 ) as 5x - 3 ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | so it all works out . x is equals to 9 . next problem . | why is x the most frequently used sign ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | so it 's a hairy looking problem . let 's multiply both sides of this equation by 9 . so if you multiply both sides of this equation by 9 , what do we get ? we get 9 over 16z is equal to -- this 9 and that 9 will cancel out -- 2 times 3z plus 1 . now let 's distribute this 2 . | why was 9 chosen in problem 2 , as the number to multiply by both sides ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | so if you multiply both sides of this equation by 9 , what do we get ? we get 9 over 16z is equal to -- this 9 and that 9 will cancel out -- 2 times 3z plus 1 . now let 's distribute this 2 . | on the second problem , i first tried to solve it myself and ended up with a slightly different answer ... z/16=2 ( 3z+1 ) /9 9/16z=2 ( 3z+1 ) 9/16z=6z+2 9z=96z+32 -87z=32 z=32/-87 where did i go wrong ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | so if you multiply both sides of this equation by 9 , what do we get ? we get 9 over 16z is equal to -- this 9 and that 9 will cancel out -- 2 times 3z plus 1 . now let 's distribute this 2 . | on another note , how , logistically , does 9*z/16 equal 9/16z instead of 9z/16 ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | so 3 times x minus 1 , that 's the same thing as 3x minus 3 -- i just distributed the 3 -- is equal to -- distribute out the 2 . 2 times x , plus 2 times 3 , which is 6 . now what i like to do is get all of my constant terms on the same side of the equation and all my variable terms on the same side of the equation . | when using the distributive property how would you express 2* ( a+3 ) would it be 2*a+2*3a or 2*a+2*3 ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | so 3 times x minus 1 , that 's the same thing as 3x minus 3 -- i just distributed the 3 -- is equal to -- distribute out the 2 . 2 times x , plus 2 times 3 , which is 6 . now what i like to do is get all of my constant terms on the same side of the equation and all my variable terms on the same side of the equation . | video : equations with variables on both sides : for problem # 2 : at the very end of solving the problem z=2 ( -16/87 ) why is the numerator multiplied by 2 and the denominator not also multiplied by 2 ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | so if you multiply both sides of this equation by 9 , what do we get ? we get 9 over 16z is equal to -- this 9 and that 9 will cancel out -- 2 times 3z plus 1 . now let 's distribute this 2 . | why does sal not put a nine over 9/16z=2 ( 3z+1 ) ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | and we can even check our answer . 3 times 9 minus 1 is what ? this is 3 times 8 . | would the 9 times 16 be 144 instead of becoming the fraction 9 over 16 ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | you 're just left with z is equal to 2 times this thing . so 2 times negative 16 is negative 32/87 . and we are done . | in multi-step equations how do you know if the answer will be negative or positive ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | but we 're still going to be doing the exact same operations , or what we could consider legitimate operations , to get our answer . so here we have 3 times x minus 1 is equal to 2 times x plus 3 . so let 's see what we can do here . | how can you solve equations such as:4/x=3*10 ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | you 're just left with z is equal to 2 times this thing . so 2 times negative 16 is negative 32/87 . and we are done . | is a negative and a negative a positive or a negative ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | and you can try it with multiple methods . maybe you can multiply both of the equation by 16 first . maybe you can distribute out the 2 9 's first . all sorts of things you can do . | if you multiply both sides by 16 first , or distribute out the 2/9ths , what do these methods look like ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | and you get x is equal to 18 over 2 . or 9 . so in either situation , andrew was thinking of the number 9 . | why did sal multiply by 9 ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | so it all works out . x is equals to 9 . next problem . z over 16 is equal to 2 times 3z plus 1 , all of that over 9 . so it 's a hairy looking problem . | why when you multiply z over 16 by 9 , does it become 9 over 16 , with the z next to it ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | you 're just left with z is equal to 2 times this thing . so 2 times negative 16 is negative 32/87 . and we are done . | why does the 16 become negative ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | these cancel out . 87 , 87 , 16 and 16 . the negative signs plus . | why is 9 over 16 and not 9z over 16 ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | now let 's get all of z 's on the same side of the equation . so let 's subtract 6z from both sides of the equation . so let 's subtract minus 6z there , and that , of course , equals minus 6z there . | why did sal decide to subtract 6z from both sides instead of 9/16z ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | so if you multiply both sides of this equation by 9 , what do we get ? we get 9 over 16z is equal to -- this 9 and that 9 will cancel out -- 2 times 3z plus 1 . now let 's distribute this 2 . | it was kind of confusing on how you subtract -6z from 9/16z is there an easier way ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | so it 's a hairy looking problem . let 's multiply both sides of this equation by 9 . so if you multiply both sides of this equation by 9 , what do we get ? | how does sal know to multiply both sides by 9 ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | like the last video , i want to start with two warm up problems . and then we 'll do an actual word . | if total keyboard sales were $ 4,998 , how many of each type were sold ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | now , i think you know how i like to operate . i like to get all of my ex-coeffients on one side . so let 's get them all on the left hand side . | why does `` ex 2 : distributive property to simplify '' precede `` ex 1 : distributive property to simplify '' ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | so if you multiply both sides of this equation by 9 , what do we get ? we get 9 over 16z is equal to -- this 9 and that 9 will cancel out -- 2 times 3z plus 1 . now let 's distribute this 2 . | after multiplying both sides by 9 , would n't be easier to convert to decimals ie -9/16z=-.5625z =2 and z=-.367816092 ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | but we 're still going to be doing the exact same operations , or what we could consider legitimate operations , to get our answer . so here we have 3 times x minus 1 is equal to 2 times x plus 3 . so let 's see what we can do here . | with the equation 3 ( x-1 ) after distributing out the problem do you ever change into a negative number ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | so if you multiply both sides of this equation by 9 , what do we get ? we get 9 over 16z is equal to -- this 9 and that 9 will cancel out -- 2 times 3z plus 1 . now let 's distribute this 2 . so we get 9 over 16z is equal to 2 times 3z is 6z plus 2 . now let 's get all of z 's on the same side of the equation . | in question 2 why did you multiply the 9 on both sides first in z/6=2 ( 3z+1 ) /9 ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | so if you multiply both sides of this equation by 9 , what do we get ? we get 9 over 16z is equal to -- this 9 and that 9 will cancel out -- 2 times 3z plus 1 . now let 's distribute this 2 . so we get 9 over 16z is equal to 2 times 3z is 6z plus 2 . now let 's get all of z 's on the same side of the equation . | why would n't you get 2 ( 3z +1 ) over 81 ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | these cancel out . 87 , 87 , 16 and 16 . the negative signs plus . | 2. why does multiplying a fraction by it 's inverse ( like 87/16 x 16/87 ) make it 1 ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | which means , let 's get rid of them on the right hand side . so let 's subtract 3x from both sides . so minus 3x minus 3x and then what do we get ? | finding perimeter with variable on both sides 6x=8x-10 help ? |
like the last video , i want to start with two warm up problems . and then we 'll do an actual word . and you 're going to see these are going to be a little bit more involved than the equations in the last video . but we 're still going to be doing the exact same operations , or what we could consider legitimate opera... | but we 're still going to be doing the exact same operations , or what we could consider legitimate operations , to get our answer . so here we have 3 times x minus 1 is equal to 2 times x plus 3 . so let 's see what we can do here . | i do n't understand how the 3 ( x-1 ) is being distributed to 3x-3 or how the 2 ( x +3 ) is being distributed to 2x +6 ? |
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