15:45 I would be interested in learning more about why division by 0 is impossible. Intuitively, it makes sense to me because when we're dividing, then we essentially have to divide by something. But to divide by nothing is to not be dividing at all; the process fundamentally disrupts its own operation and leaves us with no information at all, so I would call that undefined. Multiplication by 0 is different, because to multiply anything by 0 does still give us a real answer: 0, always. Similarly, 0 can be a numerator, and the answer here, too, is always 0. Dividing nothing by anything gives nothing, which is still a real answer, unlike if 0 were to be the denominator.
say you start with some number x and you add 3 to it, now you have x + 3. we can "undo" the addition of 3 by subtracting 3: x + 3 - 3 = x. similarly, if we multiply x by 4 we will get 4x. and we can undo that multiplication by dividing by 4: 4x/4 = x. so one way to think about subtraction and division is as inverse operations to addition and multiplication. subtracting by some value "undoes" addition by that value and dividing by some value undoes multiplication by that value. the problem is you can't undo multiplication by 0. as you pointed out, multiplication by 0 will always result in 0 no matter what the other number is and that is ultimately what prevents us from being able to undo it. here's an example to illustrate the idea: i have some secret number that only i know, and i tell you that when i multiply it by 2 i get 6. so basically 2x = 6. well we know immediately that x has to be 3. there's only one possibility. but say instead that i told you that when i multiply my secret number by 0 i get 0. well that's completely unhelpful in trying to figure out the secret number because it could be literally anything, since anything times 0 is 0.
here's another way of understanding the problem. multiplication is ultimately about area. A x B is the area of a rectangle with side lengths A and B. now given any number A it seems like it's always possible to find some other number B such that A x B = 1. in other words, if i give you one side of a rectangle it seems like you can always figure out what the other side needs to be to make the area of the rectangle 1. if i make A really big then you need to make B really small and vice versa. but if i make A be 0 then you are out of luck, there is no value of B that you can cook up that will make A x B = 1. the problem is if A is 0 then one side of our rectangle is 0, which means we dont really have a rectangle at all, we have a line of length B. and a line has no area, it has a length but no area, so its area is always 0. so every single number A has a special partner number B such that A x B = 1 except for 0. it's the exception to the rule. now a professional mathematician will tell you that division by a number A is equivalent to multiplication by A's partner number B. well if that's the case then division by 0 is meaningless, because division by 0 would be equivalent to multiplication by 0's partner number, which doesn't exist. because there is no number we can use to make a rectangle with 0 as one side have an area of 1.
15:45 I would be interested in learning more about why division by 0 is impossible. Intuitively, it makes sense to me because when we're dividing, then we essentially have to divide by something. But to divide by nothing is to not be dividing at all; the process fundamentally disrupts its own operation and leaves us with no information at all, so I would call that undefined.
Multiplication by 0 is different, because to multiply anything by 0 does still give us a real answer: 0, always. Similarly, 0 can be a numerator, and the answer here, too, is always 0. Dividing nothing by anything gives nothing, which is still a real answer, unlike if 0 were to be the denominator.
say you start with some number x and you add 3 to it, now you have x + 3.
we can "undo" the addition of 3 by subtracting 3: x + 3 - 3 = x.
similarly, if we multiply x by 4 we will get 4x.
and we can undo that multiplication by dividing by 4: 4x/4 = x.
so one way to think about subtraction and division is as inverse operations to addition and multiplication.
subtracting by some value "undoes" addition by that value and dividing by some value undoes multiplication by that value.
the problem is you can't undo multiplication by 0. as you pointed out, multiplication by 0 will always result in 0 no matter what the other number is and that is ultimately what prevents us from being able to undo it. here's an example to illustrate the idea:
i have some secret number that only i know, and i tell you that when i multiply it by 2 i get 6. so basically 2x = 6. well we know immediately that x has to be 3. there's only one possibility. but say instead that i told you that when i multiply my secret number by 0 i get 0. well that's completely unhelpful in trying to figure out the secret number because it could be literally anything, since anything times 0 is 0.
@ldov6373 Indeed, thank you!
here's another way of understanding the problem. multiplication is ultimately about area.
A x B is the area of a rectangle with side lengths A and B.
now given any number A it seems like it's always possible to find some other number B such that A x B = 1.
in other words, if i give you one side of a rectangle it seems like you can always figure out what the other side needs to be to make the area of the rectangle 1.
if i make A really big then you need to make B really small and vice versa.
but if i make A be 0 then you are out of luck, there is no value of B that you can cook up that will make A x B = 1.
the problem is if A is 0 then one side of our rectangle is 0, which means we dont really have a rectangle at all, we have a line of length B. and a line has no area, it has a length but no area, so its area is always 0. so every single number A has a special partner number B such that A x B = 1 except for 0. it's the exception to the rule.
now a professional mathematician will tell you that division by a number A is equivalent to multiplication by A's partner number B.
well if that's the case then division by 0 is meaningless, because division by 0 would be equivalent to multiplication by 0's partner number, which doesn't exist. because there is no number we can use to make a rectangle with 0 as one side have an area of 1.
Fascinating.
If it’s possible try Zoom whiteboard
Substitution